docs/BayesNets.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Bayes Nets\n",
    "*A Julia package for Bayesian Networks*\n",
    "\n",
    "A Bayesian Network (BN) represents a probability distribution over a set of variables, $P(x_1, x_2, \\ldots, x_n)$. Bayesian networks leverage variable relations in order to efficiently decompose the joint distribution into smaller conditional probability distributions.\n",
    "\n",
    "A BN is defined by a directed acyclic graph and a set of conditional probability distributions. Each node in the graph corresponds to a variable $x_i$ and is associated with a conditional probability distribution $P(x_i \\mid \\text{parents}(x_i))$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Installation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`Pkg.add(\"BayesNets\");`"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Default visualization of the network structure is provided by the GraphPlot package. However, we recommend using tex-formatted graphs provided by the TikzGraphs package. Installation requirements for TikzGraphs (e.g., PGF/Tikz and pdf2svg) are provided [here](http://nbviewer.ipython.org/github/sisl/TikzGraphs.jl/blob/master/doc/TikzGraphs.ipynb). Simply run `using TikzGraphs` in your Julia session to automatically switch to tex-formatted graphs (thanks to the Requires.jl package)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Usage"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "using Random\n",
    "Random.seed!(0) # seed the random number generator to 0, for a reproducible demonstration\n",
    "using BayesNets\n",
    "using TikzGraphs # required to plot tex-formatted graphs (recommended), otherwise GraphPlot.jl is used"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Representation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Bayesian Networks are represented with the `BayesNet` type. This type contains the directed acyclic graph (a LightTables.DiGraph) and a list of conditional probability distributions (a list of CPDs)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here we construct the BayesNet $a \\rightarrow b$, with Gaussians $a$ and $b$:\n",
    "\n",
    "$$\n",
    "a = \\mathcal{N}(0,1) \\qquad b = \\mathcal{N}(2a +3,1)\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
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      "text/plain": [
       "BayesNet{CPD}({2, 1} directed simple Int64 graph, CPD[StaticCPD{Normal{Float64}}(:a, Symbol[], Normal{Float64}(μ=1.0, σ=1.0)), LinearGaussianCPD(:b, [:a], [2.0], 3.0, 1.0)], Dict(:a => 1,:b => 2))"
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     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "bn = BayesNet()\n",
    "push!(bn, StaticCPD(:a, Normal(1.0)))\n",
    "push!(bn, LinearGaussianCPD(:b, [:a], [2.0], 3.0, 1.0))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Conditional Probability Distributions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Conditional Probablity Distributions, $P(x_i \\mid \\text{parents}(x_i))$, are defined in BayesNets.CPDs. Each CPD knows its own name, the names of its parents, and is associated with a distribution from Distributions.jl."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "| `CPDForm`                      | Description |\n",
    "| ------------------------------ | ----------- |\n",
    "| `StaticCPD`                    | Any `Distributions.distribution`; indepedent of any parents |\n",
    "| `FunctionalCPD`                | Allows for a CPD defined with a custom eval function |\n",
    "| `ParentFunctionalCPD`          | Modification to `FunctionalCPD` allowing the parent values to be passed in |\n",
    "| `CategoricalCPD`               | Categorical distribution, assumes integer parents in $1:N$ |\n",
    "| `LinearGaussianCPD`            | Linear Gaussian, assumes target and parents are numeric |\n",
    "| `ConditionalLinearGaussianCPD` | A linear Gaussian for each discrete parent instantiation|"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Each CPD can be learned from data using `fit`.\n",
    "\n",
    "Here we learn the same network as above."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
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       "BayesNet{CPD}({2, 1} directed simple Int64 graph, CPD[StaticCPD{Normal{Float64}}(:a, Symbol[], Normal{Float64}(μ=0.08532293451517128, σ=1.1218658914894157)), LinearGaussianCPD(:b, [:a], [2.126667640536174], 2.846694989226697, 2.5884318677452463)], Dict(:a => 1,:b => 2))"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = randn(100)\n",
    "b = randn(100) .+ 2*a .+ 3\n",
    "\n",
    "data = DataFrame(a=a, b=b)\n",
    "cpdA = fit(StaticCPD{Normal}, data, :a)\n",
    "cpdB = fit(LinearGaussianCPD, data, :b, [:a])\n",
    "\n",
    "bn2 = BayesNet([cpdA, cpdB])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Each `CPD` implements four functions:\n",
    "\n",
    "* `name(cpd)` - obtain the name of the variable target variable\n",
    "* `parents(cpd)` - obtain the list of parents\n",
    "* `nparams(cpd` - obtain the number of free parameters in the CPD\n",
    "* `cpd(assignment)` - allows calling `cpd()` to obtain the conditional distribution\n",
