_notebooks/2019-09-10-introduction-to-deep-gps.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Introduction to Deep Gaussian Processes\n",
    "### [Neil D. Lawrence](http://inverseprobability.com), University of Sheffield\n",
    "### 2019-09-10\n",
    "\n",
    "**Abstract**: In this talk we introduce deep Gaussian processes, describe what they\n",
    "are and what they are good for. Deep Gaussian process models make use of\n",
    "stochastic process composition to combine Gaussian processes together to\n",
    "form new models which are non-Gaussian in structure. They serve both as\n",
    "a theoretical model for deep learning and a functional model for\n",
    "regression, classification and unsupervised learning. The challenge in\n",
    "these models is propagating the uncertainty through the process.\n",
    "\n",
    "$$\n",
    "\\newcommand{\\tk}[1]{}\n",
    "%\\newcommand{\\tk}[1]{\\textbf{TK}: #1}\n",
    "\\newcommand{\\Amatrix}{\\mathbf{A}}\n",
    "\\newcommand{\\KL}[2]{\\text{KL}\\left( #1\\,\\|\\,#2 \\right)}\n",
    "\\newcommand{\\Kaast}{\\kernelMatrix_{\\mathbf{ \\ast}\\mathbf{ \\ast}}}\n",
    "\\newcommand{\\Kastu}{\\kernelMatrix_{\\mathbf{ \\ast} \\inducingVector}}\n",
    "\\newcommand{\\Kff}{\\kernelMatrix_{\\mappingFunctionVector \\mappingFunctionVector}}\n",
    "\\newcommand{\\Kfu}{\\kernelMatrix_{\\mappingFunctionVector \\inducingVector}}\n",
    "\\newcommand{\\Kuast}{\\kernelMatrix_{\\inducingVector \\bf\\ast}}\n",
    "\\newcommand{\\Kuf}{\\kernelMatrix_{\\inducingVector \\mappingFunctionVector}}\n",
    "\\newcommand{\\Kuu}{\\kernelMatrix_{\\inducingVector \\inducingVector}}\n",
    "\\newcommand{\\Kuui}{\\Kuu^{-1}}\n",
    "\\newcommand{\\Qaast}{\\mathbf{Q}_{\\bf \\ast \\ast}}\n",
    "\\newcommand{\\Qastf}{\\mathbf{Q}_{\\ast \\mappingFunction}}\n",
    "\\newcommand{\\Qfast}{\\mathbf{Q}_{\\mappingFunctionVector \\bf \\ast}}\n",
    "\\newcommand{\\Qff}{\\mathbf{Q}_{\\mappingFunctionVector \\mappingFunctionVector}}\n",
    "\\newcommand{\\aMatrix}{\\mathbf{A}}\n",
    "\\newcommand{\\aScalar}{a}\n",
    "\\newcommand{\\aVector}{\\mathbf{a}}\n",
    "\\newcommand{\\acceleration}{a}\n",
    "\\newcommand{\\bMatrix}{\\mathbf{B}}\n",
    "\\newcommand{\\bScalar}{b}\n",
    "\\newcommand{\\bVector}{\\mathbf{b}}\n",
    "\\newcommand{\\basisFunc}{\\phi}\n",
    "\\newcommand{\\basisFuncVector}{\\boldsymbol{ \\basisFunc}}\n",
    "\\newcommand{\\basisFunction}{\\phi}\n",
    "\\newcommand{\\basisLocation}{\\mu}\n",
    "\\newcommand{\\basisMatrix}{\\boldsymbol{ \\Phi}}\n",
    "\\newcommand{\\basisScalar}{\\basisFunction}\n",
    "\\newcommand{\\basisVector}{\\boldsymbol{ \\basisFunction}}\n",
    "\\newcommand{\\activationFunction}{\\phi}\n",
    "\\newcommand{\\activationMatrix}{\\boldsymbol{ \\Phi}}\n",
    "\\newcommand{\\activationScalar}{\\basisFunction}\n",
    "\\newcommand{\\activationVector}{\\boldsymbol{ \\basisFunction}}\n",
    "\\newcommand{\\bigO}{\\mathcal{O}}\n",
    "\\newcommand{\\binomProb}{\\pi}\n",
    "\\newcommand{\\cMatrix}{\\mathbf{C}}\n",
    "\\newcommand{\\cbasisMatrix}{\\hat{\\boldsymbol{ \\Phi}}}\n",
    "\\newcommand{\\cdataMatrix}{\\hat{\\dataMatrix}}\n",
    "\\newcommand{\\cdataScalar}{\\hat{\\dataScalar}}\n",
    "\\newcommand{\\cdataVector}{\\hat{\\dataVector}}\n",
    "\\newcommand{\\centeredKernelMatrix}{\\mathbf{ \\MakeUppercase{\\centeredKernelScalar}}}\n",
    "\\newcommand{\\centeredKernelScalar}{b}\n",
    "\\newcommand{\\centeredKernelVector}{\\centeredKernelScalar}\n",
    "\\newcommand{\\centeringMatrix}{\\mathbf{H}}\n",
    "\\newcommand{\\chiSquaredDist}[2]{\\chi_{#1}^{2}\\left(#2\\right)}\n",
    "\\newcommand{\\chiSquaredSamp}[1]{\\chi_{#1}^{2}}\n",
    "\\newcommand{\\conditionalCovariance}{\\boldsymbol{ \\Sigma}}\n",
    "\\newcommand{\\coregionalizationMatrix}{\\mathbf{B}}\n",
    "\\newcommand{\\coregionalizationScalar}{b}\n",
    "\\newcommand{\\coregionalizationVector}{\\mathbf{ \\coregionalizationScalar}}\n",
    "\\newcommand{\\covDist}[2]{\\text{cov}_{#2}\\left(#1\\right)}\n",
    "\\newcommand{\\covSamp}[1]{\\text{cov}\\left(#1\\right)}\n",
    "\\newcommand{\\covarianceScalar}{c}\n",
    "\\newcommand{\\covarianceVector}{\\mathbf{ \\covarianceScalar}}\n",
    "\\newcommand{\\covarianceMatrix}{\\mathbf{C}}\n",
    "\\newcommand{\\covarianceMatrixTwo}{\\boldsymbol{ \\Sigma}}\n",
    "\\newcommand{\\croupierScalar}{s}\n",
    "\\newcommand{\\croupierVector}{\\mathbf{ \\croupierScalar}}\n",
    "\\newcommand{\\croupierMatrix}{\\mathbf{ \\MakeUppercase{\\croupierScalar}}}\n",
    "\\newcommand{\\dataDim}{p}\n",
    "\\newcommand{\\dataIndex}{i}\n",
    "\\newcommand{\\dataIndexTwo}{j}\n",
    "\\newcommand{\\dataMatrix}{\\mathbf{Y}}\n",
    "\\newcommand{\\dataScalar}{y}\n",
    "\\newcommand{\\dataSet}{\\mathcal{D}}\n",
    "\\newcommand{\\dataStd}{\\sigma}\n",
    "\\newcommand{\\dataVector}{\\mathbf{ \\dataScalar}}\n",
    "\\newcommand{\\decayRate}{d}\n",
    "\\newcommand{\\degreeMatrix}{\\mathbf{ \\MakeUppercase{\\degreeScalar}}}\n",
    "\\newcommand{\\degreeScalar}{d}\n",
    "\\newcommand{\\degreeVector}{\\mathbf{ \\degreeScalar}}\n",
    "% Already defined by latex\n",
    "%\\newcommand{\\det}[1]{\\left|#1\\right|}\n",
    "\\newcommand{\\diag}[1]{\\text{diag}\\left(#1\\right)}\n",
    "\\newcommand{\\diagonalMatrix}{\\mathbf{D}}\n",
    "\\newcommand{\\diff}[2]{\\frac{\\text{d}#1}{\\text{d}#2}}\n",
    "\\newcommand{\\diffTwo}[2]{\\frac{\\text{d}^2#1}{\\text{d}#2^2}}\n",
    "\\newcommand{\\displacement}{x}\n",
    "\\newcommand{\\displacementVector}{\\textbf{\\displacement}}\n",
    "\\newcommand{\\distanceMatrix}{\\mathbf{ \\MakeUppercase{\\distanceScalar}}}\n",
    "\\newcommand{\\distanceScalar}{d}\n",
    "\\newcommand{\\distanceVector}{\\mathbf{ \\distanceScalar}}\n",
    "\\newcommand{\\eigenvaltwo}{\\ell}\n",
    "\\newcommand{\\eigenvaltwoMatrix}{\\mathbf{L}}\n",
    "\\newcommand{\\eigenvaltwoVector}{\\mathbf{l}}\n",
    "\\newcommand{\\eigenvalue}{\\lambda}\n",
    "\\newcommand{\\eigenvalueMatrix}{\\boldsymbol{ \\Lambda}}\n",
    "\\newcommand{\\eigenvalueVector}{\\boldsymbol{ \\lambda}}\n",
    "\\newcommand{\\eigenvector}{\\mathbf{ \\eigenvectorScalar}}\n",
    "\\newcommand{\\eigenvectorMatrix}{\\mathbf{U}}\n",
    "\\newcommand{\\eigenvectorScalar}{u}\n",
    "\\newcommand{\\eigenvectwo}{\\mathbf{v}}\n",
    "\\newcommand{\\eigenvectwoMatrix}{\\mathbf{V}}\n",
    "\\newcommand{\\eigenvectwoScalar}{v}\n",
    "\\newcommand{\\entropy}[1]{\\mathcal{H}\\left(#1\\right)}\n",
    "\\newcommand{\\errorFunction}{E}\n",
    "\\newcommand{\\expDist}[2]{\\left<#1\\right>_{#2}}\n",
    "\\newcommand{\\expSamp}[1]{\\left<#1\\right>}\n",
    "\\newcommand{\\expectation}[1]{\\left\\langle #1 \\right\\rangle }\n",
    "\\newcommand{\\expectationDist}[2]{\\left\\langle #1 \\right\\rangle _{#2}}\n",
    "\\newcommand{\\expectedDistanceMatrix}{\\mathcal{D}}\n",
    "\\newcommand{\\eye}{\\mathbf{I}}\n",
    "\\newcommand{\\fantasyDim}{r}\n",
    "\\newcommand{\\fantasyMatrix}{\\mathbf{ \\MakeUppercase{\\fantasyScalar}}}\n",
    "\\newcommand{\\fantasyScalar}{z}\n",
    "\\newcommand{\\fantasyVector}{\\mathbf{ \\fantasyScalar}}\n",
    "\\newcommand{\\featureStd}{\\varsigma}\n",
    "\\newcommand{\\gammaCdf}[3]{\\mathcal{GAMMA CDF}\\left(#1|#2,#3\\right)}\n",
    "\\newcommand{\\gammaDist}[3]{\\mathcal{G}\\left(#1|#2,#3\\right)}\n",
    "\\newcommand{\\gammaSamp}[2]{\\mathcal{G}\\left(#1,#2\\right)}\n",
    "\\newcommand{\\gaussianDist}[3]{\\mathcal{N}\\left(#1|#2,#3\\right)}\n",
    "\\newcommand{\\gaussianSamp}[2]{\\mathcal{N}\\left(#1,#2\\right)}\n",
    "\\newcommand{\\given}{|}\n",
    "\\newcommand{\\half}{\\frac{1}{2}}\n",
    "\\newcommand{\\heaviside}{H}\n",
    "\\newcommand{\\hiddenMatrix}{\\mathbf{ \\MakeUppercase{\\hiddenScalar}}}\n",
    "\\newcommand{\\hiddenScalar}{h}\n",
    "\\newcommand{\\hiddenVector}{\\mathbf{ \\hiddenScalar}}\n",
    "\\newcommand{\\identityMatrix}{\\eye}\n",
    "\\newcommand{\\inducingInputScalar}{z}\n",
    "\\newcommand{\\inducingInputVector}{\\mathbf{ \\inducingInputScalar}}\n",
    "\\newcommand{\\inducingInputMatrix}{\\mathbf{Z}}\n",
    "\\newcommand{\\inducingScalar}{u}\n",
    "\\newcommand{\\inducingVector}{\\mathbf{ \\inducingScalar}}\n",
    "\\newcommand{\\inducingMatrix}{\\mathbf{U}}\n",
    "\\newcommand{\\inlineDiff}[2]{\\text{d}#1/\\text{d}#2}\n",
    "\\newcommand{\\inputDim}{q}\n",
    "\\newcommand{\\inputMatrix}{\\mathbf{X}}\n",
    "\\newcommand{\\inputScalar}{x}\n",
    "\\newcommand{\\inputSpace}{\\mathcal{X}}\n",
    "\\newcommand{\\inputVals}{\\inputVector}\n",
    "\\newcommand{\\inputVector}{\\mathbf{ \\inputScalar}}\n",
    "\\newcommand{\\iterNum}{k}\n",
    "\\newcommand{\\kernel}{\\kernelScalar}\n",
    "\\newcommand{\\kernelMatrix}{\\mathbf{K}}\n",
    "\\newcommand{\\kernelScalar}{k}\n",
    "\\newcommand{\\kernelVector}{\\mathbf{ \\kernelScalar}}\n",
    "\\newcommand{\\kff}{\\kernelScalar_{\\mappingFunction \\mappingFunction}}\n",
    "\\newcommand{\\kfu}{\\kernelVector_{\\mappingFunction \\inducingScalar}}\n",
    "\\newcommand{\\kuf}{\\kernelVector_{\\inducingScalar \\mappingFunction}}\n",
    "\\newcommand{\\kuu}{\\kernelVector_{\\inducingScalar \\inducingScalar}}\n",
    "\\newcommand{\\lagrangeMultiplier}{\\lambda}\n",
    "\\newcommand{\\lagrangeMultiplierMatrix}{\\boldsymbol{ \\Lambda}}\n",
    "\\newcommand{\\lagrangian}{L}\n",
    "\\newcommand{\\laplacianFactor}{\\mathbf{ \\MakeUppercase{\\laplacianFactorScalar}}}\n",
    "\\newcommand{\\laplacianFactorScalar}{m}\n",
    "\\newcommand{\\laplacianFactorVector}{\\mathbf{ \\laplacianFactorScalar}}\n",
    "\\newcommand{\\laplacianMatrix}{\\mathbf{L}}\n",
    "\\newcommand{\\laplacianScalar}{\\ell}\n",
    "\\newcommand{\\laplacianVector}{\\mathbf{ \\ell}}\n",
    "\\newcommand{\\latentDim}{q}\n",
    "\\newcommand{\\latentDistanceMatrix}{\\boldsymbol{ \\Delta}}\n",
    "\\newcommand{\\latentDistanceScalar}{\\delta}\n",
    "\\newcommand{\\latentDistanceVector}{\\boldsymbol{ \\delta}}\n",
    "\\newcommand{\\latentForce}{f}\n",
    "\\newcommand{\\latentFunction}{u}\n",
    "\\newcommand{\\latentFunctionVector}{\\mathbf{ \\latentFunction}}\n",
    "\\newcommand{\\latentFunctionMatrix}{\\mathbf{ \\MakeUppercase{\\latentFunction}}}\n",
    "\\newcommand{\\latentIndex}{j}\n",
    "\\newcommand{\\latentScalar}{z}\n",
    "\\newcommand{\\latentVector}{\\mathbf{ \\latentScalar}}\n",
    "\\newcommand{\\latentMatrix}{\\mathbf{Z}}\n",
    "\\newcommand{\\learnRate}{\\eta}\n",
    "\\newcommand{\\lengthScale}{\\ell}\n",
    "\\newcommand{\\rbfWidth}{\\ell}\n",
    "\\newcommand{\\likelihoodBound}{\\mathcal{L}}\n",
    "\\newcommand{\\likelihoodFunction}{L}\n",
    "\\newcommand{\\locationScalar}{\\mu}\n",
    "\\newcommand{\\locationVector}{\\boldsymbol{ \\locationScalar}}\n",
    "\\newcommand{\\locationMatrix}{\\mathbf{M}}\n",
    "\\newcommand{\\variance}[1]{\\text{var}\\left( #1 \\right)}\n",
    "\\newcommand{\\mappingFunction}{f}\n",
    "\\newcommand{\\mappingFunctionMatrix}{\\mathbf{F}}\n",
    "\\newcommand{\\mappingFunctionTwo}{g}\n",
    "\\newcommand{\\mappingFunctionTwoMatrix}{\\mathbf{G}}\n",
    "\\newcommand{\\mappingFunctionTwoVector}{\\mathbf{ \\mappingFunctionTwo}}\n",
    "\\newcommand{\\mappingFunctionVector}{\\mathbf{ \\mappingFunction}}\n",
    "\\newcommand{\\scaleScalar}{s}\n",
    "\\newcommand{\\mappingScalar}{w}\n",
    "\\newcommand{\\mappingVector}{\\mathbf{ \\mappingScalar}}\n",
    "\\newcommand{\\mappingMatrix}{\\mathbf{W}}\n",
    "\\newcommand{\\mappingScalarTwo}{v}\n",
    "\\newcommand{\\mappingVectorTwo}{\\mathbf{ \\mappingScalarTwo}}\n",
    "\\newcommand{\\mappingMatrixTwo}{\\mathbf{V}}\n",
    "\\newcommand{\\maxIters}{K}\n",
    "\\newcommand{\\meanMatrix}{\\mathbf{M}}\n",
    "\\newcommand{\\meanScalar}{\\mu}\n",
    "\\newcommand{\\meanTwoMatrix}{\\mathbf{M}}\n",
    "\\newcommand{\\meanTwoScalar}{m}\n",
    "\\newcommand{\\meanTwoVector}{\\mathbf{ \\meanTwoScalar}}\n",
    "\\newcommand{\\meanVector}{\\boldsymbol{ \\meanScalar}}\n",
    "\\newcommand{\\mrnaConcentration}{m}\n",
    "\\newcommand{\\naturalFrequency}{\\omega}\n",
    "\\newcommand{\\neighborhood}[1]{\\mathcal{N}\\left( #1 \\right)}\n",
    "\\newcommand{\\neilurl}{http://inverseprobability.com/}\n",
    "\\newcommand{\\noiseMatrix}{\\boldsymbol{ E}}\n",
    "\\newcommand{\\noiseScalar}{\\epsilon}\n",
    "\\newcommand{\\noiseVector}{\\boldsymbol{ \\epsilon}}\n",
    "\\newcommand{\\norm}[1]{\\left\\Vert #1 \\right\\Vert}\n",
    "\\newcommand{\\normalizedLaplacianMatrix}{\\hat{\\mathbf{L}}}\n",
    "\\newcommand{\\normalizedLaplacianScalar}{\\hat{\\ell}}\n",
    "\\newcommand{\\normalizedLaplacianVector}{\\hat{\\mathbf{ \\ell}}}\n",
    "\\newcommand{\\numActive}{m}\n",
    "\\newcommand{\\numBasisFunc}{m}\n",
    "\\newcommand{\\numComponents}{m}\n",
    "\\newcommand{\\numComps}{K}\n",
    "\\newcommand{\\numData}{n}\n",
    "\\newcommand{\\numFeatures}{K}\n",
    "\\newcommand{\\numHidden}{h}\n",
    "\\newcommand{\\numInducing}{m}\n",
