Created
March 9, 2015 21:48
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BetaBinomial
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| { | |
| "cells": [ | |
| { | |
| "cell_type": "code", | |
| "execution_count": 1, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "name": "stdout", | |
| "output_type": "stream", | |
| "text": [ | |
| "Populating the interactive namespace from numpy and matplotlib\n" | |
| ] | |
| } | |
| ], | |
| "source": [ | |
| "%pylab inline" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 2, | |
| "metadata": { | |
| "collapsed": true | |
| }, | |
| "outputs": [], | |
| "source": [ | |
| "import pandas as pd\n", | |
| "import matplotlib.pyplot as plt\n", | |
| "import seaborn as sns\n", | |
| "from scipy import stats" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 7, | |
| "metadata": { | |
| "collapsed": true | |
| }, | |
| "outputs": [], | |
| "source": [ | |
| "n = 50000\n", | |
| "k = 60" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 8, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [], | |
| "source": [ | |
| "pSM = pd.Series(stats.beta(6+1, 14+1).rvs(n))\n", | |
| "sm = pSM.apply(lambda p: stats.binom(k, p).rvs())\n", | |
| "sm.name = 'San Miguel'" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 9, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/plain": [ | |
| "<matplotlib.axes._subplots.AxesSubplot at 0x1148b3a90>" | |
| ] | |
| }, | |
| "execution_count": 9, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| }, | |
| { | |
| "data": { | |
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| "F4E/jIjXgGvpjvF38fXcpAl8z6jbHwAHI+Io3fF9kO5R96zH2LXKJUkqyFI6VS5JkvowuCVJKojB\n", | |
| "LUlSQQxuSZIKYnBLklQQg1uSpIIY3JIkFeT/A2Mfz6QPxnxVAAAAAElFTkSuQmCC\n" | |
| ], | |
| "text/plain": [ | |
| "<matplotlib.figure.Figure at 0x114aaf750>" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| } | |
| ], | |
| "source": [ | |
| "sm.hist(bins=20)" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 11, | |
| "metadata": { | |
| "collapsed": true | |
| }, | |
| "outputs": [], | |
| "source": [ | |
| "pArg = pd.Series(stats.beta(10+1, 10+1).rvs(n))\n", | |
| "arg = pArg.apply(lambda p: stats.binom(k, p).rvs())\n", | |
| "arg.name = 'Argentinos'" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 12, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/plain": [ | |
| "<matplotlib.axes._subplots.AxesSubplot at 0x114e7a150>" | |
| ] | |
| }, | |
| "execution_count": 12, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| }, | |
| { | |
| "data": { | |
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| ], | |
| "text/plain": [ | |
| "<matplotlib.figure.Figure at 0x114e0d0d0>" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| } | |
| ], | |
| "source": [ | |
| "arg.hist(bins=20)" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 72, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
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| ], | |
| "text/plain": [ | |
| "<matplotlib.figure.Figure at 0x116b9b950>" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| } | |
| ], | |
| "source": [ | |
| "sm.hist(bins=20, alpha=0.5, label='Argentinos')\n", | |
| "arg.hist(bins=20, alpha=0.5, label='San Miguel')\n", | |
| "plt.legend()\n", | |
| "plt.title('Beta-Binomial')\n", | |
| "plt.savefig('resultsBetBinomial.png')" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 73, | |
| "metadata": { | |
| "collapsed": true | |
| }, | |
| "outputs": [], | |
| "source": [ | |
| "oldSM = pd.Series(stats.binom(60, 6.0/20).rvs(n))\n", | |
| "oldArg = pd.Series(stats.binom(60, 10.0/20).rvs(n))" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 74, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
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| "jB+1LSZzmP8DlUejEkLYnsrDWf4w4hyT193Vc2sAhwErR5p4EvtlCGGv6usDgTsmsjBjEULYFrgW\n", | |
| "+HSMcUV1cHLtMUw9kmqPEML7Qwj/VH37EtAH3JFgWwyuRwn4cUptARBjXBRjXFw9T/sb4CjgmpTa\n", | |
| "Y4g6HA38NLW2YMu86wSuraUtJm0HOODbwIoQws1Ueox+aKSHsUxS/T0PPwksDyG0AvcDl09ckerS\n", | |
| "X4+PAF8NIRSobFgdN3FFGrOlVA5PnR5C6D/nfDKwLLH2GKoe/wj8c0Lt8WPgf4cQbgLyVNrhQdL7\n", | |
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| ], | |
| "text/plain": [ | |
| "<matplotlib.figure.Figure at 0x1172372d0>" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| } | |
| ], | |
| "source": [ | |
| "oldSM.hist(bins=20, alpha=0.5, label='Argentinos')\n", | |
| "oldArg.hist(bins=20, alpha=0.5, label='San Miguel')\n", | |
| "plt.legend()\n", | |
| "plt.title('Binomial')\n", | |
| "plt.savefig('Binomial.png')" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 75, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/plain": [ | |
| "0.0089800000000000001" | |
| ] | |
| }, | |
| "execution_count": 75, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "(oldSM > oldArg).mean()" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 15, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/plain": [ | |
| "30.0253" | |
| ] | |
| }, | |
| "execution_count": 15, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "arg.mean()" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 16, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/plain": [ | |
| "19.0511" | |
| ] | |
| }, | |
| "execution_count": 16, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "sm.mean()" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 17, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/plain": [ | |
| "0.12916" | |
| ] | |
| }, | |