    "* `Distributions.fit(Type{CPD}, data, target, parents)`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Normal{Float64}(μ=3.9100288094947837, σ=2.5884318677452463)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cpdB(:a=>0.5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Several functions conveniently condition and then produce their return values:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "rand(cpdB, :a=>0.5) # condition and then sample\n",
    "pdf(cpdB, :a=>1.0, :b=>3.0) # condition and then compute pdf(distribution, 3)\n",
    "logpdf(cpdB, :a=>1.0, :b=>3.0) # condition and then compute logpdf(distribution, 3);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The NamedCategorical distribution allows for String or Symbol return values.\n",
    "The FunctionalCPD allows for crafting quick and simple CPDs:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
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      ],
      "text/plain": [
       "BayesNet{CPD}({2, 1} directed simple Int64 graph, CPD[StaticCPD{NamedCategorical{Symbol}}(:sighted, Symbol[], NamedCategorical with entries:\n",
       "\t  0.4000:  bird\n",
       "\t  0.0500:  superman\n",
       "\t  0.5500:  plane\n",
       "), FunctionalCPD{Bernoulli}(:happy, [:sighted], var\"#3#4\"())], Dict(:sighted => 1,:happy => 2))"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bn2 = BayesNet()\n",
    "push!(bn2, StaticCPD(:sighted, NamedCategorical([:bird, :plane, :superman], [0.40, 0.55, 0.05])))\n",
    "push!(bn2, FunctionalCPD{Bernoulli}(:happy, [:sighted], a->Bernoulli(a[:sighted] == :superman ? 0.95 : 0.2)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Variables can be removed by name using `delete!`. A warning will be issued when removing a CPD with children."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
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       "\n"
      ],
      "text/plain": [
       "BayesNet{CPD}({1, 0} directed simple Int64 graph, CPD[StaticCPD{NamedCategorical{Symbol}}(:sighted, Symbol[], NamedCategorical with entries:\n",
       "\t  0.4000:  bird\n",
       "\t  0.0500:  superman\n",
       "\t  0.5500:  plane\n",
       ")], Dict(:sighted => 1))"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "delete!(bn2, :happy)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Likelihood \n",
    "\n",
    "A Bayesian Network represents a joint probability distribution, $P(x_1, x_2, \\ldots, x_n)$. \n",
    "Assignments are represented as dictionaries mapping variable names (Symbols) to variable values.\n",
    "We can evaluate probabilities as we would with Distributions.jl, only we use exclamation points as we modify the internal state when we condition:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.01900834726778591"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pdf(bn, :a=>0.5, :b=>2.0) # evaluate the probability density"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can also evaluate the likelihood of a dataset:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "data = DataFrame(a=[0.5,1.0,2.0], b=[4.0,5.0,7.0])\n",
    "pdf(bn, data)    #  0.00215\n",
    "logpdf(bn, data) # -6.1386;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Or the likelihood for a particular cpd:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-5.611376074196931"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pdf(cpdB, data)    #  0.006\n",
    "logpdf(cpdB, data) # -5.201"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Sampling\n",
    "\n",
    "Assignments can be sampled from a `BayesNet`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Dict{Symbol,Any} with 2 entries:\n",
       "  :a => 1.20808\n",
       "  :b => 4.93954"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rand(bn)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"data-frame\"><thead><tr><th></th><th>a</th><th>b</th></tr><tr><th></th><th>Float64</th><th>Float64</th></tr></thead><tbody><p>5 rows × 2 columns</p><tr><th>1</th><td>0.121834</td><td>3.03183</td></tr><tr><th>2</th><td>-0.513101</td><td>2.11181</td></tr><tr><th>3</th><td>0.523282</td><td>1.93753</td></tr><tr><th>4</th><td>0.708859</td><td>5.28383</td></tr><tr><th>5</th><td>2.70031</td><td>7.53423</td></tr></tbody></table>"
      ],
      "text/latex": [
       "\\begin{tabular}{r|cc}\n",
       "\t& a & b\\\\\n",
       "\t\\hline\n",
       "\t& Float64 & Float64\\\\\n",
       "\t\\hline\n",
       "\t1 & 0.121834 & 3.03183 \\\\\n",
       "\t2 & -0.513101 & 2.11181 \\\\\n",
       "\t3 & 0.523282 & 1.93753 \\\\\n",
       "\t4 & 0.708859 & 5.28383 \\\\\n",
       "\t5 & 2.70031 & 7.53423 \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/plain": [
       "5×2 DataFrame\n",
       "│ Row │ a         │ b       │\n",
       "│     │ \u001b[90mFloat64\u001b[39m   │ \u001b[90mFloat64\u001b[39m │\n",
       "├─────┼───────────┼─────────┤\n",
       "│ 1   │ 0.121834  │ 3.03183 │\n",
       "│ 2   │ -0.513101 │ 2.11181 │\n",
       "│ 3   │ 0.523282  │ 1.93753 │\n",
       "│ 4   │ 0.708859  │ 5.28383 │\n",
       "│ 5   │ 2.70031   │ 7.53423 │"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rand(bn, 5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In general, sampling can be done according to `rand(BayesNet, BayesNetSampler, nsamples)` to produce a table of samples, `rand(BayesNet, BayesNetSampler)` to produce a single Assignment, or `rand!(Assignment, BayesNet, BayesNetSampler)` to modify an assignment in-place.\n",
    "\n",
    "New samplers need only implement `rand!`.\n",
    "\n",
