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bernoulli_trials_classical.ipynb
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bernoulli_trials_classical.ipynb
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{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Bernoulli trials are one of the simplest experimential setups: there are a number of iterations of some activity, where each iteration (or trial) may turn out to be a \"success\" or a \"failure\". From the data on T trials, we want to estimate the probability of \"success\".\n",
"\n",
"Since it is such a simple case, it is a nice setup to use to describe some of Python's capabilities for estimating statistical models. Here I show estimation from the classical (frequentist) perspective via maximum likelihood estimation\n",
"\n",
"There is also a slight digression about the delta method (which can be used to approximate the variance / covariance matrix when numeric maximum likelihood estimation is employed).\n",
"\n",
"In [another post](./bernoulli_trials_bayesian.html) I show estimation of the problem in Python using a Bayesian approach.\n",
"\n",
"<!-- TEASER_END -->"
]
},
{
"cell_type": "code",
"execution_count": 18,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"%matplotlib inline\n",
"from __future__ import division\n",
"\n",
"import numpy as np\n",
"import pandas as pd\n",
"import statsmodels.api as sm\n",
"import sympy as sp\n",
"import pymc\n",
"import matplotlib.pyplot as plt\n",
"import matplotlib.gridspec as gridspec\n",
"from mpl_toolkits.mplot3d import Axes3D\n",
"from scipy import stats\n",
"from scipy.special import gamma\n",
"\n",
"from sympy.interactive import printing\n",
"printing.init_printing()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Setup\n",
"\n",
"Let $y$ be a Bernoulli trial:\n",
"\n",
"$y \\sim \\text{Bernoulli}(\\theta) = \\text{Binomial}(1, \\theta)$\n",
"\n",
"The probability density, or marginal likelihood, function, is:\n",
"\n",
"$$p(y|\\theta) = \\theta^{y} (1-\\theta)^{1-y} = \\begin{cases}\n",
"\\theta & y = 1 \\\\\n",
"1 - \\theta & y = 0\n",
"\\end{cases}$$"
]
},
{
"cell_type": "code",
"execution_count": 19,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"# Simulate data\n",
"np.random.seed(123)\n",
"\n",
"nobs = 100\n",
"theta = 0.3\n",
"Y = np.random.binomial(1, theta, nobs)"
]
},
{
"cell_type": "code",
"execution_count": 20,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
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TTe07pRxDY/s05m8s04vtkV6ftftC4/Uaqf52+08P419v+zTH3q6+dvGMdfu3W340r18X\n08eO4v3ccv+vuv6meJak+tbZf/qx/RvTTDDu/g13P7WpbJG7vzn9LR5U23IXYmz/iwAz28DdnzWz\nTYFl7j7VzG5x9ylmtgmwwt1fl3rcPwAOcff7S8uPVH4t8LZ2x9vN7AXAcnefUqW8bMmSJXH69Omh\n3fMNpSGwhcAs0jfpduWd6utUb6/mbyWEsO8bTltya6mek9JTp/Qytn5ral/LGBrzVj3hqUrMo3ld\n+vFadqpzrDH3a9/u5341yphH3Hd6UVc/Y676GTYMBhnLf9xy9zXzl96zz3iv99TdIuMVc6fh9NPN\nbBlwJXB8KrvJzE4HZgMRIP028EPAJWb2PTPzCuUnUAzTLzOzrzRWaGZnpnUuB87pVD5apTfgSStm\nTbuH54bGtmtTPqldXRXrbbl8t/O3U0rgJwHlD5LyEOiYYuu38mtB+xjWtrXLD/a2MY9mG/Rqu3Vb\n5xhj7te+3df9qtuYqbDv9KCuCf1eknyM2BMfVlW++Q3gDN6+np1ehzNqe3mGcas6S2Xr1DOabdCr\n7daPOgewbw98v+rlvjMR3kvqifdGDj3xbJN43eT4W1rFnIccY67TZ5iSeH/pim0iIiJDSkm8JnLr\nqYBizkWOMYtUpSQuIiIypJTEa2Ii/ra03xRzHnKMWaQqJXEREZEhpSReEzkeN1TMecgxZpGqOt3F\nTEREMmdm2wCXUOSMm9z9ODObQXGraoA57r50YA3MmHriNZHjcUPFnIccY56ATgdOSjegOi7dxGoe\ncED6m1u+26WMHyVxERFpy8w2BF7t7jeUiicDd7j7GndfQ3Fr0p0G0sDMaTi9JnI8bqiY85BjzBPM\nS4AXmNm3gM2BfwYeAFab2ZlpnkeArYCVg2livpTERURkJA9RJOn3ABtSXEv+AxS3kJ4JBOBc4MF2\nFZQvnds4PDJe03Wn4fSayPG4oWLOQ44xTyTu/hRwH/Byd38SeAK4E9i5NNtkd7+zXR3lhBpjvGY8\np+tOSVxERDo5HviCmS0HLnP3xylObFsMLALmDrBtWdNwek3k9M2zQTHnIceYJxp3/wXw1qayRRQJ\nXAZIPXEREZEhpSReEzkeN1TMecgxZpGqlMRFRESGlJJ4TeR43FAx5yHHmHvNzLYYdBukP5TERUTq\n70dmdqGZ7T7ohkhvKYnXRI7HDRVzHnKMuQ8mA98BTjazW8zs78xs00E3SsZOPzETEam5dMGWbwLf\nTL3xrwOfNrMLKe5A9vuBNlBGTUm8JnI8bqiY85BjzL1mZhsD7wKOBLYEPg1cCswALgcOHFzrZCyU\nxEVE6m8l8F/Aie5+S6n8m2Z29IDaJD2gJF4T5RsM5EIx5yHHmPvgde7+WJvnlMSHmJK4iEjNjZDA\ncfefjrSsmX0R2AX4I3CRu19sZjOAOWmWOe6+tFdtle7o7PSayLGnopjzkGPMvWZmH2xRdkzFxSPw\nXnffLyXwDShufnJA+ptrZqF3rZVuKImLiNTfkS3KDu1i+XKSngzc4e5r3H0NcBew01gaJ6On4fSa\nyPG4oWLOQ44x90FsUVa1E/cY8FUzexj4B+DFwGozOzM9/wiwFcXJczLO1BMXEam/35jZ2p+Rmdm7\ngd9WWdDdj3H3vYB/BBYCDwGTgBOBk9LjB3veYqlEPfGayLGnopjzkGPMffAPwH+Y2VxgQ2Bj4B1d\n1vFH4CngTmDnUvlkd79zpAXLoymNK/CN13TdKYmLiNScu99rZlMozjIHuN3dn6myrJldCrwC+D0w\n092fNbN5wOI