    "\\newcommand{\\numLayers}{\\ell}\n",
    "\\newcommand{\\numNeighbors}{K}\n",
    "\\newcommand{\\numSequences}{s}\n",
    "\\newcommand{\\numSuccess}{s}\n",
    "\\newcommand{\\numTasks}{m}\n",
    "\\newcommand{\\numTime}{T}\n",
    "\\newcommand{\\numTrials}{S}\n",
    "\\newcommand{\\outputIndex}{j}\n",
    "\\newcommand{\\paramVector}{\\boldsymbol{ \\theta}}\n",
    "\\newcommand{\\parameterMatrix}{\\boldsymbol{ \\Theta}}\n",
    "\\newcommand{\\parameterScalar}{\\theta}\n",
    "\\newcommand{\\parameterVector}{\\boldsymbol{ \\parameterScalar}}\n",
    "\\newcommand{\\partDiff}[2]{\\frac{\\partial#1}{\\partial#2}}\n",
    "\\newcommand{\\precisionScalar}{j}\n",
    "\\newcommand{\\precisionVector}{\\mathbf{ \\precisionScalar}}\n",
    "\\newcommand{\\precisionMatrix}{\\mathbf{J}}\n",
    "\\newcommand{\\pseudotargetScalar}{\\widetilde{y}}\n",
    "\\newcommand{\\pseudotargetVector}{\\mathbf{ \\pseudotargetScalar}}\n",
    "\\newcommand{\\pseudotargetMatrix}{\\mathbf{ \\widetilde{Y}}}\n",
    "\\newcommand{\\rank}[1]{\\text{rank}\\left(#1\\right)}\n",
    "\\newcommand{\\rayleighDist}[2]{\\mathcal{R}\\left(#1|#2\\right)}\n",
    "\\newcommand{\\rayleighSamp}[1]{\\mathcal{R}\\left(#1\\right)}\n",
    "\\newcommand{\\responsibility}{r}\n",
    "\\newcommand{\\rotationScalar}{r}\n",
    "\\newcommand{\\rotationVector}{\\mathbf{ \\rotationScalar}}\n",
    "\\newcommand{\\rotationMatrix}{\\mathbf{R}}\n",
    "\\newcommand{\\sampleCovScalar}{s}\n",
    "\\newcommand{\\sampleCovVector}{\\mathbf{ \\sampleCovScalar}}\n",
    "\\newcommand{\\sampleCovMatrix}{\\mathbf{s}}\n",
    "\\newcommand{\\scalarProduct}[2]{\\left\\langle{#1},{#2}\\right\\rangle}\n",
    "\\newcommand{\\sign}[1]{\\text{sign}\\left(#1\\right)}\n",
    "\\newcommand{\\sigmoid}[1]{\\sigma\\left(#1\\right)}\n",
    "\\newcommand{\\singularvalue}{\\ell}\n",
    "\\newcommand{\\singularvalueMatrix}{\\mathbf{L}}\n",
    "\\newcommand{\\singularvalueVector}{\\mathbf{l}}\n",
    "\\newcommand{\\sorth}{\\mathbf{u}}\n",
    "\\newcommand{\\spar}{\\lambda}\n",
    "\\newcommand{\\trace}[1]{\\text{tr}\\left(#1\\right)}\n",
    "\\newcommand{\\BasalRate}{B}\n",
    "\\newcommand{\\DampingCoefficient}{C}\n",
    "\\newcommand{\\DecayRate}{D}\n",
    "\\newcommand{\\Displacement}{X}\n",
    "\\newcommand{\\LatentForce}{F}\n",
    "\\newcommand{\\Mass}{M}\n",
    "\\newcommand{\\Sensitivity}{S}\n",
    "\\newcommand{\\basalRate}{b}\n",
    "\\newcommand{\\dampingCoefficient}{c}\n",
    "\\newcommand{\\mass}{m}\n",
    "\\newcommand{\\sensitivity}{s}\n",
    "\\newcommand{\\springScalar}{\\kappa}\n",
    "\\newcommand{\\springVector}{\\boldsymbol{ \\kappa}}\n",
    "\\newcommand{\\springMatrix}{\\boldsymbol{ \\mathcal{K}}}\n",
    "\\newcommand{\\tfConcentration}{p}\n",
    "\\newcommand{\\tfDecayRate}{\\delta}\n",
    "\\newcommand{\\tfMrnaConcentration}{f}\n",
    "\\newcommand{\\tfVector}{\\mathbf{ \\tfConcentration}}\n",
    "\\newcommand{\\velocity}{v}\n",
    "\\newcommand{\\sufficientStatsScalar}{g}\n",
    "\\newcommand{\\sufficientStatsVector}{\\mathbf{ \\sufficientStatsScalar}}\n",
    "\\newcommand{\\sufficientStatsMatrix}{\\mathbf{G}}\n",
    "\\newcommand{\\switchScalar}{s}\n",
    "\\newcommand{\\switchVector}{\\mathbf{ \\switchScalar}}\n",
    "\\newcommand{\\switchMatrix}{\\mathbf{S}}\n",
    "\\newcommand{\\tr}[1]{\\text{tr}\\left(#1\\right)}\n",
    "\\newcommand{\\loneNorm}[1]{\\left\\Vert #1 \\right\\Vert_1}\n",
    "\\newcommand{\\ltwoNorm}[1]{\\left\\Vert #1 \\right\\Vert_2}\n",
    "\\newcommand{\\onenorm}[1]{\\left\\vert#1\\right\\vert_1}\n",
    "\\newcommand{\\twonorm}[1]{\\left\\Vert #1 \\right\\Vert}\n",
    "\\newcommand{\\vScalar}{v}\n",
    "\\newcommand{\\vVector}{\\mathbf{v}}\n",
    "\\newcommand{\\vMatrix}{\\mathbf{V}}\n",
    "\\newcommand{\\varianceDist}[2]{\\text{var}_{#2}\\left( #1 \\right)}\n",
    "% Already defined by latex\n",
    "%\\newcommand{\\vec}{#1:}\n",
    "\\newcommand{\\vecb}[1]{\\left(#1\\right):}\n",
    "\\newcommand{\\weightScalar}{w}\n",
    "\\newcommand{\\weightVector}{\\mathbf{ \\weightScalar}}\n",
    "\\newcommand{\\weightMatrix}{\\mathbf{W}}\n",
    "\\newcommand{\\weightedAdjacencyMatrix}{\\mathbf{A}}\n",
    "\\newcommand{\\weightedAdjacencyScalar}{a}\n",
    "\\newcommand{\\weightedAdjacencyVector}{\\mathbf{ \\weightedAdjacencyScalar}}\n",
    "\\newcommand{\\onesVector}{\\mathbf{1}}\n",
    "\\newcommand{\\zerosVector}{\\mathbf{0}}\n",
    "$$\n",
    "\n",
    "<!-- Front matter -->\n",
    "<!-- Do not edit this file locally. -->\n",
    "<!-- Do not edit this file locally. -->\n",
    "<!-- Do not edit this file locally. -->\n",
    "<!---->\n",
    "<!-- Do not edit this file locally. -->\n",
    "<!--Back matter-->\n",
    "<!-- Do not edit this file locally. -->\n",
    "<!-- The last names to be defined. Should be defined entirely in terms of macros from above-->\n",
    "## Deep Gaussian Processes\n",
    "\n",
    "-   *Deep Gaussian Processes and Variational Propagation of Uncertainty*\n",
    "    @Damianou:thesis2015\n",
    "\n",
    "Even in the early days of Gaussian processes in machine learning, it was\n",
    "understood that we were throwing something fundamental away. This is\n",
    "perhaps captured best by David MacKay in his 1997 NeurIPS tutorial on\n",
    "Gaussian processes, where he asked \"Have we thrown out the baby with the\n",
    "bathwater?\". The quote below is from his summarization paper.\n",
    "\n",
    "> According to the hype of 1987, neural networks were meant to be\n",
    "> intelligent models which discovered features and patterns in data.\n",
    "> Gaussian processes in contrast are simply smoothing devices. How can\n",
    "> Gaussian processes possibly repalce neural networks? What is going on?\n",
    ">\n",
    "> @MacKay:gpintroduction98"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import teaching_plots as plot"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/deepgp/deep-nn2.svg\" class=\"\" align=\"70%\" style=\"vertical-align:middle;\">\n",
    "\n",
    "Figure: <i>A deep neural network. Input nodes are shown at the bottom.\n",
    "Each hidden layer is the result of applying an affine transformation to\n",
    "the previous layer and placing through an activation function.</i>\n",
    "\n",
    "Mathematically, each layer of a neural network is given through\n",
    "computing the activation function, $\\basisFunction(\\cdot)$, contingent\n",
    "on the previous layer, or the inputs. In this way the activation\n",
    "functions, are composed to generate more complex interactions than would\n",
    "be possible with any single layer. $$\n",
    "\\begin{align}\n",
    "    \\hiddenVector_{1} &= \\basisFunction\\left(\\mappingMatrix_1 \\inputVector\\right)\\\\\n",
    "    \\hiddenVector_{2} &=  \\basisFunction\\left(\\mappingMatrix_2\\hiddenVector_{1}\\right)\\\\\n",
    "    \\hiddenVector_{3} &= \\basisFunction\\left(\\mappingMatrix_3 \\hiddenVector_{2}\\right)\\\\\n",
    "    \\dataVector &= \\mappingVector_4 ^\\top\\hiddenVector_{3}\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "## Overfitting \\[<a href=\"https://github.com/lawrennd/snippets/edit/main/_deepgp/includes/overfitting-low-rank.md\" target=\"_blank\" >edit</a>\\]\n",
    "\n",
    "One potential problem is that as the number of nodes in two adjacent\n",
    "layers increases, the number of parameters in the affine transformation\n",
    "between layers, $\\mappingMatrix$, increases. If there are $k_{i-1}$\n",
    "nodes in one layer, and $k_i$ nodes in the following, then that matrix\n",
    "contains $k_i k_{i-1}$ parameters, when we have layer widths in the\n",
    "1000s that leads to millions of parameters.\n",
    "\n",
    "One proposed solution is known as *dropout* where only a sub-set of the\n",
    "neural network is trained at each iteration. An alternative solution\n",
    "would be to reparameterize $\\mappingMatrix$ with its *singular value\n",
    "decomposition*. $$\n",
    "  \\mappingMatrix = \\eigenvectorMatrix\\eigenvalueMatrix\\eigenvectwoMatrix^\\top\n",
    "  $$ or $$\n",
    "  \\mappingMatrix = \\eigenvectorMatrix\\eigenvectwoMatrix^\\top\n",
    "  $$ where if $\\mappingMatrix \\in \\Re^{k_1\\times k_2}$ then\n",
    "$\\eigenvectorMatrix\\in \\Re^{k_1\\times q}$ and\n",
    "$\\eigenvectwoMatrix \\in \\Re^{k_2\\times q}$, i.e. we have a low rank\n",
    "matrix factorization for the weights."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import teaching_plots as plot"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/wisuvt.svg\" class=\"\" align=\"80%\" style=\"vertical-align:middle;\">\n",
    "\n",
    "Figure: <i>Pictorial representation of the low rank form of the matrix\n",
    "$\\mappingMatrix$.</i>\n",
    "\n",
    "## Bottleneck Layers in Deep Neural Networks \\[<a href=\"https://github.com/lawrennd/snippets/edit/main/_deepgp/includes/deep-gp.md\" target=\"_blank\" >edit</a>\\]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import teaching_plots as plot"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/deepgp/deep-nn-bottleneck2.svg\" class=\"\" align=\"70%\" style=\"vertical-align:middle;\">\n",
    "\n",
    "Figure: <i>Inserting the bottleneck layers introduces a new set of\n",
    "variables.</i>\n",
    "\n",
    "Including the low rank decomposition of $\\mappingMatrix$ in the neural\n",
    "network, we obtain a new mathematical form. Effectively, we are adding\n",
    "additional *latent* layers, $\\latentVector$, in between each of the\n",
    "existing hidden layers. In a neural network these are sometimes known as\n",
    "*bottleneck* layers. The network can now be written mathematically as $$\n",
    "\\begin{align}\n",
    "  \\latentVector_{1} &= \\eigenvectwoMatrix^\\top_1 \\inputVector\\\\\n",
    "  \\hiddenVector_{1} &= \\basisFunction\\left(\\eigenvectorMatrix_1 \\latentVector_{1}\\right)\\\\\n",
    "  \\latentVector_{2} &= \\eigenvectwoMatrix^\\top_2 \\hiddenVector_{1}\\\\\n",
    "  \\hiddenVector_{2} &= \\basisFunction\\left(\\eigenvectorMatrix_2 \\latentVector_{2}\\right)\\\\\n",
    "  \\latentVector_{3} &= \\eigenvectwoMatrix^\\top_3 \\hiddenVector_{2}\\\\\n",
    "  \\hiddenVector_{3} &= \\basisFunction\\left(\\eigenvectorMatrix_3 \\latentVector_{3}\\right)\\\\\n",
    "  \\dataVector &= \\mappingVector_4^\\top\\hiddenVector_{3}.\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "  \\latentVector_{1} &= \\eigenvectwoMatrix^\\top_1 \\inputVector\\\\\n",
    "  \\latentVector_{2} &= \\eigenvectwoMatrix^\\top_2 \\basisFunction\\left(\\eigenvectorMatrix_1 \\latentVector_{1}\\right)\\\\\n",
    "  \\latentVector_{3} &= \\eigenvectwoMatrix^\\top_3 \\basisFunction\\left(\\eigenvectorMatrix_2 \\latentVector_{2}\\right)\\\\\n",
    "  \\dataVector &= \\mappingVector_4 ^\\top \\latentVector_{3}\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "## Cascade of Gaussian Processes \\[<a href=\"https://github.com/lawrennd/snippets/edit/main/_deepgp/includes/cascade-of-gps.md\" target=\"_blank\" >edit</a>\\]\n",
    "\n",
    "Now if we replace each of these neural networks with a Gaussian process.\n",
    "This is equivalent to taking the limit as the width of each layer goes\n",
    "to infinity, while appropriately scaling down the outputs.\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "  \\latentVector_{1} &= \\mappingFunctionVector_1\\left(\\inputVector\\right)\\\\\n",
    "  \\latentVector_{2} &= \\mappingFunctionVector_2\\left(\\latentVector_{1}\\right)\\\\\n",
    "  \\latentVector_{3} &= \\mappingFunctionVector_3\\left(\\latentVector_{2}\\right)\\\\\n",
    "  \\dataVector &= \\mappingFunctionVector_4\\left(\\latentVector_{3}\\right)\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "# Deep Learning \\[<a href=\"https://github.com/lawrennd/snippets/edit/main/_ml/includes/deep-learning-overview.md\" target=\"_blank\" >edit</a>\\]\n",
    "\n",
    "<!-- No slide titles in this context -->\n",
    "### DeepFace \\[<a href=\"https://github.com/lawrennd/snippets/edit/main/_ml/includes/deep-face.md\" target=\"_blank\" >edit</a>\\]\n",
    "\n",
    "<img class=\"\" src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/deepface_neg.png\" style=\"width:100%\">\n",
    "\n",
    "Figure: <i>The DeepFace architecture [@Taigman:deepface14], visualized\n",
    "through colors to represent the functional mappings at each layer. There\n",
    "are 120 million parameters in the model.</i>\n",
    "\n",
    "The DeepFace architecture [@Taigman:deepface14] consists of layers that\n",
    "deal with *translation* and *rotational* invariances. These layers are\n",
    "followed by three locally-connected layers and two fully-connected\n",
    "layers. Color illustrates feature maps produced at each layer. The\n",
    "neural network includes more than 120 million parameters, where more\n",
    "than 95% come from the local and fully connected layers.\n",
    "\n",