    "The functions above default to the `DirectSampler`, which samples the variables in topographical order."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Rejection sampling can be used to draw samples that are consistent with a provided assignment:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
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       "<g style=\"fill:rgb(0%,0%,0%);fill-opacity:1;\">\n",
       "  <use xlink:href=\"#glyph-1585847237472004-0-3\" x=\"17.773\" y=\"37.356\"/>\n",
       "</g>\n",
       "</g>\n",
       "</svg>\n",
       "\n"
      ],
      "text/plain": [
       "BayesNet{CPD}({3, 2} directed simple Int64 graph, CPD[StaticCPD{DiscreteNonParametric{Int64,Float64,Base.OneTo{Int64},Array{Float64,1}}}(:a, Symbol[], DiscreteNonParametric{Int64,Float64,Base.OneTo{Int64},Array{Float64,1}}(support=Base.OneTo(2), p=[0.3, 0.7])), StaticCPD{DiscreteNonParametric{Int64,Float64,Base.OneTo{Int64},Array{Float64,1}}}(:b, Symbol[], DiscreteNonParametric{Int64,Float64,Base.OneTo{Int64},Array{Float64,1}}(support=Base.OneTo(2), p=[0.6, 0.4])), CategoricalCPD{Bernoulli}(:c, [:a, :b], [2, 2], Bernoulli[Bernoulli{Float64}(p=0.1), Bernoulli{Float64}(p=0.2), Bernoulli{Float64}(p=1.0), Bernoulli{Float64}(p=0.4)])], Dict(:a => 1,:b => 2,:c => 3))"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bn = BayesNet()\n",
    "push!(bn, StaticCPD(:a, Categorical([0.3,0.7])))\n",
    "push!(bn, StaticCPD(:b, Categorical([0.6,0.4])))\n",
    "push!(bn, CategoricalCPD{Bernoulli}(:c, [:a, :b], [2,2], [Bernoulli(0.1), Bernoulli(0.2), Bernoulli(1.0), Bernoulli(0.4)]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"data-frame\"><thead><tr><th></th><th>a</th><th>b</th><th>c</th></tr><tr><th></th><th>Int64</th><th>Int64</th><th>Bool</th></tr></thead><tbody><p>5 rows × 3 columns</p><tr><th>1</th><td>1</td><td>2</td><td>1</td></tr><tr><th>2</th><td>2</td><td>2</td><td>1</td></tr><tr><th>3</th><td>2</td><td>1</td><td>1</td></tr><tr><th>4</th><td>2</td><td>1</td><td>1</td></tr><tr><th>5</th><td>1</td><td>1</td><td>1</td></tr></tbody></table>"
      ],
      "text/latex": [
       "\\begin{tabular}{r|ccc}\n",
       "\t& a & b & c\\\\\n",
       "\t\\hline\n",
       "\t& Int64 & Int64 & Bool\\\\\n",
       "\t\\hline\n",
       "\t1 & 1 & 2 & 1 \\\\\n",
       "\t2 & 2 & 2 & 1 \\\\\n",
       "\t3 & 2 & 1 & 1 \\\\\n",
       "\t4 & 2 & 1 & 1 \\\\\n",
       "\t5 & 1 & 1 & 1 \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/plain": [
       "5×3 DataFrame\n",
       "│ Row │ a     │ b     │ c    │\n",
       "│     │ \u001b[90mInt64\u001b[39m │ \u001b[90mInt64\u001b[39m │ \u001b[90mBool\u001b[39m │\n",
       "├─────┼───────┼───────┼──────┤\n",
       "│ 1   │ 1     │ 2     │ 1    │\n",
       "│ 2   │ 2     │ 2     │ 1    │\n",
       "│ 3   │ 2     │ 1     │ 1    │\n",
       "│ 4   │ 2     │ 1     │ 1    │\n",
       "│ 5   │ 1     │ 1     │ 1    │"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rand(bn, RejectionSampler(:c=>1), 5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "One can also use weighted sampling:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"data-frame\"><thead><tr><th></th><th>a</th><th>b</th><th>c</th><th>p</th></tr><tr><th></th><th>Any</th><th>Any</th><th>Any</th><th>Any</th></tr></thead><tbody><p>5 rows × 4 columns</p><tr><th>1</th><td>2</td><td>2</td><td>1</td><td>0.333333</td></tr><tr><th>2</th><td>2</td><td>1</td><td>1</td><td>0.166667</td></tr><tr><th>3</th><td>2</td><td>1</td><td>1</td><td>0.166667</td></tr><tr><th>4</th><td>2</td><td>1</td><td>1</td><td>0.166667</td></tr><tr><th>5</th><td>2</td><td>1</td><td>1</td><td>0.166667</td></tr></tbody></table>"
      ],
      "text/latex": [
       "\\begin{tabular}{r|cccc}\n",
       "\t& a & b & c & p\\\\\n",
       "\t\\hline\n",
       "\t& Any & Any & Any & Any\\\\\n",
       "\t\\hline\n",
       "\t1 & 2 & 2 & 1 & 0.333333 \\\\\n",
       "\t2 & 2 & 1 & 1 & 0.166667 \\\\\n",
       "\t3 & 2 & 1 & 1 & 0.166667 \\\\\n",
       "\t4 & 2 & 1 & 1 & 0.166667 \\\\\n",
       "\t5 & 2 & 1 & 1 & 0.166667 \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/plain": [
       "5×4 DataFrame\n",
       "│ Row │ a   │ b   │ c   │ p        │\n",
       "│     │ \u001b[90mAny\u001b[39m │ \u001b[90mAny\u001b[39m │ \u001b[90mAny\u001b[39m │ \u001b[90mAny\u001b[39m      │\n",
       "├─────┼─────┼─────┼─────┼──────────┤\n",
       "│ 1   │ 2   │ 2   │ 1   │ 0.333333 │\n",
       "│ 2   │ 2   │ 1   │ 1   │ 0.166667 │\n",
       "│ 3   │ 2   │ 1   │ 1   │ 0.166667 │\n",
       "│ 4   │ 2   │ 1   │ 1   │ 0.166667 │\n",
       "│ 5   │ 2   │ 1   │ 1   │ 0.166667 │"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rand(bn, LikelihoodWeightedSampler(:c=>1), 5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "One can also use Gibbs sampling.  More options are available than are shown in the example below."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"data-frame\"><thead><tr><th></th><th>a</th><th>b</th><th>c</th></tr><tr><th></th><th>Any</th><th>Any</th><th>Float64</th></tr></thead><tbody><p>5 rows × 3 columns</p><tr><th>1</th><td>1</td><td>8.393</td><td>10.0</td></tr><tr><th>2</th><td>1</td><td>7.06636</td><td>10.0</td></tr><tr><th>3</th><td>1</td><td>7.44853</td><td>10.0</td></tr><tr><th>4</th><td>1</td><td>6.7813</td><td>10.0</td></tr><tr><th>5</th><td>1</td><td>6.7813</td><td>10.0</td></tr></tbody></table>"
      ],
      "text/latex": [
       "\\begin{tabular}{r|ccc}\n",
       "\t& a & b & c\\\\\n",
       "\t\\hline\n",
       "\t& Any & Any & Float64\\\\\n",