0y9xOdZQTanNy7fd03SmJ10SOxw0Vcx5yjLkfUtIe8edkbZY7rEXZImBRL9olY6Mk\nLiJSc2a2FfBOiuPXDdHdzxhQk6RHlMRrIseeimLOQ44x98HVwI+BVYNuiPSWkriISP095u5HDLoR\n0nv6iVlN5HjPZcWchxxj7oNbzOw1g26E9J564iIi9bcrsNjMbi2VRXd/+6AaJL2hJF4TOR43VMx5\nyDHmPpjfoqzVVdxkyCiJi4jUnLtfM+g2SH/omHhN5HjcUDHnIceY+8HMdjCzvyhNbzrI9khvKImL\niNScmb0P+CqwIE0H4KqBNkp6Qkm8JnI8bqiY85BjzH0wE9gX+B2Au+t4eE0oiYuI1N/T7v5EYyIN\npb9wgO2RHtGJbTWR4/WlFXMecoy5D240s88AW5jZIcBsiuH1SsxsY+AO4DR3P8fMZgBz0tNz3H1p\nz1sslagnLiJSfydQXHL1HuBvgHO7vG76R4BbgJiOp88DDkh/c1OZDIB64jWRY09FMechx5h7Ld3B\n7Pz01xUz2wTYH7gM2BSYDNzh7mvS83cBOwEre9ZgqUxJXERERnIM8HngZWl6K2C1mZ2Zph9JZUri\nA6AkXhM5HjdUzHnIMeZeM7Pfs/4V2qK7b95huS2AN7v7qWZ2RCp+iOKWpjOBAJwLPDhSPeVt2Pjd\n/3hN152SuIhIzbn7Ohd2MbM9gSkVFt0LeIGZfQ3YgSJnXAfsXJpnsrvfOVIl5YTanFz7PV13OrGt\nJnLbcUEx5yLHmPvN3W9g3UTcbr4r3X2Gu/8VcB5wobvfRnFi22JgETC3n22VkaknLiJSc2Y2hXWH\n018G7N5NHe7+pdLjRRQJXAZMSbwmcjxuqJjzkGPMffBZ1k3iv6P42ZkMOSVxEZGac/d9B90G6Q8d\nE6+JHHsqijkPOcYsUpV64iIiNWdmy0Z4Orr7tHFrjPSUknhN5HjcUDHnIceY++AmYDXFyWgBODSV\ne5qWIaUkLiJSf7u5+/6NCTO7BbjG3Y8fYJukB3RMvCZy7Kko5jzkGHMfvNLMXlqa3gJ48aAaI72j\nnriISP19BvihmS2h+KnZW3juVqIyxNQTr4nG9YJzopjzkGPMvZYu1DIV+CZwObCHu1882FZJL6gn\nLiKSAXf/f8C3Bt0O6S0l8ZrI8bihYs5DjjH3g5m9D9jJ3eeYWQD2dPflFZabD+wJPAt8yN3vNrMZ\nPDccP8fdl/at4TIiDaeLiNScmZ0BvBE4CMDdI3BalWXd/eT0O/I5wPHpC8A84ID0NzeVyQAoiddE\njscNFXMecoy5D97o7n8PPF4qa76/eCd7AD8DJgN3uPsad18D3AXs1JtmSrc0nC4iUn/BzNZ+3pvZ\nq4ENqy5sZtcCWwN7U9zCdLWZnZmefgTYCljZu+ZKVeqJ10SOxw0Vcx5yjLkPzgO+C2yXhtavoYv7\ngLv7W4AjgIuBh4BJwInASenxgyMtXx5NCSHsO57TdaeeuIhIzbn7l83sh8B04CngLe6+qstqHqDI\nGXdS9MYbJrv7nSMtWP4i1vylrN/TdackXhM5Xl9aMechx5j7wd1/Avyk2+XM7OsUQ+lPAke7+7Nm\nNg9YnGaZ27NGSteUxEVEas7MtnP3e0ezrLu/t0XZIoqbqciA6Zh4TeTYU1HMecgx5j749qAbIP2h\nJC4iUn9rBt0A6Q8Np9dEjscNFXMecoy5Dy4ws88Cp5QL3f3hAbVHekRJXESk/k6muLjLu0tlEdhx\nMM2RXlESr4kceyqKOQ85xtxr7r79oNsg/aFj4iIiIkNKSbwmcrpCUYNizkOOMfeKmXnp8TGDbIv0\nh5K4iEh9vaL0+F0Da4X0jZJ4TeR43FAx5yHHmEWq0oltIiL19Uoz+zgQgG1LjwGiu58xuKZJLyiJ\n10SOv6VVzHnIMeYeuhjYLD3+culxZWZ2PrALxcjtke5+t5nNAOakWea4+9KR6lj6419c3u16x2rz\nF2w44p3V6kJJXESkptx9bg/q+AiAmU0DZpnZTGAeMCPNcrWZLXP32K6O46+6893tnuuXj+/9qls3\n2WiDR8Z7veNNx8RrIseeimLOQ44xT1CPUdzJbDJwh7uvcfc1wF3ATgNtWcbUExcRkSqOAs4CtgJW\nm9mZqfyRVLZyUA3LmXriNZHjb2kVcx5yjHmiMbNDgNvd/efAQ8Ak4ETgpPQ4i+PPE5GSuIiItGVm\nU4B93P1zqeguYOfSLJPd/c7xb5mAhtNrI8fjhoo5DznGPMFcBtxnZsuA29z9Y2Y2D1icnp87sJaJ\nkriIiLTn7uvd6czdFwGLBtAcaaLh9JrI8bihYs5DjjGLVKUkLiIiMqSUxGsix+OGijkPOcYsUpWS\nuIiIyJBSEq+JHI8bKuY85BizSFVK4iIiIkNKSbwmcjxuqJjzkGPMIlUpiYuIiAypES/2YmbXUCT6\nZ4CH3f09HebfFDjW3eeXyi4C9gWOdvcrxtpgaS3Hey4r5jzkGLNIVZ2u2BaBg9z98SqVufvvgflN\nZUea2Zw2i/TN1IVLJ6WHxwKfa/F4rxWzpl2R5jsW+NyKWdNWNy2/TvnUhUvfBvwY+N9ptuVV603L\n9mL+lo+3t1l7ANekevZaMWvaFaU2Lx8ptpHalv53irlXj8uvRcsYym197THnHj514dJbu6izXczr\nrLcR53hst27rHEvM6bme79vt6h3ja7H28Shi7rjvMPK+3bGuPsW8dr2n7oZIJSHGtvdxJ10r92B3\n/0Op7EjgncAOwAXufnYqPxw4AtjU3ac21TMHuLncEzezFY35mh4fAewB7EIxCjDd3Z82s4OAT1F8\nsTjD3S9v1+4lS5bEE34QLkiT84GTWzyelf6fApwGzAZOSm/ESW3KtwO+AxxMcfu9hVXrTWW9mL9K\nPCeVPpwnlctaxdahbVtUiLlXjzvGMMbXpV3M66y3xeN+brd+7wt937dHsV9NhH1npH27H++lrmI4\ndbc4c/r06YEJwMz2Bj4LfM/dZ6WyGUCjczbH3Ze2Wz59Hve/oU0+vverbt1kow0emb/0nn3Ge92n\n7hYZr+1X5Zj4VWa2zMxOTNNfdvd3AFOBDzRmcvcvuft+PWhTBF4OzHD3fVIC3wA4Fdgf2Af4mJlt\nXLG+0OZx+Q15L8Ub8JSpC5duP0L5bIo3/Oy0fOV6ezh/lXjW9hTS4xFj69C2KjH36nHHGEbRhiqv\n7zrrrbi+Xm23fu8Lfd+3R7FfTYR9Z6R9ux/vpa5iYGLZGFjQmEifx/OAA9LfXDObEF84clQliR/k\n7vu5+6fT9FvSzeBPBjbpU7sWu/szpemtgW0ovjUvptjZ/2SkCu6+dMESijfEqrsvXbAkTa8CTik9\nXrhi1rTVIYR9b549fVeKb9Cr7r50wXVpmhWzpq2++9IF15Xmv7c0fUp5HaXH7Zbv1fwd4yn/tvbm\n2dN3Lcdw8+zpu4YQ9k0fSgs71L/w5tnTdyi3qbzuLttXZf617Wu0v8X26bb+Kq/vdY0P7C7W127/\n6Un8FbdPu/rXxlNx+1+XklGV/aHd8q3W1+/9pdv9v3n+du/ndfaHRn0tPg/6sv2ZQNz9u8DDpaLJ\nwB3uvsbd11DcmnSngTRORjWc/mPg9cCrgO+6++SmZVZUHE6/xd2nmNkmwAp3f10qP5xiSP6c0rwB\nuBZ4m7s/2imopuH0dkOkp1AMXzUPjS3sUN4YOls4inp7Nf96j+++dMGSHQ/75N409R5gnWHAkWIb\nqW1VYu7V47XtGyGGcszTu6lzhJjXWe94bbdu6+xBzP3at/vxWow6Zko6bPN2MVepq2/b/9Td4u8m\nynA6gJntQ5ELZpnZmwArPR2AS939xlbLaji9v6r0xJuz/PL0Nxt4qML8DQvM7LjS9E1mdnqqp3mZ\ndabdPQInAN9OQ/tfqdDuVnWVH5eHxrbjuaGxe0YoP41iNOC0tHzlens4f8vHOx72yf8q1bN2SK/p\nOF7L2Dq0rUrMvXpcfi1axlBuw46HffK/u6yzXczrrLfd+kZR71jib1nnWGLu1749iv2qq8ejiLnj\nvsPI+3bHuvoUc3nofyJ7iGI09ESK9k4CHhxoizI2Yk98WKVvflumyWOZAGfwjsNZzv04o3ZCn50+\nijrbxTw0Z6ePpZ703NCdnT6KmGtwdvqE64nvSzESOsvMNqQYGZ1B0Qtf7O57tVtWPfH+qm0Sn0hv\ngPGQ429pFXMecox5In2GmdnxwF9QnHD8PXf/sJkdQPFrIYB57r643fJK4v3V6XfiIiKSMXf/DPCZ\nprJFwKLBtEjKdNnVmsitpwKKORc5xixSlZK4iIjIkFISr4kc77msmPOQY8wiVSmJi4iIDCkl8ZrI\n8bihYs7qzhsJAAAG1ElEQVRDjjGLVKUkLiIiMqSUxGsix+OGijkPOcYsUpWSuIiIyJBSEq+JHI8b\nKuY85BizSFVK4iIiIkNKSbwmcjxuqJjzkGPMIlXp2ukiIjIqZjYDmJMm57j70kG2J0fqiddEjscN\nFXMecox5GJjZBsA84ID0N9fMJsSd13KiJC4iIqMxGbjD3de4+xrgLmCnAbcpOxpOr4kc77msmPOQ\nY8xD4sXAajM7M00/AmwFrGye8X27vfz+8WwYwKu3euHzf/XoE+O92nEXYoyDbkPPLVmypH5BiUhW\npk+fPqGHps1sZ+CTwEwgAOcC8939zvJ8uX4ej9f2q2USFxGR/jKzDYFrgRkUSXyxu+812FblR8fE\nRUSka+7+DMWJbYuBRcDcgTYoU+qJi4iIDCn1xEVERIaUkriIiMiQqt1PzHK4gpCZnQ/sQvEl7Eh3\nvzuHuAHMbGPgDuA0dz+n7nGb2TbAJRTv1Zvc/bgMYn4/8FHgaeBkd19Wt5jNbG/gs8D33H1WKmsZ\n43jHbmYHpPU9AXzD3c/t5/omAjO7huLz9BngYXd/T4f5NwWOdff5pbKLgH2Bo939iv61dl21Oiae\nriB0HcXZkgBXA/u4e32CLDGzacChFD/xuJ4M4jazjwH7AN8FzqPmcZvZpcDZ7n5Dmq79Pm5mtwF/\nDryIIr49qdl2Tol5M2BPd5/Varu6+1sGsb3N7L+BA9390X6tY6Ixs2XA29z98THWMwe4eTyTeN16\n4muvIARgZo0rCK138YGaeAx4kkziNrNNgP2By4BNqXnc6Sc8r24k8KTWMSc/pfii9nLgRmoYs7t/\n18z2KRWtF6OZTaboHY537P8DvNfMLmh8WTCzFe4+tcXjNwILgA2Be9398A7lBwGfAiJwhrtfnspn\nAn9D0RP+orv/W4vyi9z9wj7Gvc7vus3sSOCdwA7ABe5+dio/HDgC2LTxOoxkhNfuCGAPnhtVne7u\nT7d7jdqpWxKvfAWhmjgKOIsixhziPgb4PPCyNF33uF8CvMDMvgVsDvwz8AD1jhmKnysdC2xEcQGR\num9naP/ZFdqU9zP2j1Akzm+b2Xx3/36rmdJ10s8D3uruv65QvgFwKrAXxVD9UjP7T3d/kmJE8a3u\nvrppNe3K++EqM3uG4vfunwa+7O4XpUN4NwFnA7j7l4AvmdmKMa4vUnxRnZF+rtfuNfqOu7e99Fzd\nkvhDwCTWvYLQgwNtUZ+Y2SHA7e7+83TlpFrHbWZbAG9291PTN1io//Z+iOJD+z0UPZrlwAeoccxm\ntiNwsLu/PU1fCxxNjWNO2u3LG7Qp7xt3fxa42My+ASwDdm8z69bAr8uJukL5NsB30vQk4JXAKuB9\nwIfNbGvgW+6+PM3TrrwfDmoaTn+LmR0M/B7YpE/rXNxI4Emr1+hPKF6jluqWxO8Cdi5NT26+BGAd\nmNkUiuNin0hFOcS9F0Wv9GsUw1vPozhWWNu43f0pM7sPeLm7/9LMngDupMYxU3xZeR6s7dG9kPrG\nXB6+bfkeTodUxjV2M9sgJfINeO4XTBuk5zbhuYT2IPAKM9vG3cvXRm9X/lvgZ8A7mo+3p/k+Y2Yv\noPiyOmWk8j5pvkzqWcDrgVcBh42h3lavXTttX6MRK6+LjK4gdBkw1cyWmdlZOcTt7le6+wx3/yuK\noboL3f02ah43cDzwBTNbDlyWegq1jdndVwI3mtmVwFXAOXWM2cyOp4jjEDP7l3bv4QG9t09PJ3pd\nSbH/AdxkZqcDsymGgUnHyz8EXGJm3zMzr1B+AsUw/TIz+0pjhWZ2ZlrncuCcTuV90nyy4PL0N5ti\npKTT/A0LzOy40vR6r127OkZ6jdqp1dnpIiIiOalVT1xERCQnSuIiIiJDSklcRERkSCmJi4iIDCkl\ncRERkSGlJC4iIjKk6naxF5GOzOwvgY+lyanADynumHWluy+osPyGwBeAD7n70xXXeRzFdZh3dffN\nRtVwEZEm+p24ZM3MVgFT3P3hcVrfY0riItIr6omLNDGze4D5wAcpLvv5Dne/Jz33BeA1tOhRm9mh\nFDel2Yzi8op/5e63j1/LRSQ3SuIi64vAa919j+Yn3P1voehRt1huqbtflp4/Fvg48OF+NlRE8qYk\nLtLaKd0u4O4PmdmuwJ9S3CP4FT1vlYhIiZK4SI+Y2UXp4WXALRS3FBQR6RslcZHeeQfFvX+fpLjj\nUvOtDUVEekq/E5fctfp5RpWfbLSaZz5wG8VtI38GvKzFPC80s+vMTMfKRWTM9BMzERGRIaWeuIiI\nyJBSEhcRERlSSuIiIiJDSklcRERkSCmJi4iIDCklcRERkSGlJC4iIjKklMRFRESG1P8HgnoIhK66\npeAAAAAASUVORK5CYII=\n",
"text/plain": [
"<matplotlib.figure.Figure at 0x1156bac50>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"# Plot the data\n",
"fig = plt.figure(figsize=(7,3))\n",
"gs = gridspec.GridSpec(1, 2, width_ratios=[5, 1]) \n",
"ax1 = fig.add_subplot(gs[0])\n",
"ax2 = fig.add_subplot(gs[1])\n",
"\n",
"ax1.plot(range(nobs), Y, 'x')\n",
"ax2.hist(-Y, bins=2)\n",
"\n",
"ax1.yaxis.set(ticks=(0,1), ticklabels=('Failure', 'Success'))\n",
"ax2.xaxis.set(ticks=(-1,0), ticklabels=('Success', 'Failure'));\n",
"\n",
"ax1.set(title=r'Bernoulli Trial Outcomes $(\\theta=0.3)$', xlabel='Trial', ylim=(-0.2, 1.2))\n",
"ax2.set(ylabel='Frequency')\n",
"\n",
"fig.tight_layout()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Likelihood function\n",
"\n",
"Consider a sample of $T$ draws from the random variable $y$. The joint likelihood of observing any specific sample $Y = (y_1, ..., y_T)'$ is given by:\n",
"\n",
"$$\n",
"\\begin{align}\n",
"p(Y|\\theta) & = \\prod_{i=1}^T \\theta^{y_i} (1-\\theta)^{1-y_i} \\\\\n",
"& = \\theta^{s} (1 - \\theta)^{T-s}\n",
"\\end{align}\n",
"$$\n",
"\n",
"where $s = \\sum_i y_i$ is the number of observed \"successes\", and $T-s$ is the number of observed \"failures\"."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Maximum Likelihood Estimation\n",
"\n",
"Since we have a likelihood function, we can maximize it, as in the classical framework.\n",
"\n",
"#### Analytically\n",
"\n",
"In fact, the likelihood function here can be maximized analytically ([see details](#analytic_mle)):\n",
"\n",
"$$\n",
"\\begin{align}\n",
"L(\\theta;Y) & = \\theta^{s} (1 - \\theta)^{T-s} & \\text{Likelihood} \\\\\n",
"\\log L(\\theta;Y) & = s \\log \\theta + (T - s) \\log (1 - \\theta) & \\text{Log-likelihood} \\\\\n",
"\\frac{\\partial \\log L(\\theta;Y)}{\\partial \\theta} & = s \\theta^{-1} - (T-s)(1-\\theta)^{-1} & \\text{Score} \\\\\n",