    "### Deep Learning as Pinball \\[<a href=\"https://github.com/lawrennd/snippets/edit/main/_ml/includes/deep-learning-as-pinball.md\" target=\"_blank\" >edit</a>\\]\n",
    "\n",
    "<img class=\"\" src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/576px-Early_Pinball.jpg\" style=\"width:50%\">\n",
    "\n",
    "Figure: <i>Deep learning models are composition of simple functions. We\n",
    "can think of a pinball machine as an analogy. Each layer of pins\n",
    "corresponds to one of the layers of functions in the model. Input data\n",
    "is represented by the location of the ball from left to right when it is\n",
    "dropped in from the top. Output class comes from the position of the\n",
    "ball as it leaves the pins at the bottom.</i>\n",
    "\n",
    "Sometimes deep learning models are described as being like the brain, or\n",
    "too complex to understand, but one analogy I find useful to help the\n",
    "gist of these models is to think of them as being similar to early pin\n",
    "ball machines.\n",
    "\n",
    "In a deep neural network, we input a number (or numbers), whereas in\n",
    "pinball, we input a ball.\n",
    "\n",
    "Think of the location of the ball on the left-right axis as a single\n",
    "number. Our simple pinball machine can only take one number at a time.\n",
    "As the ball falls through the machine, each layer of pins can be thought\n",
    "of as a different layer of 'neurons'. Each layer acts to move the ball\n",
    "from left to right.\n",
    "\n",
    "In a pinball machine, when the ball gets to the bottom it might fall\n",
    "into a hole defining a score, in a neural network, that is equivalent to\n",
    "the decision: a classification of the input object.\n",
    "\n",
    "An image has more than one number associated with it, so it is like\n",
    "playing pinball in a *hyper-space*."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import pods\n",
    "from ipywidgets import IntSlider"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "pods.notebook.display_plots('pinball{sample:0>3}.svg', \n",
    "                            '../slides/diagrams',\n",
    "                            sample=IntSlider(1, 1, 2, 1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/pinball001.svg\" class=\"\" align=\"80%\" style=\"vertical-align:middle;\">\n",
    "\n",
    "Figure: <i>At initialization, the pins, which represent the parameters\n",
    "of the function, aren't in the right place to bring the balls to the\n",
    "correct decisions.</i>\n",
    "\n",
    "<img src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/pinball002.svg\" class=\"\" align=\"80%\" style=\"vertical-align:middle;\">\n",
    "\n",
    "Figure: <i>After learning the pins are now in the right place to bring\n",
    "the balls to the correct decisions.</i>\n",
    "\n",
    "Learning involves moving all the pins to be in the correct position, so\n",
    "that the ball ends up in the right place when it's fallen through the\n",
    "machine. But moving all these pins in hyperspace can be difficult.\n",
    "\n",
    "In a hyper-space you have to put a lot of data through the machine for\n",
    "to explore the positions of all the pins. Even when you feed many\n",
    "millions of data points through the machine, there are likely to be\n",
    "regions in the hyper-space where no ball has passed. When future test\n",
    "data passes through the machine in a new route unusual things can\n",
    "happen.\n",
    "\n",
    "*Adversarial examples* exploit this high dimensional space. If you have\n",
    "access to the pinball machine, you can use gradient methods to find a\n",
    "position for the ball in the hyper space where the image looks like one\n",
    "thing, but will be classified as another.\n",
    "\n",
    "Probabilistic methods explore more of the space by considering a range\n",
    "of possible paths for the ball through the machine. This helps to make\n",
    "them more data efficient and gives some robustness to adversarial\n",
    "examples.\n",
    "\n",
    "Mathematically, a deep Gaussian process can be seen as a composite\n",
    "*multivariate* function, $$\n",
    "  \\mathbf{g}(\\inputVector)=\\mappingFunctionVector_5(\\mappingFunctionVector_4(\\mappingFunctionVector_3(\\mappingFunctionVector_2(\\mappingFunctionVector_1(\\inputVector))))).\n",
    "  $$ Or if we view it from the probabilistic perspective we can see that\n",
    "a deep Gaussian process is specifying a factorization of the joint\n",
    "density, the standard deep model takes the form of a Markov chain.\n",
    "\n",
    "$$\n",
    "  p(\\dataVector|\\inputVector)= p(\\dataVector|\\mappingFunctionVector_5)p(\\mappingFunctionVector_5|\\mappingFunctionVector_4)p(\\mappingFunctionVector_4|\\mappingFunctionVector_3)p(\\mappingFunctionVector_3|\\mappingFunctionVector_2)p(\\mappingFunctionVector_2|\\mappingFunctionVector_1)p(\\mappingFunctionVector_1|\\inputVector)\n",
    "  $$\n",
    "\n",
    "<img src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/deepgp/deep-markov.svg\" class=\"\" align=\"80%\" style=\"vertical-align:middle;\">\n",
    "\n",
    "Figure: <i>Probabilistically the deep Gaussian process can be\n",
    "represented as a Markov chain.</i>\n",
    "\n",
    "<img src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/deepgp/deep-markov-vertical.svg\" class=\"\" align=\"7%\" style=\"vertical-align:middle;\">\n",
    "\n",
    "Figure: <i>More usually deep probabilistic models are written vertically\n",
    "rather than horizontally as in the Markov chain.</i>\n",
    "\n",
    "## Why Deep? \\[<a href=\"https://github.com/lawrennd/snippets/edit/main/_deepgp/includes/process-composition.md\" target=\"_blank\" >edit</a>\\]\n",
    "\n",
    "If the result of composing many functions together is simply another\n",
    "function, then why do we bother? The key point is that we can change the\n",
    "class of functions we are modeling by composing in this manner. A\n",
    "Gaussian process is specifying a prior over functions, and one with a\n",
    "number of elegant properties. For example, the derivative process (if it\n",
    "exists) of a Gaussian process is also Gaussian distributed. That makes\n",
    "it easy to assimilate, for example, derivative observations. But that\n",
    "also might raise some alarm bells. That implies that the *marginal\n",
    "derivative distribution* is also Gaussian distributed. If that's the\n",
    "case, then it means that functions which occasionally exhibit very large\n",
    "derivatives are hard to model with a Gaussian process. For example, a\n",
    "function with jumps in.\n",
    "\n",
    "A one off discontinuity is easy to model with a Gaussian process, or\n",
    "even multiple discontinuities. They can be introduced in the mean\n",
    "function, or independence can be forced between two covariance functions\n",
    "that apply in different areas of the input space. But in these cases we\n",
    "will need to specify the number of discontinuities and where they occur.\n",
    "In otherwords we need to *parameterise* the discontinuities. If we do\n",
    "not know the number of discontinuities and don't wish to specify where\n",
    "they occur, i.e. if we want a non-parametric representation of\n",
    "discontinuities, then the standard Gaussian process doesn't help.\n",
    "\n",
    "## Stochastic Process Composition\n",
    "\n",
    "The deep Gaussian process leads to *non-Gaussian* models, and\n",
    "non-Gaussian characteristics in the covariance function. In effect, what\n",
    "we are proposing is that we change the properties of the functions we\n",
    "are considering by *composing stochastic processes*. This is an approach\n",
    "to creating new stochastic processes from well known processes.\n",
    "\n",
    "Additionally, we are not constrained to the formalism of the chain. For\n",
    "example, we can easily add single nodes emerging from some point in the\n",
    "depth of the chain. This allows us to combine the benefits of the\n",
    "graphical modelling formalism, but with a powerful framework for\n",
    "relating one set of variables to another, that of Gaussian processes\n",
    "\n",
    "<img src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/deepgp/deep-markov-vertical-side.svg\" class=\"\" align=\"15%\" style=\"vertical-align:middle;\">\n",
    "\n",
    "Figure: <i>More generally we aren't constrained by the Markov chain. We\n",
    "can design structures that respect our belief about the underlying\n",
    "conditional dependencies. Here we are adding a side note from the\n",
    "chain.</i>\n",
    "\n",
    "## Difficulty for Probabilistic Approaches \\[<a href=\"https://github.com/lawrennd/snippets/edit/main/_dimred/includes/non-linear-difficulty.md\" target=\"_blank\" >edit</a>\\]\n",
    "\n",
    "The challenge for composition of probabilistic models is that you need\n",
    "to propagate a probability densities through non linear mappings. This\n",
    "allows you to create broader classes of probability density.\n",
    "Unfortunately it renders the resulting densities *intractable*.\n",
    "\n",
    "<img src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/dimred/nonlinear-mapping-3d-plot.svg\" class=\"80%\" align=\"center\" style=\"vertical-align:middle;\">\n",
    "\n",
    "Figure: <i>A two dimensional grid mapped into three dimensions to form a\n",
    "two dimensional manifold.</i>\n",
    "\n",
    "<img src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/dimred/nonlinear-mapping-2d-plot.svg\" class=\"80%\" align=\"center\" style=\"vertical-align:middle;\">\n",
    "\n",
    "Figure: <i>A one dimensional line mapped into two dimensions by two\n",
    "separate independent functions. Each point can be mapped exactly through\n",
    "the mappings.</i>\n",
    "\n",
    "<img src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/dimred/gaussian-through-nonlinear.svg\" class=\"\" align=\"80%\" style=\"vertical-align:middle;\">\n",
    "\n",
    "Figure: <i>A Gaussian density over the input of a non linear function\n",
    "leads to a very non Gaussian output. Here the output is multimodal.</i>\n",
    "\n",
    "## Standard Variational Approach Fails \\[<a href=\"https://github.com/lawrennd/snippets/edit/main/_gplvm/includes/variational-bayes-gplvm-long.md\" target=\"_blank\" >edit</a>\\]\n",
    "\n",
    "-   Standard variational bound has the form: $$\n",
    "    \\likelihoodBound = \\expDist{\\log p(\\dataVector|\\latentMatrix)}{q(\\latentMatrix)} + \\KL{q(\\latentMatrix)}{p(\\latentMatrix)}\n",
    "    $$\n",
    "\n",
    "The standard variational approach would require the expectation of\n",
    "$\\log p(\\dataVector|\\latentMatrix)$ under $q(\\latentMatrix)$. $$\n",
    "  \\begin{align}\n",
    "  \\log p(\\dataVector|\\latentMatrix) = & -\\frac{1}{2}\\dataVector^\\top\\left(\\kernelMatrix_{\\mappingFunctionVector, \\mappingFunctionVector}+\\dataStd^2\\eye\\right)^{-1}\\dataVector \\\\ & -\\frac{1}{2}\\log \\det{\\kernelMatrix_{\\mappingFunctionVector, \\mappingFunctionVector}+\\dataStd^2 \\eye} -\\frac{\\numData}{2}\\log 2\\pi\n",
    "  \\end{align}\n",
    "  $$ But this is extremely difficult to compute because\n",
    "$\\kernelMatrix_{\\mappingFunctionVector, \\mappingFunctionVector}$ is\n",
    "dependent on $\\latentMatrix$ and it appears in the inverse.\n",
    "\n",
    "## Variational Bayesian GP-LVM\n",
    "\n",
    "The alternative approach is to consider the collapsed variational bound\n",
    "(used for low rank (sparse is a misnomer) Gaussian process\n",
    "approximations. $$\n",
    "    p(\\dataVector)\\geq \\prod_{i=1}^\\numData c_i \\int \\gaussianDist{\\dataVector}{\\expSamp{\\mappingFunctionVector}}{\\dataStd^2\\eye}p(\\inducingVector) \\text{d}\\inducingVector\n",
    "  $$ $$\n",
    "    p(\\dataVector|\\latentMatrix )\\geq \\prod_{i=1}^\\numData c_i \\int \\gaussianDist{\\dataVector}{\\expDist{\\mappingFunctionVector}{p(\\mappingFunctionVector|\\inducingVector, \\latentMatrix)}}{\\dataStd^2\\eye}p(\\inducingVector) \\text{d}\\inducingVector\n",
    "  $$ $$\n",
    "      \\int p(\\dataVector|\\latentMatrix)p(\\latentMatrix) \\text{d}\\latentMatrix \\geq \\int \\prod_{i=1}^\\numData c_i \\gaussianDist{\\dataVector}{\\expDist{\\mappingFunctionVector}{p(\\mappingFunctionVector|\\inducingVector, \\latentMatrix)}}{\\dataStd^2\\eye} p(\\latentMatrix)\\text{d}\\latentMatrix p(\\inducingVector) \\text{d}\\inducingVector\n",
    "  $$\n",
    "\n",
    "To integrate across $\\latentMatrix$ we apply the lower bound to the\n",
    "inner integral. $$\n",
    "    \\begin{align}\n",
    "    \\int \\prod_{i=1}^\\numData c_i \\gaussianDist{\\dataVector}{\\expDist{\\mappingFunctionVector}{p(\\mappingFunctionVector|\\inducingVector, \\latentMatrix)}}{\\dataStd^2\\eye} p(\\latentMatrix)\\text{d}\\latentMatrix \\geq & \\expDist{\\sum_{i=1}^\\numData\\log  c_i}{q(\\latentMatrix)}\\\\ & +\\expDist{\\log\\gaussianDist{\\dataVector}{\\expDist{\\mappingFunctionVector}{p(\\mappingFunctionVector|\\inducingVector, \\latentMatrix)}}{\\dataStd^2\\eye}}{q(\\latentMatrix)}\\\\& + \\KL{q(\\latentMatrix)}{p(\\latentMatrix)}    \n",