       "\t\\hline\n",
       "\t1 & 1 & 8.393 & 10.0 \\\\\n",
       "\t2 & 1 & 7.06636 & 10.0 \\\\\n",
       "\t3 & 1 & 7.44853 & 10.0 \\\\\n",
       "\t4 & 1 & 6.7813 & 10.0 \\\\\n",
       "\t5 & 1 & 6.7813 & 10.0 \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/plain": [
       "5×3 DataFrame\n",
       "│ Row │ a   │ b       │ c       │\n",
       "│     │ \u001b[90mAny\u001b[39m │ \u001b[90mAny\u001b[39m     │ \u001b[90mFloat64\u001b[39m │\n",
       "├─────┼─────┼─────────┼─────────┤\n",
       "│ 1   │ 1   │ 8.393   │ 10.0    │\n",
       "│ 2   │ 1   │ 7.06636 │ 10.0    │\n",
       "│ 3   │ 1   │ 7.44853 │ 10.0    │\n",
       "│ 4   │ 1   │ 6.7813  │ 10.0    │\n",
       "│ 5   │ 1   │ 6.7813  │ 10.0    │"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bn_gibbs = BayesNet()\n",
    "push!(bn_gibbs, StaticCPD(:a, Categorical([0.999,0.001])))\n",
    "push!(bn_gibbs, StaticCPD(:b, Normal(1.0)))\n",
    "push!(bn_gibbs, LinearGaussianCPD(:c, [:a, :b], [3.0, 1.0], 0.0, 1.0))\n",
    "\n",
    "evidence = Assignment(:c => 10.0)\n",
    "initial_sample = Assignment(:a => 1, :b => 1, :c => 10.0)\n",
    "gsampler = GibbsSampler(evidence, burn_in=500, thinning=1, initial_sample=initial_sample)\n",
    "rand(bn_gibbs, gsampler, 5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Parameter Learning\n",
    "\n",
    "BayesNets.jl supports parameter learning for an entire graph."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"UTF-8\"?>\n",
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       "<defs>\n",
       "<g>\n",
       "<symbol overflow=\"visible\" id=\"glyph-1585847237472005-0-0\">\n",
       "<path style=\"stroke:none;\" d=\"\"/>\n",
       "</symbol>\n",
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       "</symbol>\n",
       "</g>\n",
       "</defs>\n",
       "<g id=\"surface1\">\n",
       "<path style=\"fill:none;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;\" d=\"M -0.00015625 -5.807063 L -0.00015625 -20.857844 \" transform=\"matrix(1,0,0,-1,6.09,5.607)\"/>\n",
       "<path style=\"fill:none;stroke-width:0.31879;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;\" d=\"M -1.194299 1.593594 C -1.096643 0.995938 0.00101375 0.101406 0.297889 -0.00015625 C 0.00101375 -0.0978125 -1.096643 -0.99625 -1.194299 -1.593906 \" transform=\"matrix(0,1,1,0,6.09,26.46383)\"/>\n",
       "<g style=\"fill:rgb(0%,0%,0%);fill-opacity:1;\">\n",
       "  <use xlink:href=\"#glyph-1585847237472005-0-1\" x=\"3.599\" y=\"7.784\"/>\n",
       "</g>\n",
       "<g style=\"fill:rgb(0%,0%,0%);fill-opacity:1;\">\n",
       "  <use xlink:href=\"#glyph-1585847237472005-0-2\" x=\"3.321\" y=\"37.356\"/>\n",
       "</g>\n",
       "</g>\n",
       "</svg>\n",
       "\n"
      ],
      "text/plain": [
       "BayesNet{CPD}({2, 1} directed simple Int64 graph, CPD[StaticCPD{Normal{Float64}}(:a, Symbol[], Normal{Float64}(μ=1.1666666666666667, σ=0.6236095644623235)), LinearGaussianCPD(:b, [:a], [2.000000000000001], 2.9999999999999987, 1.5275252316519465)], Dict(:a => 1,:b => 2))"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# specify each node's CPD type individually\n",
    "fit(BayesNet, data, (:a=>:b), [StaticCPD{Normal}, LinearGaussianCPD])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"UTF-8\"?>\n",
       "<svg xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"12.18pt\" height=\"40.786pt\" viewBox=\"0 0 12.18 40.786\" version=\"1.1\">\n",
       "<defs>\n",
       "<g>\n",
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       "<g style=\"fill:rgb(0%,0%,0%);fill-opacity:1;\">\n",
       "  <use xlink:href=\"#glyph-1585847237472006-0-1\" x=\"3.599\" y=\"7.784\"/>\n",
       "</g>\n",
       "<g style=\"fill:rgb(0%,0%,0%);fill-opacity:1;\">\n",
       "  <use xlink:href=\"#glyph-1585847237472006-0-2\" x=\"3.321\" y=\"37.356\"/>\n",
       "</g>\n",
       "</g>\n",
       "</svg>\n",
       "\n"
      ],
      "text/plain": [
       "BayesNet{LinearGaussianCPD}({2, 1} directed simple Int64 graph, LinearGaussianCPD[LinearGaussianCPD(:a, Symbol[], Float64[], 1.1666666666666667, 0.7637626158259733), LinearGaussianCPD(:b, [:a], [2.000000000000001], 2.9999999999999987, 1.5275252316519465)], Dict(:a => 1,:b => 2))"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# specify a single CPD type for all nodes\n",
    "fit(BayesNet, data, (:a=>:b), LinearGaussianCPD)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Fitting can be done for specific BayesNets types as well:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "\n"
      ],
      "text/plain": [
       "BayesNet{CategoricalCPD{DiscreteNonParametric{Int64,Float64,Base.OneTo{Int64},Array{Float64,1}}}}({3, 3} directed simple Int64 graph, CategoricalCPD{DiscreteNonParametric{Int64,Float64,Base.OneTo{Int64},Array{Float64,1}}}[2 instantiations:\n",
       "  a (2), 4 instantiations:\n",
       "  b (2)\n",
       "  a (2), 12 instantiations:\n",
       "  c (3)\n",
       "  b (2)\n",
       "  a (2)], Dict(:a => 1,:b => 2,:c => 3))"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data = DataFrame(c=[1,1,1,1,2,2,2,2,3,3,3,3], \n",
    "                 b=[1,1,1,2,2,2,2,1,1,2,1,1],\n",
    "                 a=[1,1,1,2,1,1,2,1,1,2,1,1])\n",
    "\n",
    "fit(DiscreteBayesNet, data, (:a=>:b, :a=>:c, :b=>:c))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Fitting a `DiscreteCPD`, which is a `CategoricalCPD{Categorical}`, can be done with a specified number of categories. This prevents cases where your test data does not provide an example for every category."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "cpd = fit(DiscreteCPD, DataFrame(a=[1,2,1,2,2]), :a, ncategories=3);\n",