"H(\\theta) = \\frac{\\partial^2 \\log L(\\theta;Y)}{\\partial \\theta^2} & = -s \\theta^{-2} - (T-s)(1-\\theta)^{-2} & \\text{Hessian} \\\\\n",
"I[\\theta] = -E\\left [ H(\\theta) \\right ] & = \\frac{T}{\\theta (1 - \\theta)} & \\text{Information} \\\\\n",
"I[\\theta]^{-1} & = \\frac{\\theta (1-\\theta)}{T} & \\text{Variance/Covariance}\n",
"\\end{align}\n",
"$$"
]
},
{
"cell_type": "code",
"execution_count": 21,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"t, T, s = sp.symbols('theta, T, s')\n",
"\n",
"# Create the functions symbolically\n",
"likelihood = (t**s)*(1-t)**(T-s)\n",
"loglike = s*sp.log(t) + (T-s)*sp.log(1-t)\n",
"score = (s/t) - (T-s)/(1-t)\n",
"hessian = -s/(t**2) - (T-s)/((1-t)**2)\n",
"information = T / (s*(1-s))\n",
"var = 1/information\n",
"\n",
"# Convert them to Numpy-callable functions\n",
"_likelihood = sp.lambdify((t,T,s), likelihood, modules='numpy')\n",
"_loglike = sp.lambdify((t,T,s), loglike, modules='numpy')\n",
"_score = sp.lambdify((t,T,s), score, modules='numpy')\n",
"_hessian = sp.lambdify((t,T,s), hessian, modules='numpy')"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The first-order condition tells us that the log-likelihood is maximized ([see details](#foc)) when:\n",
"$$\n",
"\\begin{align}\n",
"0 & = \\left . \\frac{\\partial \\log L(\\theta;Y)}{\\partial \\theta} \\right |_{\\hat \\theta_\\text{MLE}} \\implies \\hat \\theta_\\text{MLE} = \\frac{s}{T} \\\\\n",
"\\end{align}\n",
"$$\n",
"\n",
"To make sure we're at a maximum, we can check the second order condition (note that $T > s$):\n",
"$$\n",
"\\begin{align}\n",
"\\left . \\frac{\\partial^2 \\log L(\\theta;Y)}{\\partial \\theta^2} \\right |_{\\hat \\theta_\\text{MLE}} & = -s (s/T)^{-2} - (T-s)(1-s/T)^{-2} < 0\\\\\n",
"\\end{align}\n",
"$$"
]
},
{
"cell_type": "code",
"execution_count": 22,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Analytic MLE Results: 0.22 (0.0414246304)\n"
]
}
],
"source": [
"# Find the optimal theta_hat using the analytic solution\n",
"theta_hat_analytic = Y.sum() / nobs\n",
"var_analytic = (theta_hat_analytic * (1 - theta_hat_analytic)) / nobs\n",
"print('Analytic MLE Results: %.2f (%.10f)' % (theta_hat_analytic, var_analytic**0.5))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### Numeric MLE: Transformations and the The Delta Method\n",
"\n",
"In many situations, we won't be able to analytically solve the maximization. In those cases we can use numeric optimization techniques. In Python we can use function from `scipy.optimize`.\n",
"\n",
"The first thing to note is that these functions are minimizers, so we'll actually be minimizing the negative of the likelihood function.\n",
"\n",
"The second thing to note about numeric optimization is that the optimizers are often over an unconstrained parameter space, meaning that they will find the vector in $\\mathbb{R^k}$ (if you're optimizing over $k$ parameters). This is a problem in many cases, including this one since our likelihood function is a function of $\\theta \\in (0,1)$. To get around this, we need to optimize over a transformed parameter that can take any value on the real line, and then map the parameter given to us by the optimizer to an appropriate value. We will use:\n",
"\n",
"$$\\theta = g(\\varphi) = \\frac{e^\\varphi}{1 + e^\\varphi}, \\quad \\varphi \\in \\mathbb{R}$$\n",
"\n",
"The next thing to note is that we need to use the negative inverse of the hessian to calculate the variance / covariance matrix, but in general we won't have the analytic hessian matrix (if we did, we wouldn't need to do numeric optimization). Fortunately, some optimization routines (e.g. `scipy.optimize.fmin_bfgs`) will return the numeric approximation to the hessian, which we can use to get the variance / covariance matrix.\n",
"\n",
"The final issue is that the variance / covariance matrix from the approximated hessian will be for the _unconstrained_ parameter $\\varphi$ rather than the parameter of interest $\\theta$. To get around this, we can use the delta method to recover the variance / covariance matrix of interest:\n",
"\n",
"$$\n",
"\\text{Cov}(\\theta) = \\left . \\frac{\\partial g(\\varphi)}{\\partial \\varphi} \\right |_{\\varphi_\\text{MLE}} \\times \\text{Cov}({\\varphi}) \\times \\left . \\frac{\\partial g(\\varphi)}{\\partial \\varphi}^T \\right |_{\\varphi_\\text{MLE}}\n",
"$$\n",
"\n",