| "execution_count": 17, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "(sm > arg).mean()" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 19, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/plain": [ | |
| "0.39300000000000002" | |
| ] | |
| }, | |
| "execution_count": 19, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "(sm >= 21).mean()" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 20, | |
| "metadata": { | |
| "collapsed": true | |
| }, | |
| "outputs": [], | |
| "source": [ | |
| "z = 1000" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 31, | |
| "metadata": { | |
| "collapsed": true | |
| }, | |
| "outputs": [], | |
| "source": [ | |
| "def bootsrap(n):\n", | |
| " comparison = sm > arg\n", | |
| " results = []\n", | |
| " for i in range(n):\n", | |
| " index = random.choice(comparison.index, sm.count(), replace=True)\n", | |
| " results.append(comparison.ix[index].mean())\n", | |
| " return pd.Series(results)\n", | |
| " " | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 32, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [], | |
| "source": [ | |
| "bootsrapped = bootsrap(500)" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 84, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
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| ], | |
| "text/plain": [ | |
| "<matplotlib.figure.Figure at 0x116bb3a50>" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| } | |
| ], | |
| "source": [ | |
| "bootsrapped.hist(bins=20, alpha=0.5)\n", | |
| "plt.title('Bootstrapping the proability of SM winning the match')\n", | |
| "plt.savefig('bootstrap.png')" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 34, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/plain": [ | |
| "count 500.000000\n", | |
| "mean 0.129136\n", | |
| "std 0.001469\n", | |
| "min 0.124100\n", | |
| "25% 0.128175\n", | |
| "50% 0.129110\n", | |
| "75% 0.130125\n", | |
| "max 0.133680\n", | |
| "dtype: float64" | |
| ] | |
| }, | |
| "execution_count": 34, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "bootsrapped.describe()" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 79, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/plain": [ | |
| "(0.12609900000000002, 0.1318)" | |
| ] | |
| }, | |
| "execution_count": 79, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "bootsrapped.quantile(0.025), bootsrapped.quantile(0.975)" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 82, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/plain": [ | |
| "16.125" | |
| ] | |
| }, | |
| "execution_count": 82, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "12.9/.8" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 39, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [], | |
| "source": [ | |
| "def betaBinomial(alpha, beta, n, k):\n", | |
| " p = pd.Series(stats.beta(alpha, beta).rvs(n))\n", | |
| " return p.apply(lambda p: stats.binom(k, p).rvs())" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 57, | |
| "metadata": { | |
| "collapsed": true | |
| }, | |
| "outputs": [], | |
| "source": [ | |
| "def resultsForXTimesBigger(x=1):\n", | |
| " n = 10000\n", | |
| " sm = betaBinomial(6*x + 1, 14*x +1, n, k)\n", | |
| " arg = betaBinomial(10*x + 1, 10*x +1, n, k)\n", | |
| " return (sm > arg).mean()" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 85, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [], | |
| "source": [ | |
| "space = np.logspace(1, 4, 10)\n", | |
| "resultsByIncrement = pd.Series([resultsForXTimesBigger(x) for x in space], index=space)" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 94, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
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| ], | |
| "text/plain": [ | |
| "<matplotlib.figure.Figure at 0x116937990>" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| } | |
| ], | |
| "source": [ | |
| "resultsByIncrement.plot(label='Beta-Binomial')\n", | |
| "resultsByIncrement.apply(lambda x: 0.008).plot(label='Binomial')\n", | |
| "plt.yscale('log')\n", | |
| "plt.xscale('log')\n", | |
| "plt.xlabel('x times original data size')\n", | |
| "plt.ylabel('probability of SM winning the match')\n", | |
| "plt.legend()\n", | |
| "plt.savefig('convergence.png')" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 91, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/plain": [ | |
| "<matplotlib.axes._subplots.AxesSubplot at 0x11544c390>" | |
| ] | |
| }, | |
| "execution_count": 91, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| }, | |
| { | |
| "data": { | |
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| ], | |
| "text/plain": [ | |
| "<matplotlib.figure.Figure at 0x1175c2c90>" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| } | |
| ], | |
| "source": [ | |
| "resultsByIncrement.plot()\n", | |
| "resultsByIncrement.apply(lambda x: 0.008).plot()" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": null, | |
| "metadata": { | |
| "collapsed": true | |
| }, | |
| "outputs": [], | |
| "source": [] | |
| } | |
| ], | |
| "metadata": { | |
| "kernelspec": { | |
| "display_name": "Python 2", | |
| "language": "python", | |
| "name": "python2" | |
| }, | |
| "language_info": { | |
| "codemirror_mode": { | |
| "name": "ipython", | |
| "version": 2 | |
| }, | |
| "file_extension": ".py", | |
| "mimetype": "text/x-python", | |
| "name": "python", | |
| "nbconvert_exporter": "python", | |
| "pygments_lexer": "ipython2", | |
| "version": "2.7.9" | |
| } | |
| }, | |
| "nbformat": 4, | |
| "nbformat_minor": 0 | |
| } |
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