    "cpd = fit(DiscreteCPD, data, :b, [:a], parental_ncategories=[3], target_ncategories=3);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Inference\n",
    "\n",
    "Inference methods for discrete Bayesian networks can be used via the `infer` method:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
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       "\n"
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      "text/plain": [
       "BayesNet{CategoricalCPD{DiscreteNonParametric{Int64,Float64,Base.OneTo{Int64},Array{Float64,1}}}}({3, 2} directed simple Int64 graph, CategoricalCPD{DiscreteNonParametric{Int64,Float64,Base.OneTo{Int64},Array{Float64,1}}}[2 instantiations:\n",
       "  a (2), 2 instantiations:\n",
       "  b (2), 8 instantiations:\n",
       "  c (2)\n",
       "  a (2)\n",
       "  b (2)], Dict(:a => 1,:b => 2,:c => 3))"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bn = DiscreteBayesNet()\n",
    "push!(bn, DiscreteCPD(:a, [0.3,0.7]))\n",
    "push!(bn, DiscreteCPD(:b, [0.2,0.8]))\n",
    "push!(bn, DiscreteCPD(:c, [:a, :b], [2,2], \n",
    "        [Categorical([0.1,0.9]),\n",
    "         Categorical([0.2,0.8]),\n",
    "         Categorical([1.0,0.0]),\n",
    "         Categorical([0.4,0.6]),\n",
    "        ]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"data-frame\"><thead><tr><th></th><th>c</th><th>potential</th></tr><tr><th></th><th>Int64</th><th>Float64</th></tr></thead><tbody><p>2 rows × 2 columns</p><tr><th>1</th><td>1</td><td>0.17</td></tr><tr><th>2</th><td>2</td><td>0.83</td></tr></tbody></table>"
      ],
      "text/plain": [
       "2 instantiations:\n",
       "  c (2)"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ϕ = infer(bn, :c, evidence=Assignment(:b=>1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Several inference methods are available. Exact inference is the default.\n",
    "\n",
    "| `InfereceMethod`               | Description |\n",
    "| ------------------------------ | ----------- |\n",
    "| `ExactInference`               | Performs exact inference using discrete factors and variable elimination |\n",
    "| `LikelihoodWeightingInference` | Approximates p(query \\ evidence) with N weighted samples using likelihood weighted sampling |\n",
    "| `LoopyBelief`                  | The loopy belief propagation algorithm |\n",
    "| `GibbsSamplingNodewise`        | Gibbs sampling where each iteration changes one node |\n",
    "| `GibbsSamplingFull`            | Gibbs sampling where each iteration changes all nodes | \n",
    "\n",
    "All inference methods inherit from the `InferenceMethod` type."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"data-frame\"><thead><tr><th></th><th>a</th><th>b</th><th>potential</th></tr><tr><th></th><th>Int64</th><th>Int64</th><th>Float64</th></tr></thead><tbody><p>4 rows × 3 columns</p><tr><th>1</th><td>1</td><td>1</td><td>0.134</td></tr><tr><th>2</th><td>2</td><td>1</td><td>0.214</td></tr><tr><th>3</th><td>1</td><td>2</td><td>0.0</td></tr><tr><th>4</th><td>2</td><td>2</td><td>0.652</td></tr></tbody></table>"
      ],
      "text/plain": [
       "4 instantiations:\n",
       "  a (2)\n",
       "  b (2)"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ϕ = infer(GibbsSamplingNodewise(), bn, [:a, :b], evidence=Assignment(:c=>2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Inference produces a `Factor` type. It can be converted to a DataFrame."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"data-frame\"><thead><tr><th></th><th>a</th><th>b</th><th>potential</th></tr><tr><th></th><th>Int64</th><th>Int64</th><th>Float64</th></tr></thead><tbody><p>4 rows × 3 columns</p><tr><th>1</th><td>1</td><td>1</td><td>0.134</td></tr><tr><th>2</th><td>2</td><td>1</td><td>0.214</td></tr><tr><th>3</th><td>1</td><td>2</td><td>0.0</td></tr><tr><th>4</th><td>2</td><td>2</td><td>0.652</td></tr></tbody></table>"
      ],
      "text/latex": [
       "\\begin{tabular}{r|ccc}\n",
       "\t& a & b & potential\\\\\n",
       "\t\\hline\n",
       "\t& Int64 & Int64 & Float64\\\\\n",
       "\t\\hline\n",
       "\t1 & 1 & 1 & 0.134 \\\\\n",
       "\t2 & 2 & 1 & 0.214 \\\\\n",
       "\t3 & 1 & 2 & 0.0 \\\\\n",
       "\t4 & 2 & 2 & 0.652 \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/plain": [
       "4×3 DataFrame\n",
       "│ Row │ a     │ b     │ potential │\n",
       "│     │ \u001b[90mInt64\u001b[39m │ \u001b[90mInt64\u001b[39m │ \u001b[90mFloat64\u001b[39m   │\n",
       "├─────┼───────┼───────┼───────────┤\n",
       "│ 1   │ 1     │ 1     │ 0.134     │\n",
       "│ 2   │ 2     │ 1     │ 0.214     │\n",
       "│ 3   │ 1     │ 2     │ 0.0       │\n",
       "│ 4   │ 2     │ 2     │ 0.652     │"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "convert(DataFrame, ϕ)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Structure Learning\n",
    "\n",
    "Structure learning can be done as well."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"data-frame\"><thead><tr><th></th><th>SepalLength</th><th>SepalWidth</th><th>PetalLength</th><th>PetalWidth</th><th>Species</th></tr><tr><th></th><th>Float64</th><th>Float64</th><th>Float64</th><th>Float64</th><th>Int64</th></tr></thead><tbody><p>3 rows × 5 columns</p><tr><th>1</th><td>5.1</td><td>3.5</td><td>1.4</td><td>0.2</td><td>1</td></tr><tr><th>2</th><td>4.9</td><td>3.0</td><td>1.4</td><td>0.2</td><td>1</td></tr><tr><th>3</th><td>4.7</td><td>3.2</td><td>1.3</td><td>0.2</td><td>1</td></tr></tbody></table>"