"For the transformation function above, the derivative we need ([see details](#trans_grad)) is:\n",
"\n",
"$$\n",
"\\begin{align}\n",
"\\frac{\\partial g(\\varphi)}{\\partial \\varphi} = \\frac{e^\\varphi}{(1 + e^\\varphi)^2}\\\\\n",
"\\end{align}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Since we have analytic results available, we can actually prove that the Delta method works:\n",
"\n",
"First, note that there exists an inverse ([see details](#trans_inv)) to $g$:\n",
"\n",
"$$\n",
"\\begin{align}\n",
"\\varphi = g^{-1}(\\theta) = \\log \\left (\\frac{\\theta}{1 - \\theta} \\right ) \\\\\n",
"\\end{align}\n",
"$$\n",
"\n",
"Since we solved the MLE analytically, we can get the optimizing parameters:\n",
"\n",
"$$\n",
"\\begin{align}\n",
"\\hat \\theta_\\text{MLE} & = \\frac{s}{T} \\\\\n",
"\\hat \\varphi_\\text{MLE} & = g^{-1}(\\hat \\theta_\\text{MLE}) = \\log \\left (\\frac{\\hat \\theta_\\text{MLE}}{1 - \\hat \\theta_\\text{MLE}} \\right ) = \\log \\left (\\frac{s}{T-s} \\right )\n",
"\\end{align}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We can restate the system in terms of the unconstrained parameter $\\varphi$ ([see details](#mle_uncon)):\n",
"\n",
"$$\n",
"\\begin{align}\n",
"L(g(\\varphi);Y) & = g(\\varphi)^{s} (1 - g(\\varphi))^{T-s} & \\text{Likelihood} \\\\\n",
"\\log L(g(\\varphi);Y) & = s \\log g(\\varphi) + (T - s) \\log (1 - g(\\varphi)) & \\text{Log-likelihood} \\\\\n",
"\\frac{\\partial \\log L(g(\\varphi);Y)}{\\partial g(\\varphi)} & = s - T g(\\varphi) & \\text{Score}\\\\\n",
"H(\\varphi) = \\frac{\\partial^2 \\log L(g(\\varphi);Y)}{\\partial g(\\varphi)^2} & = -T & \\text{Hessian}\\\\\n",
"I[\\varphi] = -E\\left [ H(\\varphi) \\right ] & = T & \\text{Information} \\\\\n",
"I[\\varphi]^{-1} & = \\frac{1}{T} & \\text{Variance/Covariance}\n",
"\\end{align}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Then the delta method gives ([see details](#delta))\n",
"\n",
"$$\n",
"\\begin{align}\n",
"I_\\text{Delta}[\\hat \\theta_\\text{MLE}]^{-1} & = \\left . \\frac{\\partial g(\\varphi)}{\\partial \\varphi} \\right |_{\\varphi_\\text{MLE}} \\times I[\\varphi]^{-1} \\times \\left . \\frac{\\partial g(\\varphi)}{\\partial \\varphi}^T \\right |_{\\varphi_\\text{MLE}} = I[\\hat \\theta_\\text{MLE}]^{-1} \\left (\\hat \\theta_\\text{MLE} (1-\\hat \\theta_\\text{MLE}) \\right ) \\\\\n",
"\\end{align}\n",
"$$\n",
"\n",
"In this case the delta method will underestimate the variance / covariance matrix, and we can assess the error due to approximation.\n",
"\n",
"The absolute error is:\n",
"\n",
"$$\n",
"\\begin{align}\n",
"I[\\theta]^{-1} - I_\\text{Delta}[\\theta]^{-1} & = \\frac{\\theta (1 - \\theta) - [\\theta (1 - \\theta)]^2}{T}\n",
"\\end{align}\n",
"$$\n",
"\n",
"The relative error is:\n",
"\n",
"$$\n",
"\\begin{align}\n",
"\\frac{\\left \\vert I[\\theta]^{-1} - I_\\text{Delta}[\\theta]^{-1} \\right \\vert}{I[\\theta]^{-1}} & = \\frac{\\theta (1 - \\theta) - [\\theta (1 - \\theta)]^2}{T} \\Bigg / \\frac{\\theta (1 - \\theta)}{T} = 1 - \\theta (1 - \\theta)\n",
"\\end{align}\n",
"$$"
]
},
{
"cell_type": "code",
"execution_count": 23,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
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VQinVbc3/NuBoigf6b2MM/wzwW2v8aUqpbktWG/BX9HldDpyCLl30s9SlgJeir6X/sM7H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05Lgb6taD7RRFtWMY9iRFUdOf+9znNpw+fVqIYdguj8czfbf7arJ4WIzC9p1oRbgdDYFLViu7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\nROQJ/w9n1bcKQG4INwAAAABJRU5ErkJggg==\n",
"text/plain": [
"<matplotlib.figure.Figure at 0x11598f050>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"fig = plt.figure(figsize=(12,4))\n",
"\n",
"# Absolute Error\n",
"ax = fig.add_subplot(121, projection='3d')\n",
"delta = 0.025\n",
"T_range = np.arange(0, 100, 10)\n",
"TH_range = np.linspace(0, 1)\n",
"T_grid, TH_grid = np.meshgrid(T_range, TH_range)\n",
"E_grid = (TH_grid*(1-TH_grid) - (TH_grid*(1-TH_grid))**2)/T_grid\n",
"ax.plot_wireframe(T_grid, TH_grid, E_grid, color='#348ABD')\n",
"ax.set(title='Absolute Error from the Delta Method', xlabel='T', ylabel=r\"$\\theta$\")\n",
"ax.xaxis.set(pane_color=(1,1,1))\n",
"ax.yaxis.set(pane_color=(1,1,1))\n",
"\n",
"# Relative Error\n",
"ax = fig.add_subplot(122)\n",
"X = np.linspace(0,1)\n",
"ax.plot(X, 1 - X*(1-X))\n",
"ax.set(title='Relative Error from the Delta Method', xlabel=r\"$\\theta$\", ylim=(0.7, 1))\n",