    "    \\end{align}\n",
    "  $$ \\* Which is analytically tractable for Gaussian $q(\\latentMatrix)$\n",
    "and some covariance functions.\n",
    "\n",
    "-   Need expectations under $q(\\latentMatrix)$ of: $$\n",
    "    \\log c_i = \\frac{1}{2\\dataStd^2} \\left[\\kernelScalar_{i, i} - \\kernelVector_{i, \\inducingVector}^\\top \\kernelMatrix_{\\inducingVector, \\inducingVector}^{-1} \\kernelVector_{i, \\inducingVector}\\right]\n",
    "    $$ and $$\n",
    "    \\log \\gaussianDist{\\dataVector}{\\expDist{\\mappingFunctionVector}{p(\\mappingFunctionVector|\\inducingVector,\\dataMatrix)}}{\\dataStd^2\\eye} = -\\frac{1}{2}\\log 2\\pi\\dataStd^2 - \\frac{1}{2\\dataStd^2}\\left(\\dataScalar_i - \\kernelMatrix_{\\mappingFunctionVector, \\inducingVector}\\kernelMatrix_{\\inducingVector,\\inducingVector}^{-1}\\inducingVector\\right)^2\n",
    "    $$\n",
    "\n",
    "-   This requires the expectations $$\n",
    "    \\expDist{\\kernelMatrix_{\\mappingFunctionVector,\\inducingVector}}{q(\\latentMatrix)}\n",
    "    $$ and $$\n",
    "    \\expDist{\\kernelMatrix_{\\mappingFunctionVector,\\inducingVector}\\kernelMatrix_{\\inducingVector,\\inducingVector}^{-1}\\kernelMatrix_{\\inducingVector,\\mappingFunctionVector}}{q(\\latentMatrix)}\n",
    "    $$ which can be computed analytically for some covariance functions\n",
    "    [@Damianou:variational15] or through sampling\n",
    "    [@Damianou:thesis2015; @Salimbeni:doubly2017].\n",
    "\n",
    "Variational approximations aren't the only approach to approximate\n",
    "inference. The original work on deep Gaussian processes made use of MAP\n",
    "approximations [@Lawrence:hgplvm07], which couldn't propagate the\n",
    "uncertainty through the model at the data points but sustain uncertainty\n",
    "elsewhere. Since the variational approximation was proposed researchers\n",
    "have also considered sampling approaches [@Havasi:deepgp18] and\n",
    "expectation propagation [@Bui:deep16].\n",
    "\n",
    "<img class=\"\" src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/deepgp/neural-network-uncertainty.png\" style=\"width:90%\">\n",
    "\n",
    "Figure: <i>Even the latest work on Bayesian neural networks has severe\n",
    "problems handling uncertainty. In this example, [@Izmailov:subspace19],\n",
    "methods even fail to interpolate through the data correctly or provide\n",
    "well calibrated error bars in regions where data is observed.</i>\n",
    "\n",
    "The argument in the deep learning revolution is that deep architectures\n",
    "allow us to develop an abstraction of the feature set through model\n",
    "composition. Composing Gaussian processes is analytically intractable.\n",
    "To form deep Gaussian processes we use a variational approach to stack\n",
    "the models."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import pods"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "pods.notebook.display_plots('stack-gp-sample-Linear-{sample:0>1}.svg', \n",
    "                            directory='../../slides/diagrams/deepgp', sample=(0,4))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Stacked PCA \\[<a href=\"https://github.com/lawrennd/snippets/edit/main/_deepgp/includes/stacked-pca.md\" target=\"_blank\" >edit</a>\\]\n",
    "\n",
    "<img src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/stack-pca-sample-4.svg\" class=\"\" align=\"20%\" style=\"vertical-align:middle;\">\n",
    "\n",
    "Figure: <i>Composition of linear functions just leads to a new linear\n",
    "function. Here you see the result of multiple affine transformations\n",
    "applied to a square in two dimensions.</i>\n",
    "\n",
    "Stacking a series of linear functions simply leads to a new linear\n",
    "function. The use of multiple linear function merely changes the\n",
    "covariance of the resulting Gaussian. If $$\n",
    "\\latentMatrix \\sim \\gaussianSamp{\\zerosVector}{\\eye}\n",
    "$$ and the $i$th hidden layer is a multivariate linear transformation\n",
    "defined by $\\weightMatrix_i$, $$\n",
    "\\dataMatrix = \\latentMatrix\\weightMatrix_1 \\weightMatrix_2 \\dots \\weightMatrix_\\numLayers\n",
    "$$ then the rules of multivariate Gaussians tell us that $$\n",
    "\\dataMatrix \\sim \\gaussianSamp{\\zerosVector}{\\weightMatrix_\\numLayers \\dots \\weightMatrix_1 \\weightMatrix^\\top_1 \\dots \\weightMatrix^\\top_\\numLayers}.\n",
    "$$ So the model can be replaced by one where we set\n",
    "$\\vMatrix = \\weightMatrix_\\numLayers \\dots \\weightMatrix_2 \\weightMatrix_1$.\n",
    "So is such a model trivial? The answer is that it depends. There are two\n",
    "cases in which such a model remaisn interesting. Firstly, if we make\n",
    "intermediate observations stemming from the chain. So, for example, if\n",
    "we decide that, $$\n",
    "\\latentMatrix_i = \\weightMatrix_i \\latentMatrix_{i-1}\n",
    "$$ and set\n",
    "$\\latentMatrix_{0} = \\inputMatrix \\sim \\gaussianSamp{\\zerosVector}{\\eye}$,\n",
    "then the matrices $\\weightMatrix$ inter-relate a series of jointly\n",
    "Gaussian observations in an interesting way, stacking the full data\n",
    "matrix to give $$\n",
    "\\latentMatrix = \\begin{bmatrix}\n",
    "\\latentMatrix_0 \\\\\n",
    "\\latentMatrix_1 \\\\\n",
    "\\vdots \\\\\n",
    "\\latentMatrix_\\numLayers\n",
    "\\end{bmatrix}\n",
    "$$ we can obtain\n",
    "$$\\latentMatrix \\sim \\gaussianSamp{\\zerosVector}{\\begin{bmatrix}\n",
    "\\eye & \\weightMatrix^\\top_1 & \\weightMatrix_1^\\top\\weightMatrix_2^\\top & \\dots & \\vMatrix^\\top \\\\\n",
    "\\weightMatrix_1 & \\weightMatrix_1 \\weightMatrix_1^\\top & \\weightMatrix_1 \\weightMatrix_1^\\top \\weightMatrix_2^\\top & \\dots & \\weightMatrix_1 \\vMatrix^\\top \\\\\n",
    "\\weightMatrix_2 \\weightMatrix_1 & \\weightMatrix_2 \\weightMatrix_1 \\weightMatrix_1^\\top & \\weightMatrix_2 \\weightMatrix_1 \\weightMatrix_1^\\top \\weightMatrix_2^\\top & \\dots & \\weightMatrix_2 \\weightMatrix_1 \\vMatrix^\\top \\\\\n",
    "\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n",
    "\\vMatrix & \\vMatrix   \\weightMatrix_1^\\top  & \\vMatrix \\weightMatrix_1^\\top \\weightMatrix_2^\\top& \\dots & \\vMatrix\\vMatrix^\\top\n",
    "\\end{bmatrix}}$$ which is a highly structured Gaussian covariance with\n",
    "hierarchical dependencies between the variables $\\latentMatrix_i$.\n",
    "\n",
    "## Stacked GP \\[<a href=\"https://github.com/lawrennd/snippets/edit/main/_deepgp/includes/stacked-gp.md\" target=\"_blank\" >edit</a>\\]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "pods.notebook.display_plots('stack-gp-sample-RBF-{sample:0>1}.svg', \n",
    "                            directory='../../slides/diagrams/deepgp', sample=(0,4))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/stack-gp-sample-4.svg\" class=\"\" align=\"20%\" style=\"vertical-align:middle;\">\n",
    "\n",
    "Figure: <i>Stacking Gaussian process models leads to non linear mappings\n",
    "at each stage. Here we are mapping from two dimensions to two dimensions\n",
    "in each layer.</i>\n",
    "\n",
    "Note that once the box has folded over on itself, it cannot be unfolded.\n",
    "So a feature that is generated near the top of the model cannot be\n",
    "removed further down the model.\n",
    "\n",
    "This folding over effect happens in low dimensions. In higher dimensions\n",
    "it is less common.\n",
    "\n",
    "Observation of this effect at a talk in Cambridge was one of the things\n",
    "that caused David Duvenaud (and collaborators) to consider the behavior\n",
    "of deeper Gaussian process models [@Duvenaud:pathologies14].\n",
    "\n",
    "Such folding over in the latent spaces necessarily forces the density to\n",
    "be non-Gaussian. Indeed, since folding-over is avoided as we increase\n",
    "the dimensionality of the latent spaces, such processes become more\n",
    "Gaussian. If we take the limit of the latent space dimensionality as it\n",
    "tends to infinity, the entire deep Gaussian process returns to a\n",
    "standard Gaussian process, with a covariance function given as a deep\n",
    "kernel (such as those described by @Cho:deep09).\n",
    "\n",
    "Further analysis of these deep networks has been conducted by\n",
    "@Dunlop:deep2017, who use analysis of the deep network's stationary\n",
    "density (treating it as a Markov chain across layers), to explore the\n",
    "nature of the implied process prior for a deep GP.\n",
    "\n",
    "Both of these works, however, make constraining assumptions on the form\n",
    "of the Gaussian process prior at each layer (e.g. same covariance at\n",
    "each layer). In practice, the form of this covariance can be learnt and\n",
    "the densities described by the deep GP are more general than those\n",
    "mentioned in either of these papers.\n",
    "\n",
    "## Stacked GPs (video by David Duvenaud) \\[<a href=\"https://github.com/lawrennd/snippets/edit/main/_deepgp/includes/deep-pathologies.md\" target=\"_blank\" >edit</a>\\]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from IPython.lib.display import YouTubeVideo\n",
    "YouTubeVideo('XhIvygQYFFQ')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Figure: <i>Visualization of mapping of a two dimensional space through a\n",
    "deep Gaussian process.</i>\n",
    "\n",
    "David Duvenaud also created a YouTube video to help visualize what\n",
    "happens as you drop through the layers of a deep GP.\n",
    "\n",
    "## GPy: A Gaussian Process Framework in Python \\[<a href=\"https://github.com/lawrennd/snippets/edit/main/_gp/includes/gpy-software.md\" target=\"_blank\" >edit</a>\\]\n",
    "\n",
    "<img class=\"\" src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/gp/gpy.png\" style=\"width:70%\">\n",
    "\n",
    "Figure: <i>GPy is a BSD licensed software code base for implementing\n",
    "Gaussian process models in Python. It is designed for teaching and\n",
    "modelling. We welcome contributions which can be made through the Github\n",
    "repository <https://github.com/SheffieldML/GPy></i>\n",
    "\n",
    "GPy is a BSD licensed software code base for implementing Gaussian\n",
    "process models in python. This allows GPs to be combined with a wide\n",
    "variety of software libraries.\n",
    "\n",
    "The software itself is available on\n",
    "[GitHub](https://github.com/SheffieldML/GPy) and the team welcomes\n",
    "contributions.\n",
    "\n",
    "The aim for GPy is to be a probabilistic-style programming language,\n",
    "i.e. you specify the model rather than the algorithm. As well as a large\n",
    "range of covariance functions the software allows for non-Gaussian\n",
    "likelihoods, multivariate outputs, dimensionality reduction and\n",
    "approximations for larger data sets.\n",
    "\n",
    "The GPy library can be installed via pip:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "pip install GPy"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This notebook depends on PyDeepGP. These libraries can be installed via\n",
    "pip:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "pip install git+https://github.com/SheffieldML/PyDeepGP.git"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Olympic Marathon Data \\[<a href=\"https://github.com/lawrennd/snippets/edit/main/_ml/includes/olympic-marathon-data.md\" target=\"_blank\" >edit</a>\\]\n",
    "\n",
    "<table>\n",
    "<tr>\n",
    "<td width=\"70%\">\n",
    "-   Gold medal times for Olympic Marathon since 1896.\n",
    "-   Marathons before 1924 didn't have a standardised distance.\n",
    "-   Present results using pace per km.\n",
    "-   In 1904 Marathon was badly organised leading to very slow times.\n",
    "\n",
    "</td>\n",
    "<td width=\"30%\">\n",
    "<img class=\"\" src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/Stephen_Kiprotich.jpg\" style=\"width:100%\">\n",
    "<small>Image from Wikimedia Commons <http://bit.ly/16kMKHQ></small>\n",
    "</td>\n",
    "</tr>\n",
    "</table>\n",
    "The first thing we will do is load a standard data set for regression\n",
    "modelling. The data consists of the pace of Olympic Gold Medal Marathon\n",
    "winners for the Olympics from 1896 to present. First we load in the data\n",
    "and plot."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import pods"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "data = pods.datasets.olympic_marathon_men()\n",
    "x = data['X']\n",
    "y = data['Y']\n",
    "\n",
    "offset = y.mean()\n",
    "scale = np.sqrt(y.var())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "import teaching_plots as plot\n",