      ],
      "text/latex": [
       "\\begin{tabular}{r|ccccc}\n",
       "\t& SepalLength & SepalWidth & PetalLength & PetalWidth & Species\\\\\n",
       "\t\\hline\n",
       "\t& Float64 & Float64 & Float64 & Float64 & Int64\\\\\n",
       "\t\\hline\n",
       "\t1 & 5.1 & 3.5 & 1.4 & 0.2 & 1 \\\\\n",
       "\t2 & 4.9 & 3.0 & 1.4 & 0.2 & 1 \\\\\n",
       "\t3 & 4.7 & 3.2 & 1.3 & 0.2 & 1 \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/plain": [
       "3×5 DataFrame\n",
       "│ Row │ SepalLength │ SepalWidth │ PetalLength │ PetalWidth │ Species │\n",
       "│     │ \u001b[90mFloat64\u001b[39m     │ \u001b[90mFloat64\u001b[39m    │ \u001b[90mFloat64\u001b[39m     │ \u001b[90mFloat64\u001b[39m    │ \u001b[90mInt64\u001b[39m   │\n",
       "├─────┼─────────────┼────────────┼─────────────┼────────────┼─────────┤\n",
       "│ 1   │ 5.1         │ 3.5        │ 1.4         │ 0.2        │ 1       │\n",
       "│ 2   │ 4.9         │ 3.0        │ 1.4         │ 0.2        │ 1       │\n",
       "│ 3   │ 4.7         │ 3.2        │ 1.3         │ 0.2        │ 1       │"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using Discretizers\n",
    "using RDatasets\n",
    "iris = dataset(\"datasets\", \"iris\")\n",
    "names(iris)\n",
    "data = DataFrame(\n",
    "    SepalLength = iris[!,:SepalLength],\n",
    "    SepalWidth = iris[!,:SepalWidth],\n",
    "    PetalLength = iris[!,:PetalLength],\n",
    "    PetalWidth = iris[!,:PetalWidth],\n",
    "    Species = encode(CategoricalDiscretizer(iris[!,:Species]), iris[!,:Species]),\n",
    ")\n",
    "data[1:3,:] # only display a subset..."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here we use the K2 structure learning algorithm which runs in polynomial time but requires that we specify a topological node ordering."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
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       "BayesNet{CPD}({5, 7} directed simple Int64 graph, CPD[ConditionalLinearGaussianCPD(:Species, Symbol[], Symbol[], Int64[], LinearGaussianCPD[LinearGaussianCPD(:Species, Symbol[], Float64[], 2.0, 0.8192319205190405)]), ConditionalLinearGaussianCPD(:SepalLength, [:Species], [:Species], [3], LinearGaussianCPD[LinearGaussianCPD(:SepalLength, Symbol[], Float64[], 5.006, 0.3524896872134513), LinearGaussianCPD(:SepalLength, Symbol[], Float64[], 5.935999999999999, 0.5161711470638634), LinearGaussianCPD(:SepalLength, Symbol[], Float64[], 6.587999999999999, 0.6358795932744321)]), ConditionalLinearGaussianCPD(:PetalLength, [:Species, :SepalLength], [:Species], [3], LinearGaussianCPD[LinearGaussianCPD(:PetalLength, [:SepalLength], [0.13163168095660227], 0.8030518051312493, 0.17366399648018407), LinearGaussianCPD(:PetalLength, [:SepalLength], [0.686469759176422], 0.18511550952875835, 0.46991097723995795), LinearGaussianCPD(:PetalLength, [:SepalLength], [0.7500807558749694], 0.6104679802957019, 0.5518946956639834)]), ConditionalLinearGaussianCPD(:PetalWidth, [:Species, :PetalLength], [:Species], [3], LinearGaussianCPD[LinearGaussianCPD(:PetalWidth, [:PetalLength], [0.20124509405874183], -0.0482203275138806, 0.10538558938004566), LinearGaussianCPD(:PetalWidth, [:PetalLength], [0.3310536044362199], -0.08428835489829607, 0.19775268000454405), LinearGaussianCPD(:PetalWidth, [:PetalLength], [0.1602969554031031], 1.1360313036019714, 0.27465005563666733)]), ConditionalLinearGaussianCPD(:SepalWidth, [:Species, :SepalLength], [:Species], [3], LinearGaussianCPD[LinearGaussianCPD(:SepalWidth, [:SepalLength], [0.7985283006471842], -0.5694326730398034, 0.3790643690962886), LinearGaussianCPD(:SepalWidth, [:SepalLength], [0.31971934554813936], 0.8721459648262444, 0.3137983233784114), LinearGaussianCPD(:SepalWidth, [:SepalLength], [0.23189049503351394], 1.4463054187192101, 0.32249663817263746)])], Dict(:SepalLength => 2,:PetalWidth => 4,:SepalWidth => 5,:PetalLength => 3,:Species => 1))"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "parameters = K2GraphSearch([:Species, :SepalLength, :SepalWidth, :PetalLength, :PetalWidth], \n",
    "                       ConditionalLinearGaussianCPD,\n",
    "                       max_n_parents=2)\n",
    "fit(BayesNet, data, parameters)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "CPD types can also be specified per-node. Note that complete CPD definitions are required - simply using `StaticCPD` is insufficient as you need the target distribution type as well, as in `StaticCPD{Categorical}`.\n",
    "\n",
    "Changing the ordering will change the structure."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
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      "text/plain": [