"ax.vlines(theta_hat_analytic, 0.7, 1, 'r');"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"So, although the absolute error decreases with the sample size T, the relative error is independent of T. At the analytic solution to the MLE problem (0.22), the relative error is about 82%."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### Numeric MLE\n",
"Setup for the optimization:"
]
},
{
"cell_type": "code",
"execution_count": 24,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"# MLE via scipy optimization\n",
"from scipy import optimize\n",
"\n",
"# Create transformation functions\n",
"transform = lambda phi: np.exp(phi)/(1 + np.exp(phi))\n",
"transform_grad = lambda phi: np.exp(phi) / ((1 + np.exp(phi))**2)\n",
"\n",
"# Create functions for the optimizer to call\n",
"# (negative because we are using minimizing optimizers)\n",
"f = lambda params: -_loglike(transform(params[0]), nobs, Y.sum())\n",
"fprime = lambda params: np.matrix(-_score(transform(params[0]), nobs, Y.sum()))\n",
"fhess = lambda params: np.matrix(-_hessian(transform(params[0]), nobs, Y.sum()))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"There are many different optimization algorithms in `scipy.optimize`; this just demonstrates two of them.\n",
"\n",
"- The [Newton-CG method](http://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.fmin_ncg.html) requires at least the gradient of the function to optimize, and can optionally accept the hessian (otherwise the hessian is numerically approximated from the gradient). It does not return the hessian evaluated at the optimizing parameters (so the delta method cannot be used on this output).\n",
"- The [BFGS method](http://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.fmin_bfgs.html) does not require the gradient, and does not accept the hessian. If the gradient is not provided, it is numerically approximated. It returns the inverse of the hessian, which either can be used as the variance / covariance matrix, or can be used to calculate the it via the delta method if there is a transformation involved."
]
},
{
"cell_type": "code",
"execution_count": 25,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Timing: 10 loops, best of 3: 52.2 ms per loop\n",
"Optimization terminated successfully.\n",
" Current function value: 52.690796\n",
" Iterations: 35\n",
" Function evaluations: 51\n",
" Gradient evaluations: 85\n",
" Hessian evaluations: 35\n",
"Newton-CG MLE Estimate: 0.22 (0.0414252425)\n"
]
}
],
"source": [
"# Run MLE via Newton-CG Method\n",
"#%timeit res_ncg = optimize.fmin_ncg(f, np.array([0.5]), fprime, fhess=fhess, disp=False)\n",
"print 'Timing: 10 loops, best of 3: 52.2 ms per loop'\n",
"res_ncg = optimize.fmin_ncg(f, np.array([0.5]), fprime, fhess=fhess)[0]\n",
"var_ncg = np.linalg.inv(-_hessian(transform(res_ncg), nobs, Y.sum()))[0,0]\n",
"\n",
"print 'Newton-CG MLE Estimate: %.2f (%.10f)' % (transform(res_ncg), var_ncg**0.5)"
]
},
{
"cell_type": "code",
"execution_count": 26,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Timing: 100 loops, best of 3: 6.43 ms per loop\n",
"Optimization terminated successfully.\n",
" Current function value: 52.690796\n",
" Iterations: 5\n",
" Function evaluations: 14\n",
" Gradient evaluations: 14\n",
"BFGS MLE Estimate: 0.22 (0.0414246308)\n",
"BFGS MLE Estimate (delta): 0.22 (0.0171532173)\n"
]
}
],
"source": [
"# Run MLE via BFGS Method\n",
"#%timeit res_bfgs = optimize.fmin_bfgs(f, np.array([0.5]), fprime=fprime, disp=False)\n",
"print 'Timing: 100 loops, best of 3: 6.43 ms per loop'\n",
"res_bfgs_full = optimize.fmin_bfgs(f, np.array([0.5]), fprime=fprime, full_output=True)\n",
"res_bfgs = res_bfgs_full[0]\n",
"est_var_bfgs = res_bfgs_full[3]\n",
"\n",
"# Get variance/covariance matrix by inverting the information matrix\n",
"var_bfgs = np.linalg.inv(-_hessian(transform(res_bfgs), nobs, Y.sum()))[0,0]\n",
"# Get variance/covariance matrix by the delta method\n",
"var_bfgs_delta = (transform_grad(res_bfgs)**2 * est_var_bfgs)[0,0]\n",
"\n",
"print 'BFGS MLE Estimate: %.2f (%.10f)' % (transform(res_bfgs), var_bfgs**0.5)\n",
"print 'BFGS MLE Estimate (delta): %.2f (%.10f)' % (transform(res_bfgs), var_bfgs_delta**0.5)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Notice the error in the approximated variance of the error term matches what we calculated analytically for the error of the delta method, above."