    "import mlai"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "xlim = (1875,2030)\n",
    "ylim = (2.5, 6.5)\n",
    "yhat = (y-offset)/scale\n",
    "\n",
    "fig, ax = plt.subplots(figsize=plot.big_wide_figsize)\n",
    "_ = ax.plot(x, y, 'r.',markersize=10)\n",
    "ax.set_xlabel('year', fontsize=20)\n",
    "ax.set_ylabel('pace min/km', fontsize=20)\n",
    "ax.set_xlim(xlim)\n",
    "ax.set_ylim(ylim)\n",
    "\n",
    "mlai.write_figure(figure=fig, \n",
    "                  filename='../slides/diagrams/datasets/olympic-marathon.svg', \n",
    "                  transparent=True, \n",
    "                  frameon=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/datasets/olympic-marathon.svg\" class=\"\" align=\"\" style=\"vertical-align:middle;\">\n",
    "\n",
    "Figure: <i>Olympic marathon pace times since 1892.</i>\n",
    "\n",
    "Things to notice about the data include the outlier in 1904, in this\n",
    "year, the olympics was in St Louis, USA. Organizational problems and\n",
    "challenges with dust kicked up by the cars following the race meant that\n",
    "participants got lost, and only very few participants completed.\n",
    "\n",
    "More recent years see more consistently quick marathons.\n",
    "\n",
    "## Alan Turing \\[<a href=\"https://github.com/lawrennd/snippets/edit/main/_ml/includes/alan-turing-marathon.md\" target=\"_blank\" >edit</a>\\]\n",
    "\n",
    "<table>\n",
    "<tr>\n",
    "<td width=\"50%\">\n",
    "<img class=\"\" src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/turing-times.gif\" style=\"width:100%\">\n",
    "</td>\n",
    "<td width=\"50%\">\n",
    "<img class=\"\" src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/turing-run.jpg\" style=\"width:50%\">\n",
    "</td>\n",
    "</tr>\n",
    "</table>\n",
    "Figure: <i>Alan Turing, in 1946 he was only 11 minutes slower than the\n",
    "winner of the 1948 games. Would he have won a hypothetical games held in\n",
    "1946? Source: [Alan Turing Internet\n",
    "Scrapbook](http://www.turing.org.uk/scrapbook/run.html).</i>\n",
    "\n",
    "If we had to summarise the objectives of machine learning in one word, a\n",
    "very good candidate for that word would be *generalization*. What is\n",
    "generalization? From a human perspective it might be summarised as the\n",
    "ability to take lessons learned in one domain and apply them to another\n",
    "domain. If we accept the definition given in the first session for\n",
    "machine learning, $$\n",
    "\\text{data} + \\text{model} \\xrightarrow{\\text{compute}} \\text{prediction}\n",
    "$$ then we see that without a model we can't generalise: we only have\n",
    "data. Data is fine for answering very specific questions, like \"Who won\n",
    "the Olympic Marathon in 2012?\", because we have that answer stored,\n",
    "however, we are not given the answer to many other questions. For\n",
    "example, Alan Turing was a formidable marathon runner, in 1946 he ran a\n",
    "time 2 hours 46 minutes (just under four minutes per kilometer, faster\n",
    "than I and most of the other [Endcliffe Park\n",
    "Run](http://www.parkrun.org.uk/sheffieldhallam/) runners can do 5 km).\n",
    "What is the probability he would have won an Olympics if one had been\n",
    "held in 1946?\n",
    "\n",
    "To answer this question we need to generalize, but before we formalize\n",
    "the concept of generalization let's introduce some formal representation\n",
    "of what it means to generalize in machine learning.\n",
    "\n",
    "Our first objective will be to perform a Gaussian process fit to the\n",
    "data, we'll do this using the [GPy\n",
    "software](https://github.com/SheffieldML/GPy)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import GPy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "m_full = GPy.models.GPRegression(x,yhat)\n",
    "_ = m_full.optimize() # Optimize parameters of covariance function"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The first command sets up the model, then `m_full.optimize()` optimizes\n",
    "the parameters of the covariance function and the noise level of the\n",
    "model. Once the fit is complete, we'll try creating some test points,\n",
    "and computing the output of the GP model in terms of the mean and\n",
    "standard deviation of the posterior functions between 1870 and 2030. We\n",
    "plot the mean function and the standard deviation at 200 locations. We\n",
    "can obtain the predictions using `y_mean, y_var = m_full.predict(xt)`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "xt = np.linspace(1870,2030,200)[:,np.newaxis]\n",
    "yt_mean, yt_var = m_full.predict(xt)\n",
    "yt_sd=np.sqrt(yt_var)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we plot the results using the helper function in `teaching_plots`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import teaching_plots as plot"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig, ax = plt.subplots(figsize=plot.big_wide_figsize)\n",
    "plot.model_output(m_full, scale=scale, offset=offset, ax=ax, xlabel='year', ylabel='pace min/km', fontsize=20, portion=0.2)\n",
    "ax.set_xlim(xlim)\n",
    "ax.set_ylim(ylim)\n",
    "mlai.write_figure(figure=fig,\n",
    "                  filename='../slides/diagrams/gp/olympic-marathon-gp.svg', \n",
    "                  transparent=True, frameon=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/gp/olympic-marathon-gp.svg\" class=\"\" align=\"\" style=\"vertical-align:middle;\">\n",
    "\n",
    "Figure: <i>Gaussian process fit to the Olympic Marathon data. The error\n",
    "bars are too large, perhaps due to the outlier from 1904.</i>\n",
    "\n",
    "## Fit Quality\n",
    "\n",
    "In the fit we see that the error bars (coming mainly from the noise\n",
    "variance) are quite large. This is likely due to the outlier point in\n",
    "1904, ignoring that point we can see that a tighter fit is obtained. To\n",
    "see this making a version of the model, `m_clean`, where that point is\n",
    "removed."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x_clean=np.vstack((x[0:2, :], x[3:, :]))\n",
    "y_clean=np.vstack((y[0:2, :], y[3:, :]))\n",
    "\n",
    "m_clean = GPy.models.GPRegression(x_clean,y_clean)\n",
    "_ = m_clean.optimize()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Deep GP Fit \\[<a href=\"https://github.com/lawrennd/snippets/edit/main/_deepgp/includes/deep-olympic.md\" target=\"_blank\" >edit</a>\\]\n",
    "\n",
    "Let's see if a deep Gaussian process can help here. We will construct a\n",
    "deep Gaussian process with one hidden layer (i.e. one Gaussian process\n",
    "feeding into another).\n",
    "\n",
    "Build a Deep GP with an additional hidden layer (one dimensional) to fit\n",
    "the model."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import GPy\n",
    "import deepgp"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "hidden = 1\n",
    "m = deepgp.DeepGP([y.shape[1],hidden,x.shape[1]],Y=yhat, X=x, inits=['PCA','PCA'], \n",
    "                  kernels=[GPy.kern.RBF(hidden,ARD=True),\n",
    "                           GPy.kern.RBF(x.shape[1],ARD=True)], # the kernels for each layer\n",
    "                  num_inducing=50, back_constraint=False)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import deepgp"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Call the initalization\n",
    "m.initialize()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now optimize the model."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "for layer in m.layers:\n",
    "    layer.likelihood.variance.constrain_positive(warning=False)\n",
    "m.optimize(messages=True,max_iters=10000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "m.staged_optimize(messages=(True,True,True))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Olympic Marathon Data Deep GP\n",
    "\n",
    "<img src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/deepgp/olympic-marathon-deep-gp.svg\" class=\"\" align=\"100%\" style=\"vertical-align:middle;\">\n",
    "\n",
    "Figure: <i>Deep GP fit to the Olympic marathon data. Error bars now\n",
    "change as the prediction evolves.</i>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig, ax = plt.subplots(figsize=plot.big_wide_figsize)\n",
    "plot.model_sample(m, scale=scale, offset=offset, samps=10, ax=ax, \n",
    "                  xlabel='year', ylabel='pace min/km', portion = 0.225)\n",
    "ax.set_xlim(xlim)\n",
    "ax.set_ylim(ylim)\n",
    "mlai.write_figure(figure=fig, filename='../slides/diagrams/deepgp/olympic-marathon-deep-gp-samples.svg', \n",
    "                  transparent=True, frameon=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Olympic Marathon Data Deep GP\n",
    "\n",
    "<img src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/deepgp/olympic-marathon-deep-gp-samples.svg\" class=\"\" align=\"\" style=\"vertical-align:middle;\">\n",
    "\n",
    "Figure: <i>Point samples run through the deep Gaussian process show the\n",
    "distribution of output locations.</i>\n",
    "\n",
    "## Fitted GP for each layer\n",
    "\n",
    "Now we explore the GPs the model has used to fit each layer. First of\n",
    "all, we look at the hidden layer."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "m.visualize(scale=scale, offset=offset, xlabel='year',\n",
    "            ylabel='pace min/km',xlim=xlim, ylim=ylim,\n",
    "            dataset='olympic-marathon',\n",
    "            diagrams='../slides/diagrams/deepgp')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import pods"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "pods.notebook.display_plots('olympic-marathon-deep-gp-layer-{sample:0>1}.svg', \n",
    "                            '../slides/diagrams/deepgp', sample=(0,1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/deepgp/olympic-marathon-deep-gp-layer-0.svg\" class=\"\" align=\"\" style=\"vertical-align:middle;\">\n",
    "\n",
    "Figure: <i>The mapping from input to the latent layer is broadly, with\n",
    "some flattening as time goes on. Variance is high across the input\n",
    "range.</i>\n",
    "\n",
    "<img src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/deepgp/olympic-marathon-deep-gp-layer-1.svg\" class=\"\" align=\"\" style=\"vertical-align:middle;\">\n",
    "\n",
    "Figure: <i>The mapping from the latent layer to the output layer.</i>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig, ax = plt.subplots(figsize=plot.big_wide_figsize)\n",
    "m.visualize_pinball(ax=ax, scale=scale, offset=offset, points=30, portion=0.1,\n",
    "                    xlabel='year', ylabel='pace km/min', vertical=True)\n",
    "mlai.write_figure(figure=fig, filename='../slides/diagrams/deepgp/olympic-marathon-deep-gp-pinball.svg', \n",
    "                  transparent=True, frameon=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Olympic Marathon Pinball Plot\n",
    "\n",
    "<img src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/deepgp/olympic-marathon-deep-gp-pinball.svg\" class=\"\" align=\"80%\" style=\"vertical-align:middle;\">\n",
    "\n",
    "Figure: <i>A pinball plot shows the movement of the 'ball' as it passes\n",
    "through each layer of the Gaussian processes. Mean directions of\n",
    "movement are shown by lines. Shading gives one standard deviation of\n",
    "movement position. At each layer, the uncertainty is reset. The overal\n",
    "uncertainty is the cumulative uncertainty from all the layers. There is\n",
    "some grouping of later points towards the right in the first layer,\n",
    "which also injects a large amount of uncertainty. Due to flattening of\n",
    "the curve in the second layer towards the right the uncertainty is\n",
    "reduced in final output.</i>\n",
    "\n",
    "The pinball plot shows the flow of any input ball through the deep\n",
    "Gaussian process. In a pinball plot a series of vertical parallel lines\n",
    "would indicate a purely linear function. For the olypmic marathon data\n",
    "we can see the first layer begins to shift from input towards the right.\n",
    "Note it also does so with some uncertainty (indicated by the shaded\n",
    "backgrounds). The second layer has less uncertainty, but bunches the\n",
    "inputs more strongly to the right. This input layer of uncertainty,\n",
    "followed by a layer that pushes inputs to the right is what gives the\n",
    "heteroschedastic noise.\n",
    "\n",
    "## Gene Expression Example \\[<a href=\"https://github.com/lawrennd/snippets/edit/main/_gp/includes/della-gatta-gene-gp.md\" target=\"_blank\" >edit</a>\\]\n",
    "\n",
    "We now consider an example in gene expression. Gene expression is the\n",
    "measurement of mRNA levels expressed in cells. These mRNA levels show\n",
    "which genes are 'switched on' and producing data. In the example we will\n",