       "BayesNet{CPD}({5, 7} directed simple Int64 graph, CPD[StaticCPD{DiscreteNonParametric{Int64,Float64,Base.OneTo{Int64},Array{Float64,1}}}(:Species, Symbol[], DiscreteNonParametric{Int64,Float64,Base.OneTo{Int64},Array{Float64,1}}(support=Base.OneTo(3), p=[0.3333333333333333, 0.3333333333333333, 0.3333333333333333])), ConditionalLinearGaussianCPD(:PetalLength, [:Species], [:Species], [3], LinearGaussianCPD[LinearGaussianCPD(:PetalLength, Symbol[], Float64[], 1.462, 0.17366399648018407), LinearGaussianCPD(:PetalLength, Symbol[], Float64[], 4.26, 0.46991097723995795), LinearGaussianCPD(:PetalLength, Symbol[], Float64[], 5.552, 0.5518946956639834)]), ConditionalLinearGaussianCPD(:SepalLength, [:Species, :PetalLength], [:Species], [3], LinearGaussianCPD[LinearGaussianCPD(:SepalLength, [:PetalLength], [0.5422925971037884], 4.213168223034262, 0.3524896872134513), LinearGaussianCPD(:SepalLength, [:PetalLength], [0.8282809611830106], 2.407523105360375, 0.5161711470638634), LinearGaussianCPD(:SepalLength, [:PetalLength], [0.995738636363621], 1.0596590909091785, 0.6358795932744321)]), ConditionalLinearGaussianCPD(:SepalWidth, [:Species, :SepalLength], [:Species], [3], LinearGaussianCPD[LinearGaussianCPD(:SepalWidth, [:SepalLength], [0.7985283006471842], -0.5694326730398034, 0.3790643690962886), LinearGaussianCPD(:SepalWidth, [:SepalLength], [0.31971934554813936], 0.8721459648262444, 0.3137983233784114), LinearGaussianCPD(:SepalWidth, [:SepalLength], [0.23189049503351394], 1.4463054187192101, 0.32249663817263746)]), ConditionalLinearGaussianCPD(:PetalWidth, [:Species, :PetalLength], [:Species], [3], LinearGaussianCPD[LinearGaussianCPD(:PetalWidth, [:PetalLength], [0.20124509405874183], -0.0482203275138806, 0.10538558938004566), LinearGaussianCPD(:PetalWidth, [:PetalLength], [0.3310536044362199], -0.08428835489829607, 0.19775268000454405), LinearGaussianCPD(:PetalWidth, [:PetalLength], [0.1602969554031031], 1.1360313036019714, 0.27465005563666733)])], Dict(:SepalLength => 3,:SepalWidth => 4,:PetalWidth => 5,:PetalLength => 2,:Species => 1))"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "CLG = ConditionalLinearGaussianCPD\n",
    "parameters = K2GraphSearch([:Species, :PetalLength, :PetalWidth, :SepalLength, :SepalWidth], \n",
    "                        [StaticCPD{Categorical}, CLG, CLG, CLG, CLG],\n",
    "                        max_n_parents=2)\n",
    "fit(BayesNet, data, parameters)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A `ScoringFunction` allows for extracting a scoring metric for a CPD given data.\n",
    "The negative BIC score is implemented in NegativeBayesianInformationCriterion.\n",
    "\n",
    "A `GraphSearchStrategy` defines a structure learning algorithm.\n",
    "The K2 algorithm is defined through `K2GraphSearch` and `GreedyHillClimbing` is implemented for discrete Bayesian networks and the Bayesian score:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
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      "text/plain": [
       "BayesNet{CategoricalCPD{DiscreteNonParametric{Int64,Float64,Base.OneTo{Int64},Array{Float64,1}}}}({3, 3} directed simple Int64 graph, CategoricalCPD{DiscreteNonParametric{Int64,Float64,Base.OneTo{Int64},Array{Float64,1}}}[2 instantiations:\n",
       "  a (2), 4 instantiations:\n",
       "  b (2)\n",
       "  a (2), 12 instantiations:\n",
       "  c (3)\n",
       "  b (2)\n",
       "  a (2)], Dict(:a => 1,:b => 2,:c => 3))"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data = DataFrame(c=[1,1,1,1,2,2,2,2,3,3,3,3], \n",
    "                 b=[1,1,1,2,2,2,2,1,1,2,1,1],\n",
    "                 a=[1,1,1,2,1,1,2,1,1,2,1,1])\n",
    "parameters = GreedyHillClimbing(ScoreComponentCache(data), max_n_parents=3, prior=UniformPrior())\n",
    "bn = fit(DiscreteBayesNet, data, parameters)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "One can specify the number of categories for each variable in case it cannot be correctly inferred:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
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      ],
      "text/plain": [
       "BayesNet{CategoricalCPD{DiscreteNonParametric{Int64,Float64,Base.OneTo{Int64},Array{Float64,1}}}}({3, 3} directed simple Int64 graph, CategoricalCPD{DiscreteNonParametric{Int64,Float64,Base.OneTo{Int64},Array{Float64,1}}}[2 instantiations:\n",
       "  a (2), 6 instantiations:\n",
       "  b (3)\n",
       "  a (2), 18 instantiations:\n",
       "  c (3)\n",
       "  b (3)\n",
       "  a (2)], Dict(:a => 1,:b => 2,:c => 3))"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bn = fit(DiscreteBayesNet, data, parameters, ncategories=[3,3,2])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A whole suite of features are supported for DiscreteBayesNets."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-31.28804624550449"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bayesian_score(bn, data, parameters.prior) # compute the Bayesian score of the data under the BayesNet"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"data-frame\"><thead><tr><th></th><th>a</th><th>count</th></tr><tr><th></th><th>Int64</th><th>Int64</th></tr></thead><tbody><p>2 rows × 2 columns</p><tr><th>1</th><td>1</td><td>9</td></tr><tr><th>2</th><td>2</td><td>3</td></tr></tbody></table>"
      ],
      "text/latex": [
       "\\begin{tabular}{r|cc}\n",
       "\t& a & count\\\\\n",
       "\t\\hline\n",
       "\t& Int64 & Int64\\\\\n",
       "\t\\hline\n",
       "\t1 & 1 & 9 \\\\\n",
       "\t2 & 2 & 3 \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/plain": [
       "2×2 DataFrame\n",
       "│ Row │ a     │ count │\n",
       "│     │ \u001b[90mInt64\u001b[39m │ \u001b[90mInt64\u001b[39m │\n",
       "├─────┼───────┼───────┤\n",
       "│ 1   │ 1     │ 9     │\n",
       "│ 2   │ 2     │ 3     │"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "count(bn, :a, data) # obtain a list of counts for the node"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3-element Array{Array{Int64,2},1}:\n",