]
},
{
"cell_type": "code",
"execution_count": 27,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"0.00142176717404\n",
"0.828535631545\n"
]
}
],
"source": [
"absolute_error = var_bfgs - var_bfgs_delta\n",
"relative_error = (var_bfgs - var_bfgs_delta) / var_bfgs\n",
"\n",
"print absolute_error\n",
"print relative_error"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### Numeric MLE: Statsmodels"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We can also perform MLE using the [Statsmodels](https://github.com/statsmodels/statsmodels) package. It can use any of the `scipy.optimize` methods (e.g. ncg and bfgs, above), but by default it uses its own implementation of the simple Newton-Raphson method. The Newton-Raphson method is very fast but less robust."
]
},
{
"cell_type": "code",
"execution_count": 28,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Timing: 1000 loops, best of 3: 1.19 ms per loop\n",
"Optimization terminated successfully.\n",
" Current function value: 0.526908\n",
" Iterations 2\n",
"Statsmodels MLE Estimate: 0.22 (0.0414246304)\n"
]
}
],
"source": [
"# MLE via Statsmodels framework\n",
"from statsmodels.base.model import LikelihoodModel\n",
"\n",
"class Bernoulli(LikelihoodModel):\n",
" _loglike = lambda self, theta, T, s: _loglike(theta, T, s)\n",
" _score = lambda self, theta, T, s: _score(theta, T, s)\n",
" _hessian = lambda self, theta, T, s: _hessian(theta, T, s)\n",
" \n",
" def __init__(self, endog, exog=None, **kwargs):\n",
" endog = np.asarray(endog)\n",
" super(Bernoulli, self).__init__(endog, exog, **kwargs)\n",
" self.T = self.endog.shape[0]\n",
" self.s = self.endog.sum()\n",
" \n",
" def loglike(self, params):\n",
" \"\"\"\n",
" Joint log-likelihood for Bernoulli trials\n",
" \"\"\"\n",
" return self._loglike(params[0], self.T, self.s)\n",
" \n",
" def score(self, params):\n",
" \"\"\"\n",
" Gradient of the joint log-likelihood for Bernoulli trials\n",
" \"\"\"\n",
" return self._score(params[0], self.T, self.s)\n",
" \n",
" def hessian(self, params):\n",
" \"\"\"\n",
" Hessian of the joint log-likelihood for Bernoulli trials\n",
" \"\"\"\n",
" return np.matrix(self._hessian(params[0], self.T, self.s))\n",
" \n",
"mod = Bernoulli(Y)\n",
"#%timeit res = mod.fit(start_params=[0.5], disp=False)\n",
"print 'Timing: 1000 loops, best of 3: 1.19 ms per loop'\n",
"res = mod.fit(start_params=[0.5])\n",
"# Run MLE via Newton-Raphson method\n",
"print 'Statsmodels MLE Estimate: %.2f (%.10f)' % (res.params[0,0], res.bse[0])"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Appendix"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### Analytic MLE: Model\n",
"\n",
"<a name=\"analytic_mle\"></a>\n",
"\n",
"$$\n",
"\\begin{align}\n",
"L(\\theta;Y) & = \\theta^{s} (1 - \\theta)^{T-s} & \\text{Likelihood} \\\\\n",
"\\log L(\\theta;Y) & = s \\log \\theta + (T - s) \\log (1 - \\theta) & \\text{Log-likelihood} \\\\\n",
"\\frac{\\partial \\log L(\\theta;Y)}{\\partial \\theta} & = \\frac{s}{\\theta} - \\frac{T-s}{1 - \\theta} & \\text{Score} \\\\\n",
"& = s \\theta^{-1} - (T-s)(1-\\theta)^{-1} & \\\\\n",
"H(\\theta) = \\frac{\\partial^2 \\log L(\\theta;Y)}{\\partial \\theta^2} & = -s \\theta^{-2} - (T-s)(1-\\theta)^{-2} & \\text{Hessian} \\\\\n",
"I[\\theta] = -E\\left [ H(\\theta) \\right ] & = -E[-s \\theta^{-2} - (T-s)(1-\\theta)^{-2}] & \\text{Information} \\\\\n",
"& = (\\theta T) \\theta^{-2} + (T-(\\theta T))(1-\\theta)^{-2} & \\\\\n",
"& = T \\frac{1 - \\theta + \\theta}{\\theta (1 - \\theta)} & \\\\\n",
"& = \\frac{T}{\\theta (1 - \\theta)} & \\\\\n",
"I[\\theta]^{-1} & = \\frac{\\theta (1-\\theta)}{T} & \\text{Variance/Covariance}\n",
"\\end{align}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### Analytic MLE: First-Order Conditions\n",
"\n",
"<a name=\"foc\"></a>\n",
"\n",
"$$\n",
"\\begin{align}\n",
"0 & = \\left . \\frac{\\partial \\log L(\\theta;Y)}{\\partial \\theta} \\right |_{\\hat \\theta_\\text{MLE}} \\\\\n",
"0 & = s \\hat \\theta_\\text{MLE}^{-1} - (T-s)(1-\\hat \\theta_\\text{MLE})^{-1} \\\\\n",
"(T-s)(1- \\hat \\theta_\\text{MLE})^{-1} & = s \\hat \\theta_\\text{MLE}^{-1} \\\\\n",
"(T-s) \\hat \\theta_\\text{MLE} & = s (1 - \\hat \\theta_\\text{MLE}) \\\\\n",
"T \\hat \\theta_\\text{MLE} - s \\hat \\theta_\\text{MLE} & = s - s \\hat \\theta_\\text{MLE} \\\\\n",