    "use a Gaussian process to determine whether a given gene is active, or\n",
    "we are merely observing a noise response.\n",
    "\n",
    "## Della Gatta Gene Data \\[<a href=\"https://github.com/lawrennd/snippets/edit/main/_ml/includes/della-gatta-gene-data.md\" target=\"_blank\" >edit</a>\\]\n",
    "\n",
    "-   Given given expression levels in the form of a time series from\n",
    "    @DellaGatta:direct08."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import pods"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "data = pods.datasets.della_gatta_TRP63_gene_expression(data_set='della_gatta',gene_number=937)\n",
    "\n",
    "x = data['X']\n",
    "y = data['Y']\n",
    "\n",
    "offset = y.mean()\n",
    "scale = np.sqrt(y.var())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "import teaching_plots as plot\n",
    "import mlai"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "xlim = (-20,260)\n",
    "ylim = (5, 7.5)\n",
    "yhat = (y-offset)/scale\n",
    "\n",
    "fig, ax = plt.subplots(figsize=plot.big_wide_figsize)\n",
    "_ = ax.plot(x, y, 'r.',markersize=10)\n",
    "ax.set_xlabel('time/min', fontsize=20)\n",
    "ax.set_ylabel('expression', fontsize=20)\n",
    "ax.set_xlim(xlim)\n",
    "ax.set_ylim(ylim)\n",
    "\n",
    "mlai.write_figure(figure=fig, \n",
    "                  filename='../slides/diagrams/datasets/della-gatta-gene.svg', \n",
    "                  transparent=True, \n",
    "                  frameon=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/datasets/della-gatta-gene.svg\" class=\"\" align=\"80%\" style=\"vertical-align:middle;\">\n",
    "\n",
    "Figure: <i>Gene expression levels over time for a gene from data\n",
    "provided by @DellaGatta:direct08. We would like to understand whethere\n",
    "there is signal in the data, or we are only observing noise.</i>\n",
    "\n",
    "-   Want to detect if a gene is expressed or not, fit a GP to each gene\n",
    "    @Kalaitzis:simple11.\n",
    "\n",
    "<img class=\"\" src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/health/1471-2105-12-180_1.png\" style=\"width:80%\">\n",
    "\n",
    "Figure: <i>The example is taken from the paper \"A Simple Approach to\n",
    "Ranking Differentially Expressed Gene Expression Time Courses through\n",
    "Gaussian Process Regression.\" @Kalaitzis:simple11.</i>\n",
    "\n",
    "<center>\n",
    "<http://www.biomedcentral.com/1471-2105/12/180>\n",
    "</center>\n",
    "Our first objective will be to perform a Gaussian process fit to the\n",
    "data, we'll do this using the [GPy\n",
    "software](https://github.com/SheffieldML/GPy)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import GPy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "m_full = GPy.models.GPRegression(x,yhat)\n",
    "m_full.kern.lengthscale=50\n",
    "_ = m_full.optimize() # Optimize parameters of covariance function"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Initialize the length scale parameter (which here actually represents a\n",
    "*time scale* of the covariance function) to a reasonable value. Default\n",
    "would be 1, but here we set it to 50 minutes, given points are arriving\n",
    "across zero to 250 minutes."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "xt = np.linspace(-20,260,200)[:,np.newaxis]\n",
    "yt_mean, yt_var = m_full.predict(xt)\n",
    "yt_sd=np.sqrt(yt_var)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we plot the results using the helper function in `teaching_plots`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import teaching_plots as plot"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig, ax = plt.subplots(figsize=plot.big_wide_figsize)\n",
    "plot.model_output(m_full, scale=scale, offset=offset, ax=ax, xlabel='time/min', ylabel='expression', fontsize=20, portion=0.2)\n",
    "ax.set_xlim(xlim)\n",
    "ax.set_ylim(ylim)\n",
    "ax.set_title('log likelihood: {ll:.3}'.format(ll=m_full.log_likelihood()), fontsize=20)\n",
    "mlai.write_figure(figure=fig,\n",
    "                  filename='../slides/diagrams/gp/della-gatta-gene-gp.svg', \n",
    "                  transparent=True, frameon=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/gp/della-gatta-gene-gp.svg\" class=\"\" align=\"80%\" style=\"vertical-align:middle;\">\n",
    "\n",
    "Figure: <i>Result of the fit of the Gaussian process model with the time\n",
    "scale parameter initialized to 50 minutes.</i>\n",
    "\n",
    "Now we try a model initialized with a longer length scale."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "m_full2 = GPy.models.GPRegression(x,yhat)\n",
    "m_full2.kern.lengthscale=2000\n",
    "_ = m_full2.optimize() # Optimize parameters of covariance function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import teaching_plots as plot"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig, ax = plt.subplots(figsize=plot.big_wide_figsize)\n",
    "plot.model_output(m_full2, scale=scale, offset=offset, ax=ax, xlabel='time/min', ylabel='expression', fontsize=20, portion=0.2)\n",
    "ax.set_xlim(xlim)\n",
    "ax.set_ylim(ylim)\n",
    "ax.set_title('log likelihood: {ll:.3}'.format(ll=m_full2.log_likelihood()), fontsize=20)\n",
    "mlai.write_figure(figure=fig,\n",
    "                  filename='../slides/diagrams/gp/della-gatta-gene-gp2.svg', \n",
    "                  transparent=True, frameon=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/gp/della-gatta-gene-gp2.svg\" class=\"\" align=\"80%\" style=\"vertical-align:middle;\">\n",
    "\n",
    "Figure: <i>Result of the fit of the Gaussian process model with the time\n",
    "scale parameter initialized to 2000 minutes.</i>\n",
    "\n",
    "Now we try a model initialized with a lower noise."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "m_full3 = GPy.models.GPRegression(x,yhat)\n",
    "m_full3.kern.lengthscale=20\n",
    "m_full3.likelihood.variance=0.001\n",
    "_ = m_full3.optimize() # Optimize parameters of covariance function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import teaching_plots as plot"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig, ax = plt.subplots(figsize=plot.big_wide_figsize)\n",
    "plot.model_output(m_full3, scale=scale, offset=offset, ax=ax, xlabel='time/min', ylabel='expression', fontsize=20, portion=0.2)\n",
    "ax.set_xlim(xlim)\n",
    "ax.set_ylim(ylim)\n",
    "ax.set_title('log likelihood: {ll:.3}'.format(ll=m_full3.log_likelihood()), fontsize=20)\n",
    "mlai.write_figure(figure=fig,\n",
    "                  filename='../slides/diagrams/gp/della-gatta-gene-gp3.svg', \n",
    "                  transparent=True, frameon=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/gp/della-gatta-gene-gp3.svg\" class=\"\" align=\"80%\" style=\"vertical-align:middle;\">\n",
    "\n",
    "Figure: <i>Result of the fit of the Gaussian process model with the\n",
    "noise initialized low (standard deviation 0.1) and the time scale\n",
    "parameter initialized to 20 minutes.</i>\n",
    "\n",
    "<img src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/gp/multiple-optima000.svg\" class=\"\" align=\"50%\" style=\"vertical-align:middle;\">\n",
    "\n",
    "Figure: <i></i>\n",
    "\n",
    "<!--\n",
    "\n",
    "<img src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/gp/multiple-optima001.svg\" class=\"\" align=\"\" style=\"vertical-align:middle;\">-->"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "layers = [y.shape[1], 1,x.shape[1]]\n",
    "inits = ['PCA']*(len(layers)-1)\n",
    "kernels = []\n",
    "for i in layers[1:]:\n",
    "    kernels += [GPy.kern.RBF(i)]\n",
    "m = deepgp.DeepGP(layers,Y=yhat, X=x, \n",
    "                  inits=inits, \n",
    "                  kernels=kernels, # the kernels for each layer\n",
    "                  num_inducing=20, back_constraint=False)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "m.initialize()\n",
    "m.staged_optimize()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig, ax=plt.subplots(figsize=plot.big_wide_figsize)\n",
    "plot.model_output(m, scale=scale, offset=offset, ax=ax, fontsize=20, portion=0.5)\n",
    "ax.set_ylim(ylim)\n",
    "ax.set_xlim(xlim)\n",
    "mlai.write_figure(filename='../slides/diagrams/deepgp/della-gatta-gene-deep-gp.svg', \n",
    "            transparent=True, frameon=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Della Gatta Gene Data Deep GP\n",
    "\n",
    "<img src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/deepgp/della-gatta-gene-deep-gp.svg\" class=\"\" align=\"80%\" style=\"vertical-align:middle;\">\n",
    "\n",
    "Figure: <i>Deep Gaussian process fit to the Della Gatta gene expression\n",
    "data.</i>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig, ax=plt.subplots(figsize=plot.big_wide_figsize)\n",
    "plot.model_sample(m, scale=scale, offset=offset, samps=10, ax=ax, portion = 0.5)\n",
    "ax.set_ylim(ylim)\n",
    "ax.set_xlim(xlim)\n",
    "mlai.write_figure(figure=fig, filename='../slides/diagrams/deepgp/della-gatta-gene-deep-gp-samples.svg', \n",
    "                  transparent=True, frameon=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Della Gatta Gene Data Deep GP\n",
    "\n",
    "<img src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/deepgp/della-gatta-gene-deep-gp-samples.svg\" class=\"\" align=\"\" style=\"vertical-align:middle;\">\n",
    "\n",
    "Figure: <i>Deep Gaussian process samples fitted to the Della Gatta gene\n",
    "expression data.</i>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "m.visualize(offset=offset, scale=scale, xlim=xlim, ylim=ylim,\n",
    "            dataset='della-gatta-gene',\n",
    "            diagrams='../slides/diagrams/deepgp')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Della Gatta Gene Data Latent 1\n",
    "\n",
    "<img src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/deepgp/della-gatta-gene-deep-gp-layer-0.svg\" class=\"\" align=\"50%\" style=\"vertical-align:middle;\">\n",
    "\n",
    "Figure: <i>Gaussian process mapping from input to latent layer for the\n",
    "della Gatta gene expression data.</i>\n",
    "\n",
    "## Della Gatta Gene Data Latent 2\n",
    "\n",
    "<img src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/deepgp/della-gatta-gene-deep-gp-layer-1.svg\" class=\"\" align=\"50%\" style=\"vertical-align:middle;\">\n",
    "\n",
    "Figure: <i>Gaussian process mapping from latent to output layer for the\n",
    "della Gatta gene expression data.</i>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig, ax=plt.subplots(figsize=plot.big_wide_figsize)\n",
    "m.visualize_pinball(offset=offset, ax=ax, scale=scale, xlim=xlim, ylim=ylim, portion=0.1, points=50)\n",
    "mlai.write_figure(figure=fig, filename='../slides/diagrams/deepgp/della-gatta-gene-deep-gp-pinball.svg', \n",
    "                  transparent=True, frameon=True, ax=ax)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## TP53 Gene Pinball Plot\n",
    "\n",
    "<img src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/deepgp/della-gatta-gene-deep-gp-pinball.svg\" class=\"\" align=\"60%\" style=\"vertical-align:middle;\">\n",
    "\n",
    "Figure: <i>A pinball plot shows the movement of the 'ball' as it passes\n",
    "through each layer of the Gaussian processes. Mean directions of\n",
    "movement are shown by lines. Shading gives one standard deviation of\n",
    "movement position. At each layer, the uncertainty is reset. The overal\n",
    "uncertainty is the cumulative uncertainty from all the layers. Pinball\n",
    "plot of the della Gatta gene expression data.</i>\n",
    "\n",
    "## Step Function \\[<a href=\"https://github.com/lawrennd/snippets/edit/main/_ml/includes/step-function-data.md\" target=\"_blank\" >edit</a>\\]\n",
    "\n",
    "Next we consider a simple step function data set."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "num_low=25\n",
    "num_high=25\n",
    "gap = -.1\n",
    "noise=0.0001\n",
    "x = np.vstack((np.linspace(-1, -gap/2.0, num_low)[:, np.newaxis],\n",
    "              np.linspace(gap/2.0, 1, num_high)[:, np.newaxis]))\n",
    "y = np.vstack((np.zeros((num_low, 1)), np.ones((num_high,1))))\n",
    "scale = np.sqrt(y.var())\n",
    "offset = y.mean()\n",
    "yhat = (y-offset)/scale"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig, ax = plt.subplots(figsize=plot.big_wide_figsize)\n",
    "_ = ax.plot(x, y, 'r.',markersize=10)\n",
    "_ = ax.set_xlabel('$x$', fontsize=20)\n",
    "_ = ax.set_ylabel('$y$', fontsize=20)\n",
    "xlim = (-2, 2)\n",
    "ylim = (-0.6, 1.6)\n",
    "ax.set_ylim(ylim)\n",
    "ax.set_xlim(xlim)\n",
    "mlai.write_figure(figure=fig, filename='../../slides/diagrams/datasets/step-function.svg', \n",
    "            transparent=True, frameon=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Step Function Data\n",
    "\n",