       " [4; 4; 4]\n",
       " [3 1 3; 1 3 1]\n",
       " [3 1 … 2 0; 0 0 … 1 1]"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "statistics(bn.dag, data) # sufficient statistics from a discrete dataset"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"data-frame\"><thead><tr><th></th><th>a</th><th>b</th><th>p</th></tr><tr><th></th><th>Int64</th><th>Int64</th><th>Float64</th></tr></thead><tbody><p>6 rows × 3 columns</p><tr><th>1</th><td>1</td><td>1</td><td>0.666667</td></tr><tr><th>2</th><td>2</td><td>1</td><td>0.166667</td></tr><tr><th>3</th><td>1</td><td>2</td><td>0.25</td></tr><tr><th>4</th><td>2</td><td>2</td><td>0.666667</td></tr><tr><th>5</th><td>1</td><td>3</td><td>0.0833333</td></tr><tr><th>6</th><td>2</td><td>3</td><td>0.166667</td></tr></tbody></table>"
      ],
      "text/plain": [
       "Table(6×3 DataFrame\n",
       "│ Row │ a     │ b     │ p         │\n",
       "│     │ \u001b[90mInt64\u001b[39m │ \u001b[90mInt64\u001b[39m │ \u001b[90mFloat64\u001b[39m   │\n",
       "├─────┼───────┼───────┼───────────┤\n",
       "│ 1   │ 1     │ 1     │ 0.666667  │\n",
       "│ 2   │ 2     │ 1     │ 0.166667  │\n",
       "│ 3   │ 1     │ 2     │ 0.25      │\n",
       "│ 4   │ 2     │ 2     │ 0.666667  │\n",
       "│ 5   │ 1     │ 3     │ 0.0833333 │\n",
       "│ 6   │ 2     │ 3     │ 0.166667  │)"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "table(bn, :b) # obtain the factor table for a node"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"data-frame\"><thead><tr><th></th><th>b</th><th>a</th><th>c</th><th>p</th></tr><tr><th></th><th>Int64</th><th>Int64</th><th>Int64</th><th>Float64</th></tr></thead><tbody><p>9 rows × 4 columns</p><tr><th>1</th><td>1</td><td>1</td><td>1</td><td>0.4</td></tr><tr><th>2</th><td>2</td><td>1</td><td>1</td><td>0.2</td></tr><tr><th>3</th><td>3</td><td>1</td><td>1</td><td>0.333333</td></tr><tr><th>4</th><td>1</td><td>1</td><td>2</td><td>0.2</td></tr><tr><th>5</th><td>2</td><td>1</td><td>2</td><td>0.6</td></tr><tr><th>6</th><td>3</td><td>1</td><td>2</td><td>0.333333</td></tr><tr><th>7</th><td>1</td><td>1</td><td>3</td><td>0.4</td></tr><tr><th>8</th><td>2</td><td>1</td><td>3</td><td>0.2</td></tr><tr><th>9</th><td>3</td><td>1</td><td>3</td><td>0.333333</td></tr></tbody></table>"
      ],
      "text/plain": [
       "Table(9×4 DataFrame\n",
       "│ Row │ b     │ a     │ c     │ p        │\n",
       "│     │ \u001b[90mInt64\u001b[39m │ \u001b[90mInt64\u001b[39m │ \u001b[90mInt64\u001b[39m │ \u001b[90mFloat64\u001b[39m  │\n",
       "├─────┼───────┼───────┼───────┼──────────┤\n",
       "│ 1   │ 1     │ 1     │ 1     │ 0.4      │\n",
       "│ 2   │ 2     │ 1     │ 1     │ 0.2      │\n",
       "│ 3   │ 3     │ 1     │ 1     │ 0.333333 │\n",
       "│ 4   │ 1     │ 1     │ 2     │ 0.2      │\n",
       "│ 5   │ 2     │ 1     │ 2     │ 0.6      │\n",
       "│ 6   │ 3     │ 1     │ 2     │ 0.333333 │\n",
       "│ 7   │ 1     │ 1     │ 3     │ 0.4      │\n",
       "│ 8   │ 2     │ 1     │ 3     │ 0.2      │\n",
       "│ 9   │ 3     │ 1     │ 3     │ 0.333333 │)"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "table(bn, :c, :a=>1) # obtain a factor table matching a particular assignment"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "# Reading from XDSL\n",
    "\n",
    "One can read discrete Bayesian networks from the .XDSL file format."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
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       "  <use xlink:href=\"#glyph-1585847237472013-0-4\" x=\"18.713279\" y=\"38.621\"/>\n",
       "  <use xlink:href=\"#glyph-1585847237472013-0-3\" x=\"23.136691\" y=\"38.621\"/>\n",
       "  <use xlink:href=\"#glyph-1585847237472013-0-9\" x=\"27.560103\" y=\"38.621\"/>\n",
       "  <use xlink:href=\"#glyph-1585847237472013-0-5\" x=\"32.541423\" y=\"38.621\"/>\n",
       "  <use xlink:href=\"#glyph-1585847237472013-0-10\" x=\"36.466703\" y=\"38.621\"/>\n",
       "</g>\n",
       "</g>\n",
       "</svg>\n",
       "\n"
      ],
      "text/plain": [
       "BayesNet{CategoricalCPD{DiscreteNonParametric{Int64,Float64,Base.OneTo{Int64},Array{Float64,1}}}}({2, 1} directed simple Int64 graph, CategoricalCPD{DiscreteNonParametric{Int64,Float64,Base.OneTo{Int64},Array{Float64,1}}}[2 instantiations:\n",
       "  Success (2), 6 instantiations:\n",
       "  Forecast (3)\n",
       "  Success (2)], Dict(:Forecast => 2,:Success => 1))"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bn = readxdsl(joinpath(dirname(pathof(BayesNets)), \"..\", \"test\", \"sample_bn.xdsl\"))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "# Bayesian Score for a Network Structure\n",
    "\n",
    "The bayesian score for a discrete-valued BayesNet can can be calculated based only on the structure and data (the CPDs do not need to be defined beforehand). This is implemented with a method of `bayesian_score` that takes in a directed graph, the names of the nodes and data."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-29.642240688899513"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data = DataFrame(c=[1,1,1,1,2,2,2,2,3,3,3,3], \n",
    "                 b=[1,1,1,2,2,2,2,1,1,2,1,1],\n",
    "                 a=[1,1,1,2,1,1,2,1,1,2,1,1])\n",
    "g = DAG(3)\n",
    "add_edge!(g,1,2); add_edge!(g,2,3); add_edge!(g,1,3)\n",
    "bayesian_score(g, [:a,:b,:c], data)"
   ]
  }
 ],
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