"\\hat \\theta_\\text{MLE} & = \\frac{s}{T} \\\\\n",
"\\end{align}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### Analytic MLE: Transformation Gradient\n",
"\n",
"<a name=\"trans_grad\"></a>\n",
"\n",
"$$\n",
"\\begin{align}\n",
"\\frac{\\partial g(\\varphi)}{\\partial \\varphi} & = \\frac{\\partial e^\\varphi (1 + e^\\varphi)^{-1}}{\\partial \\varphi} = e^\\varphi (-1) (1 + e^\\varphi)^{-2} e^\\varphi + (1 + e^\\varphi)^{-1} e^\\varphi \\\\\n",
"& = \\frac{e^\\varphi}{1 + e^\\varphi} - \\frac{e^\\varphi e^\\varphi}{(1 + e^\\varphi)^2} = \\frac{(1 + e^\\varphi) e^\\varphi - e^\\varphi e^\\varphi}{(1 + e^\\varphi)^2} = e^\\varphi \\frac{(1 + e^\\varphi) - e^\\varphi}{(1 + e^\\varphi)^2} \\\\\n",
"& = \\frac{e^\\varphi}{(1 + e^\\varphi)^2}\\\\\n",
"\\end{align}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### Analytic MLE: Transformation Inverse\n",
"\n",
"<a name=\"trans_inv\"></a>\n",
"\n",
"$$\n",
"\\begin{align}\n",
"\\theta & = g(\\varphi) = \\frac{e^\\varphi}{1 + e^\\varphi}\\\\\n",
"(1 + e^\\varphi) \\theta & = e^\\varphi \\\\\n",
"\\theta & = e^\\varphi (1 - \\theta) \\\\\n",
"\\varphi & = \\log \\left (\\frac{\\theta}{1 - \\theta} \\right ) = g^{-1}(\\theta) \\\\\n",
"\\end{align}\n",
"$$\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### Analytic MLE: Model (Unconstrained Parameter)\n",
"\n",
"<a name=\"mle_uncon\"></a>\n",
"\n",
"$$\n",
"\\begin{align}\n",
"L(g(\\varphi);Y) & = g(\\varphi)^{s} (1 - g(\\varphi))^{T-s} & \\text{Likelihood} \\\\\n",
"\\log L(g(\\varphi);Y) & = s \\log g(\\varphi) + (T - s) \\log (1 - g(\\varphi)) & \\text{Log-likelihood} \\\\\n",
"\\frac{\\partial \\log L(g(\\varphi);Y)}{\\partial g(\\varphi)} & = \\frac{\\partial \\log L(g(\\varphi);Y)}{\\partial \\theta} \\frac{\\partial g(\\varphi)}{\\partial \\varphi} & \\text{Score}\\\\\n",
"& = \\left ( \\frac{s}{g(\\varphi)} - \\frac{T-s}{1 - g(\\varphi)} \\right ) \\frac{e^\\varphi}{(1 + e^\\varphi)^2} & \\\\\n",
"& = \\left ( \\frac{s(1 + e^\\varphi)}{e^\\varphi} - \\frac{(T-s)}{1 - e^\\varphi/(1 + e^\\varphi)} \\right ) \\frac{e^\\varphi}{(1 + e^\\varphi)^2} & \\\\\n",
"& = \\left ( \\frac{s(1 + e^\\varphi)}{e^\\varphi} - \\frac{(T-s)(1 + e^\\varphi)}{1 + e^\\varphi - e^\\varphi} \\right ) \\frac{e^\\varphi}{(1 + e^\\varphi)^2} & \\\\\n",
"& = \\left ( \\frac{s - (T-s) e^\\varphi}{1 + e^\\varphi} \\right ) & \\\\\n",
"& = \\left ( \\frac{s (1 + e^\\varphi) - T e^\\varphi}{1 + e^\\varphi} \\right ) & \\\\\n",
"& = \\left ( s - T \\frac{e^\\varphi}{1 + e^\\varphi} \\right ) & \\\\\n",
"& = s - T g(\\varphi) & \\\\\n",
"H(\\varphi) = \\frac{\\partial^2 \\log L(g(\\varphi);Y)}{\\partial g(\\varphi)^2} & = -T & \\text{Hessian}\\\\\n",
"I[\\varphi] = -E\\left [ H(\\varphi) \\right ] & = T & \\text{Information} \\\\\n",
"I[\\varphi]^{-1} & = \\frac{1}{T} & \\text{Variance/Covariance}\n",
"\\end{align}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### Analytic MLE: Delta Method\n",
"\n",
"<a name=\"delta\"></a>\n",
"\n",
"$$\n",
"\\begin{align}\n",
"I_\\text{Delta}[\\hat \\theta_\\text{MLE}]^{-1} & = \\left . \\frac{\\partial g(\\varphi)}{\\partial \\varphi} \\right |_{\\varphi_\\text{MLE}} \\times I[\\varphi]^{-1} \\times \\left . \\frac{\\partial g(\\varphi)}{\\partial \\varphi}^T \\right |_{\\varphi_\\text{MLE}} \\\\\n",
"& = \\left ( \\frac{\\exp \\left [ \\log \\left (\\frac{\\hat \\theta_\\text{MLE}}{1-\\hat \\theta_\\text{MLE}} \\right ) \\right ]}{\\left (1 + \\exp \\left [ \\log \\left (\\frac{\\hat \\theta_\\text{MLE}}{1-\\hat \\theta_\\text{MLE}} \\right ) \\right ] \\right )^2} \\right )^2 \\frac{1}{T} = \\left ( \\frac{\\hat \\theta_\\text{MLE}}{1-\\hat \\theta_\\text{MLE}} \\left (1 + \\frac{\\hat \\theta_\\text{MLE}}{1-\\hat \\theta_\\text{MLE}} \\right )^{-2} \\right )^2 \\frac{1}{T} \\\\\n",
"& = \\left ( \\frac{\\hat \\theta_\\text{MLE}}{1-\\hat \\theta_\\text{MLE}} \\left (1-\\hat \\theta_\\text{MLE} \\right )^2 \\right )^2 \\frac{1}{T} = \\left (\\hat \\theta_\\text{MLE} (1-\\hat \\theta_\\text{MLE}) \\right )^2 \\frac{1}{T} \\\\\n",
"& = I[\\hat \\theta_\\text{MLE}]^{-1} \\left (\\hat \\theta_\\text{MLE} (1-\\hat \\theta_\\text{MLE}) \\right ) \\\\\n",
"\\end{align}\n",
"$$"
]
}
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