    "<img src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/datasets/step-function.svg\" class=\"\" align=\"80%\" style=\"vertical-align:middle;\">\n",
    "\n",
    "Figure: <i>Simulation study of step function data artificially\n",
    "generated. Here there is a small overlap between the two lines.</i>\n",
    "\n",
    "## Step Function Data GP\n",
    "\n",
    "We can fit a Gaussian process to the step function data using `GPy` as\n",
    "follows."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "m_full = GPy.models.GPRegression(x,yhat)\n",
    "_ = m_full.optimize() # Optimize parameters of covariance function"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Where `GPy.models.GPRegression()` gives us a standard GP regression\n",
    "model with exponentiated quadratic covariance function.\n",
    "\n",
    "The model is optimized using `m_full.optimize()` which calls an L-BGFS\n",
    "gradient based solver in python."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig, ax=plt.subplots(figsize=plot.big_wide_figsize)\n",
    "plot.model_output(m_full, scale=scale, offset=offset, ax=ax, fontsize=20, portion=0.5)\n",
    "ax.set_ylim(ylim)\n",
    "ax.set_xlim(xlim)\n",
    "\n",
    "mlai.write_figure(figure=fig,filename='../slides/diagrams/gp/step-function-gp.svg', \n",
    "            transparent=True, frameon=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/gp/step-function-gp.svg\" class=\"\" align=\"80%\" style=\"vertical-align:middle;\">\n",
    "\n",
    "Figure: <i>Gaussian process fit to the step function data. Note the\n",
    "large error bars and the over-smoothing of the discontinuity. Error bars\n",
    "are shown at two standard deviations.</i>\n",
    "\n",
    "The resulting fit to the step function data shows some challenges. In\n",
    "particular, the over smoothing at the discontinuity. If we know how many\n",
    "discontinuities there are, we can parameterize them in the step\n",
    "function. But by doing this, we form a semi-parametric model. The\n",
    "parameters indicate how many discontinuities are, and where they are.\n",
    "They can be optimized as part of the model fit. But if new, unforeseen,\n",
    "discontinuities arise when the model is being deployed in practice,\n",
    "these won't be accounted for in the predictions.\n",
    "\n",
    "## Step Function Data Deep GP\n",
    "\n",
    "First we initialize a deep Gaussian process with three latent layers\n",
    "(four layers total). Within each layer we create a GP with an\n",
    "exponentiated quadratic covariance (`GPy.kern.RBF`).\n",
    "\n",
    "At each layer we use 20 inducing points for the variational\n",
    "approximation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "layers = [y.shape[1], 1, 1, 1,x.shape[1]]\n",
    "inits = ['PCA']*(len(layers)-1)\n",
    "kernels = []\n",
    "for i in layers[1:]:\n",
    "    kernels += [GPy.kern.RBF(i)]\n",
    "    \n",
    "m = deepgp.DeepGP(layers,Y=yhat, X=x, \n",
    "                  inits=inits, \n",
    "                  kernels=kernels, # the kernels for each layer\n",
    "                  num_inducing=20, back_constraint=False)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Once the model is constructed we initialize the parameters, and perform\n",
    "the staged optimization which starts by optimizing variational\n",
    "parameters with a low noise and proceeds to optimize the whole model."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "m.initialize()\n",
    "m.staged_optimize()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We plot the output of the deep Gaussian process fitted to the stpe data\n",
    "as follows."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig, ax=plt.subplots(figsize=plot.big_wide_figsize)\n",
    "plot.model_output(m, scale=scale, offset=offset, ax=ax, fontsize=20, portion=0.5)\n",
    "ax.set_ylim(ylim)\n",
    "ax.set_xlim(xlim)\n",
    "mlai.write_figure(filename='../slides/diagrams/deepgp/step-function-deep-gp.svg', \n",
    "            transparent=True, frameon=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The deep Gaussian process does a much better job of fitting the data. It\n",
    "handles the discontinuity easily, and error bars drop to smaller values\n",
    "in the regions of data.\n",
    "\n",
    "<img src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/deepgp/step-function-deep-gp.svg\" class=\"\" align=\"80%\" style=\"vertical-align:middle;\">\n",
    "\n",
    "Figure: <i>Deep Gaussian process fit to the step function data.</i>\n",
    "\n",
    "## Step Function Data Deep GP\n",
    "\n",
    "The samples of the model can be plotted with the helper function from\n",
    "`teaching_plots.py`, `model_sample`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig, ax=plt.subplots(figsize=plot.big_wide_figsize)\n",
    "\n",
    "plot.model_sample(m, scale=scale, offset=offset, samps=10, ax=ax, portion = 0.5)\n",
    "ax.set_ylim(ylim)\n",
    "ax.set_xlim(xlim)\n",
    "mlai.write_figure(figure=fig, filename='../slides/diagrams/deepgp/step-function-deep-gp-samples.svg', \n",
    "                  transparent=True, frameon=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The samples from the model show that the error bars, which are\n",
    "informative for Gaussian outputs, are less informative for this model.\n",
    "They make clear that the data points lie, in output mainly at 0 or 1, or\n",
    "occasionally in between.\n",
    "\n",
    "<img src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/deepgp/step-function-deep-gp-samples.svg\" class=\"\" align=\"80%\" style=\"vertical-align:middle;\">\n",
    "\n",
    "Figure: <i>Samples from the deep Gaussian process model for the step\n",
    "function fit.</i>\n",
    "\n",
    "The visualize code allows us to inspect the intermediate layers in the\n",
    "deep GP model to understand how it has reconstructed the step function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "m.visualize(offset=offset, scale=scale, xlim=xlim, ylim=ylim,\n",
    "            dataset='step-function',\n",
    "            diagrams='../slides/diagrams/deepgp')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/deepgp/step-function-deep-gp-layer-0.svg\" class=\"\" align=\"60%\" style=\"vertical-align:middle;\">\n",
    "<img src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/deepgp/step-function-deep-gp-layer-1.svg\" class=\"\" align=\"60%\" style=\"vertical-align:middle;\">\n",
    "<img src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/deepgp/step-function-deep-gp-layer-2.svg\" class=\"\" align=\"60%\" style=\"vertical-align:middle;\">\n",
    "<img src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/deepgp/step-function-deep-gp-layer-3.svg\" class=\"\" align=\"60%\" style=\"vertical-align:middle;\">\n",
    "\n",
    "Figure: <i>From top to bottom, the Gaussian process mapping function\n",
    "that makes up each layer of the resulting deep Gaussian process.</i>\n",
    "\n",
    "A pinball plot can be created for the resulting model to understand how\n",
    "the input is being translated to the output across the different layers."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig, ax=plt.subplots(figsize=plot.big_wide_figsize)\n",
    "m.visualize_pinball(offset=offset, ax=ax, scale=scale, xlim=xlim, ylim=ylim, portion=0.1, points=50)\n",
    "mlai.write_figure(figure=fig, filename='../slides/diagrams/deepgp/step-function-deep-gp-pinball.svg', \n",
    "                  transparent=True, frameon=True, ax=ax)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/deepgp/step-function-deep-gp-pinball.svg\" class=\"\" align=\"60%\" style=\"vertical-align:middle;\">\n",
    "\n",
    "Figure: <i>Pinball plot of the deep GP fitted to the step function data.\n",
    "Each layer of the model pushes the 'ball' towards the left or right,\n",
    "saturating at 1 and 0. This causes the final density to be be peaked at\n",
    "0 and 1. Transitions occur driven by the uncertainty of the mapping in\n",
    "each layer.</i>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import pods"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "data = pods.datasets.mcycle()\n",
    "x = data['X']\n",
    "y = data['Y']\n",
    "scale=np.sqrt(y.var())\n",
    "offset=y.mean()\n",
    "yhat = (y - offset)/scale"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig, ax = plt.subplots(figsize=plot.big_wide_figsize)\n",
    "_ = ax.plot(x, y, 'r.',markersize=10)\n",
    "_ = ax.set_xlabel('time', fontsize=20)\n",
    "_ = ax.set_ylabel('acceleration', fontsize=20)\n",
    "xlim = (-20, 80)\n",
    "ylim = (-175, 125)\n",
    "ax.set_xlim(xlim)\n",
    "ax.set_ylim(ylim)\n",
    "mlai.write_figure(filename='../slides/diagrams/datasets/motorcycle-helmet.svg', \n",
    "            transparent=True, frameon=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Motorcycle Helmet Data \\[<a href=\"https://github.com/lawrennd/snippets/edit/main/_ml/includes/motorcycle-helmet-data.md\" target=\"_blank\" >edit</a>\\]\n",
    "\n",
    "<img src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/datasets/motorcycle-helmet.svg\" class=\"\" align=\"80%\" style=\"vertical-align:middle;\">\n",
    "\n",
    "Figure: <i>Motorcycle helmet data. The data consists of acceleration\n",
    "readings on a motorcycle helmet undergoing a collision. The data\n",
    "exhibits heteroschedastic (time varying) noise levles and\n",
    "non-stationarity.</i>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "m_full = GPy.models.GPRegression(x,yhat)\n",
    "_ = m_full.optimize() # Optimize parameters of covariance function"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Motorcycle Helmet Data GP\n",
    "\n",
    "<img src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/gp/motorcycle-helmet-gp.svg\" class=\"\" align=\"80%\" style=\"vertical-align:middle;\">\n",
    "\n",
    "Figure: <i>Gaussian process fit to the motorcycle helmet accelerometer\n",
    "data.</i>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import deepgp"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "layers = [y.shape[1], 1, x.shape[1]]\n",
    "inits = ['PCA']*(len(layers)-1)\n",
    "kernels = []\n",
    "for i in layers[1:]:\n",
    "    kernels += [GPy.kern.RBF(i)]\n",
    "m = deepgp.DeepGP(layers,Y=yhat, X=x, \n",
    "                  inits=inits, \n",
    "                  kernels=kernels, # the kernels for each layer\n",
    "                  num_inducing=20, back_constraint=False)\n",
    "\n",
    "\n",
    "\n",
    "m.initialize()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "m.staged_optimize(iters=(1000,1000,10000), messages=(True, True, True))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import teaching_plots as plot\n",
    "import mlai"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig, ax=plt.subplots(figsize=plot.big_wide_figsize)\n",
    "plot.model_output(m, scale=scale, offset=offset, ax=ax, xlabel='time', ylabel='acceleration/$g$', fontsize=20, portion=0.5)\n",
    "ax.set_ylim(ylim)\n",
    "ax.set_xlim(xlim)\n",
    "mlai.write_figure(filename='../slides/diagrams/deepgp/motorcycle-helmet-deep-gp.svg', \n",
    "            transparent=True, frameon=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Motorcycle Helmet Data Deep GP\n",
    "\n",
    "<img src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/deepgp/motorcycle-helmet-deep-gp.svg\" class=\"\" align=\"80%\" style=\"vertical-align:middle;\">\n",
    "\n",
    "Figure: <i>Deep Gaussian process fit to the motorcycle helmet\n",
    "accelerometer data.</i>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import teaching_plots as plot\n",
    "import mlai"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig, ax=plt.subplots(figsize=plot.big_wide_figsize)\n",
    "plot.model_sample(m, scale=scale, offset=offset, samps=10, ax=ax, xlabel='time', ylabel='acceleration/$g$', portion = 0.5)\n",
    "ax.set_ylim(ylim)\n",
    "ax.set_xlim(xlim)\n",
    "\n",
    "mlai.write_figure(figure=fig, filename='../slides/diagrams/deepgp/motorcycle-helmet-deep-gp-samples.svg', \n",
    "                  transparent=True, frameon=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Motorcycle Helmet Data Deep GP\n",
    "\n",
    "<img src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/deepgp/motorcycle-helmet-deep-gp-samples.svg\" class=\"\" align=\"80%\" style=\"vertical-align:middle;\">\n",
    "\n",
    "Figure: <i>Samples from the deep Gaussian process as fitted to the\n",
    "motorcycle helmet accelerometer data.</i>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "m.visualize(xlim=xlim, ylim=ylim, scale=scale,offset=offset, \n",
    "            xlabel=\"time\", ylabel=\"acceleration/$g$\", portion=0.5,\n",
    "            dataset='motorcycle-helmet',\n",
    "            diagrams='../slides/diagrams/deepgp')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Motorcycle Helmet Data Latent 1\n",
    "\n",
    "<img src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/deepgp/motorcycle-helmet-deep-gp-layer-0.svg\" class=\"\" align=\"60%\" style=\"vertical-align:middle;\">\n",
    "\n",
    "Figure: <i>Mappings from the input to the latent layer for the\n",
    "motorcycle helmet accelerometer data.</i>\n",
    "\n",
    "## Motorcycle Helmet Data Latent 2\n",
    "\n",
    "<img src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/deepgp/motorcycle-helmet-deep-gp-layer-1.svg\" class=\"\" align=\"60%\" style=\"vertical-align:middle;\">\n",
    "\n",
    "Figure: <i>Mappings from the latent layer to the output layer for the\n",
    "motorcycle helmet accelerometer data.</i>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig, ax=plt.subplots(figsize=plot.big_wide_figsize)\n",
    "m.visualize_pinball(ax=ax, xlabel='time', ylabel='acceleration/g', \n",
    "                    points=50, scale=scale, offset=offset, portion=0.1)\n",
    "mlai.write_figure(figure=fig, filename='../slides/diagrams/deepgp/motorcycle-helmet-deep-gp-pinball.svg', \n",
    "                  transparent=True, frameon=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Motorcycle Helmet Pinball Plot\n",
    "\n",
    "<img src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/deepgp/motorcycle-helmet-deep-gp-pinball.svg\" class=\"\" align=\"60%\" style=\"vertical-align:middle;\">\n",
    "\n",
    "Figure: <i>Pinball plot for the mapping from input to output layer for\n",
    "the motorcycle helmet accelerometer data.</i>\n",
    "\n",
    "## Motion Capture \\[<a href=\"https://github.com/lawrennd/snippets/edit/main/_deepgp/includes/deep-high-five.md\" target=\"_blank\" >edit</a>\\]\n",
    "\n",
    "-   'High five' data.\n",
    "-   Model learns structure between two interacting subjects.\n",
    "\n",
    "## Shared LVM\n",
    "\n",
    "<img src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/shared.svg\" class=\"\" align=\"60%\" style=\"vertical-align:middle;\">\n",
    "\n",
    "Figure: <i>Shared latent variable model structure. Here two related data\n",
    "sets are brought together with a set of latent variables that are\n",
    "partially shared and partially specific to one of the data sets.</i>\n",
    "\n",
    "<img class=\"\" src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/deep-gp-high-five2.png\" style=\"width:80%\">\n",
    "\n",
    "Figure: <i>Latent spaces of the 'high five' data. The structure of the\n",
    "model is automatically learnt. One of the latent spaces is coordinating\n",
    "how the two figures walk together, the other latent spaces contain\n",
    "latent variables that are specific to each of the figures\n",
    "separately.</i>\n",
    "\n",
    "## Fitting a GP to the USPS Digits Data \\[<a href=\"https://github.com/lawrennd/snippets/edit/main/_deepgp/includes/deep-usps-digits.md\" target=\"_blank\" >edit</a>\\]\n",
    "\n",
    "<small>[Thanks to: Zhenwen Dai and Neil D.\n",
    "Lawrence]{style=\"text-align:right\"}</small>\n",
    "\n",
    "We now look at the deep Gaussian processes' capacity to perform\n",
    "unsupervised learning.\n",
    "\n",
    "We will look at a sub-sample of the MNIST digit data set.\n",
    "\n",
    "First load in the MNIST data set from scikit learn. This can take a\n",
    "little while because it's large to download."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sklearn.datasets import fetch_mldata"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "mnist = fetch_mldata('MNIST original')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Sub-sample the dataset to make the training faster."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "np.random.seed(0)\n",
    "digits = [0,1,2,3,4]\n",
    "N_per_digit = 100\n",
    "Y = []\n",
    "labels = []\n",
    "for d in digits:\n",
    "    imgs = mnist['data'][mnist['target']==d]\n",
    "    Y.append(imgs[np.random.permutation(imgs.shape[0])][:N_per_digit])\n",
    "    labels.append(np.ones(N_per_digit)*d)\n",
    "Y = np.vstack(Y).astype(np.float64)\n",
    "labels = np.hstack(labels)\n",
    "Y /= 255."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Fit a Deep GP\n",
    "\n",
    "We're going to fit a Deep Gaussian process model to the MNIST data with\n",
    "two hidden layers. Each of the two Gaussian processes (one from the\n",
    "first hidden layer to the second, one from the second hidden layer to\n",
    "the data) has an exponentiated quadratic covariance."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import deepgp\n",
    "import GPy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "num_latent = 2\n",
    "num_hidden_2 = 5\n",
    "m = deepgp.DeepGP([Y.shape[1],num_hidden_2,num_latent],\n",
    "                  Y,\n",
    "                  kernels=[GPy.kern.RBF(num_hidden_2,ARD=True), \n",
    "                           GPy.kern.RBF(num_latent,ARD=False)], \n",
    "                  num_inducing=50, back_constraint=False, \n",
    "                  encoder_dims=[[200],[200]])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Initialization\n",
    "\n",
    "Just like deep neural networks, there are some tricks to intitializing\n",
    "these models. The tricks we use here include some early training of the\n",
    "model with model parameters constrained. This gives the variational\n",
    "inducing parameters some scope to tighten the bound for the case where\n",
    "the noise variance is small and the variances of the Gaussian processes\n",
    "are around 1."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "m.obslayer.likelihood.variance[:] = Y.var()*0.01\n",
    "for layer in m.layers:\n",
    "    layer.kern.variance.fix(warning=False)\n",
    "    layer.likelihood.variance.fix(warning=False)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We now we optimize for a hundred iterations with the constrained model."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "m.optimize(messages=False,max_iters=100)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we remove the fixed constraint on the kernel variance parameters,\n",
    "but keep the noise output constrained, and run for a further 100\n",
    "iterations."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "for layer in m.layers:\n",
    "    layer.kern.variance.constrain_positive(warning=False)\n",
    "m.optimize(messages=False,max_iters=100)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally we unconstrain the layer likelihoods and allow the full model to\n",
    "be trained for 1000 iterations."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "for layer in m.layers:\n",
    "    layer.likelihood.variance.constrain_positive(warning=False)\n",
    "m.optimize(messages=True,max_iters=10000)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Visualize the latent space of the top layer\n",
    "\n",
    "Now the model is trained, let's plot the mean of the posterior\n",
    "distributions in the top latent layer of the model."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "from matplotlib import rc\n",
    "import teaching_plots as plot\n",
    "import mlai"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "rc(\"font\", **{'family':'sans-serif','sans-serif':['Helvetica'],'size':20})\n",
    "fig, ax = plt.subplots(figsize=plot.big_figsize)\n",
    "for d in digits:\n",
    "    ax.plot(m.layer_1.X.mean[labels==d,0],m.layer_1.X.mean[labels==d,1],'.',label=str(d))\n",
    "_ = plt.legend()\n",
    "mlai.write_figure(figure=fig, filename=\"../slides/diagrams/deepgp/usps-digits-latent.svg\", transparent=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/usps-digits-latent.svg\" class=\"\" align=\"60%\" style=\"vertical-align:middle;\">\n",
    "\n",
    "Figure: <i>Latent space for the deep Gaussian process learned through\n",
    "unsupervised learning and fitted to a subset of the USPS digit data.</i>\n",
    "\n",
    "## Visualize the latent space of the intermediate layer\n",
    "\n",
    "We can also visualize dimensions of the intermediate layer. First the\n",
    "lengthscale of those dimensions is given by"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "m.obslayer.kern.lengthscale"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "import mlai"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig, ax = plt.subplots(figsize=plot.big_figsize)\n",
    "for i in range(5):\n",
    "    for j in range(i):\n",
    "        dims=[i, j]\n",
    "        ax.cla()\n",
    "        for d in digits:\n",
    "            ax.plot(m.obslayer.X.mean[labels==d,dims[0]],\n",
    "                 m.obslayer.X.mean[labels==d,dims[1]],\n",
    "                 '.', label=str(d))\n",
    "        plt.legend()\n",
    "        plt.xlabel('dimension ' + str(dims[0]))\n",
    "        plt.ylabel('dimension ' + str(dims[1]))\n",
    "        mlai.write_figure(figure=fig, filename=\"../slides/diagrams/deepgp/usps-digits-hidden-\" + str(dims[0]) + '-' + str(dims[1]) + '.svg', transparent=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/usps-digits-hidden-1-0.svg\" class=\"\" align=\"60%\" style=\"vertical-align:middle;\">\n",
    "\n",
    "Figure: <i>Visualisation of the intermediate layer, plot of dimension 1\n",
    "vs dimension 0.</i>\n",
    "\n",
    "<img src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/usps-digits-hidden-2-0.svg\" class=\"\" align=\"60%\" style=\"vertical-align:middle;\">\n",
    "\n",
    "Figure: <i>Visualisation of the intermediate layer, plot of dimension 1\n",
    "vs dimension 0.</i>\n",
    "\n",
    "<img src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/usps-digits-hidden-3-0.svg\" class=\"\" align=\"60%\" style=\"vertical-align:middle;\">\n",
    "\n",
    "Figure: <i>Visualisation of the intermediate layer, plot of dimension 1\n",
    "vs dimension 0.</i>\n",
    "\n",
    "<img src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/usps-digits-hidden-4-0.svg\" class=\"\" align=\"60%\" style=\"vertical-align:middle;\">\n",
    "\n",
    "Figure: <i>Visualisation of the intermediate layer, plot of dimension 1\n",
    "vs dimension 0.</i>\n",
    "\n",
    "## Generate From Model\n",
    "\n",
    "Now we can take a look at a sample from the model, by drawing a Gaussian\n",
    "random sample in the latent space and propagating it through the model."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "rows = 10\n",
    "cols = 20\n",
    "t=np.linspace(-1, 1, rows*cols)[:, None]\n",
    "kern = GPy.kern.RBF(1,lengthscale=0.05)\n",
    "cov = kern.K(t, t)\n",
    "x = np.random.multivariate_normal(np.zeros(rows*cols), cov, num_latent).T"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "import mlai"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "yt = m.predict(x)\n",
    "fig, axs = plt.subplots(rows,cols,figsize=(10,6))\n",
    "for i in range(rows):\n",
    "    for j in range(cols):\n",
    "        #v = np.random.normal(loc=yt[0][i*cols+j, :], scale=np.sqrt(yt[1][i*cols+j, :]))\n",
    "        v = yt[0][i*cols+j, :]\n",
    "        axs[i,j].imshow(v.reshape(28,28), \n",
    "                        cmap='gray', interpolation='none',\n",
    "                        aspect='equal')\n",
    "        axs[i,j].set_axis_off()\n",
    "mlai.write_figure(figure=fig, filename=\"../slides/diagrams/deepgp/digit-samples-deep-gp.svg\", transparent=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/digit-samples-deep-gp.svg\" class=\"\" align=\"80%\" style=\"vertical-align:middle;\">\n",
    "\n",
    "Figure: <i>These digits are produced by taking a tour of the two\n",
    "dimensional latent space (as described by a Gaussian process sample) and\n",
    "mapping the tour into the data space. We visualize the mean of the\n",
    "mapping in the images.</i>\n",
    "\n",
    "## Deep Health \\[<a href=\"https://github.com/lawrennd/snippets/edit/main/_health/includes/deep-health-model.md\" target=\"_blank\" >edit</a>\\]\n",
    "\n",
    "<img src=\"http://inverseprobability.com/talks/slides/../slides/diagrams/deep-health.svg\" class=\"\" align=\"70%\" style=\"vertical-align:middle;\">\n",
    "\n",
    "Figure: <i>The deep health model uses different layers of abstraction in\n",
    "the deep Gaussian process to represent information about diagnostics and\n",
    "treatment to model interelationships between a patients different data\n",
    "modalities.</i>\n",
    "\n",
    "From a machine learning perspective, we'd like to be able to interrelate\n",
    "all the different modalities that are informative about the state of the\n",
    "disease. For deep health, the notion is that the state of the disease is\n",
    "appearing at the more abstract levels, as we descend the model, we\n",
    "express relationships between the more abstract concept, that sits\n",
    "within the physician's mind, and the data we can measure.\n",
    "\n",
    "# References {#references .unnumbered}"
   ]
  }
 ],
 "metadata": {},
 "nbformat": 4,
 "nbformat_minor": 2
}