みどり本9章

midori-chap-9.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 9 章: GLM のベイズモデル化と事後分布の推定\n",
    "\n",
    "* 線形予測子をもつ統計モデルのベイズ化\n",
    "* 複数のパラメータの事後分布からの MCMC サンプリング"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 9.1 例題: 種子数のポアソン回帰（個体差なし)\n",
    "\n",
    "* 架空植物の個体 i では体サイズ $x_i$ に依存して種子数 $y_i$ の平均が増減する"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 597,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Populating the interactive namespace from numpy and matplotlib\n"
     ]
    }
   ],
   "source": [
    "%pylab inline --no-import-all\n",
    "\n",
    "import matplotlib.pyplot as plt\n",
    "import pandas as pd\n",
    "import numpy as np\n",
    "import scipy as sp\n",
    "import statsmodels.api as sm\n",
    "from pymc import Normal, Poisson, deterministic, MCMC, Model\n",
    "from statsmodels.formula.api import glm"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 598,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "df = pd.read_csv(\"http://hosho.ees.hokudai.ac.jp/~kubo/stat/iwanamibook/fig/poisson/data3a.csv\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 599,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7fc888469e10>]"
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     },
     "execution_count": 599,
     "metadata": {},
     "output_type": "execute_result"
    },
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fc8226c40d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(df.x, df.y, 'bo')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 600,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "        <td>Model:</td>               <td>GLM</td>            <td>AIC:</td>       <td>474.7725</td> \n",
       "</tr>\n",
       "<tr>\n",
       "    <td>Link Function:</td>           <td>log</td>            <td>BIC:</td>       <td>-366.3137</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <td>Dependent Variable:</td>         <td>y</td>        <td>Log-Likelihood:</td>  <td>-235.39</td> \n",
       "</tr>\n",
       "<tr>\n",
       "         <td>Date:</td>        <td>2015-05-13 08:07</td>    <td>LL-Null:</td>      <td>-237.64</td> \n",
       "</tr>\n",
       "<tr>\n",
       "   <td>No. Observations:</td>         <td>100</td>          <td>Deviance:</td>     <td>84.993</td>  \n",
       "</tr>\n",
       "<tr>\n",
       "       <td>Df Model:</td>              <td>1</td>         <td>Pearson chi2:</td>    <td>83.8</td>   \n",
       "</tr>\n",
       "<tr>\n",
       "     <td>Df Residuals:</td>           <td>98</td>            <td>Scale:</td>       <td>1.0000</td>  \n",
       "</tr>\n",
       "<tr>\n",
       "        <td>Method:</td>             <td>IRLS</td>              <td></td>             <td></td>     \n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "      <td></td>       <th>Coef.</th> <th>Std.Err.</th>    <th>z</th>    <th>P>|z|</th> <th>[0.025</th> <th>0.975]</th>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Intercept</th> <td>1.2917</td>  <td>0.3637</td>  <td>3.5517</td> <td>0.0004</td> <td>0.5789</td> <td>2.0045</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>x</th>         <td>0.0757</td>  <td>0.0356</td>  <td>2.1251</td> <td>0.0336</td> <td>0.0059</td> <td>0.1454</td>\n",
       "</tr>\n",
       "</table>"
      ],
      "text/plain": [
       "<class 'statsmodels.iolib.summary2.Summary'>\n",
       "\"\"\"\n",
       "               Results: Generalized linear model\n",
       "==============================================================\n",
       "Model:              GLM              AIC:            474.7725 \n",
       "Link Function:      log              BIC:            -366.3137\n",
       "Dependent Variable: y                Log-Likelihood: -235.39  \n",
       "Date:               2015-05-13 08:07 LL-Null:        -237.64  \n",
       "No. Observations:   100              Deviance:       84.993   \n",
       "Df Model:           1                Pearson chi2:   83.8     \n",
       "Df Residuals:       98               Scale:          1.0000   \n",
       "Method:             IRLS                                      \n",
       "---------------------------------------------------------------\n",
       "               Coef.   Std.Err.    z     P>|z|   [0.025  0.975]\n",
       "---------------------------------------------------------------\n",
       "Intercept      1.2917    0.3637  3.5517  0.0004  0.5789  2.0045\n",
       "x              0.0757    0.0356  2.1251  0.0336  0.0059  0.1454\n",
       "==============================================================\n",
       "\n",
       "\"\"\""
      ]
     },
     "execution_count": 600,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def calc_x_related_model(df):\n",
    "    return glm('y ~ x', data=df, family=sm.families.Poisson(sm.families.links.log)).fit()\n",
    "\n",
    "calc_x_related_model(df).summary2()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 9.2 GLM のベイズモデル化\n",
    "\n",
    "* 個体 $i$ の種子数 $y_i$ のばらつきを平均 $\\lambda_i$ のポアソン分布 $p(y_i | \\lambda_i )$ に従うとする\n",
    "* 平均を $\\lambda_i = exp (\\beta_1+\\beta_2 x_i)$ と指定\n",
    "* 個体差はないとしているのでランダム効果を考慮した項はない\n",
    "\n",
    "尤度関数は\n",
    "\n",
    "\\begin{equation}\n",
    "L(\\beta_1, \\beta_2) = \\Pi_i p(y_i | \\lambda_i ) = \\Pi_i p ( y_i | (\\beta_1, \\beta_2, x_i )\n",
    "\\end{equation}\n",
    "\n",
    "パラメーター $ \\{ \\beta_1, \\beta_2 \\} $ が与えられた時に $ Y $ が得られる確率は $ p(Y | \\beta_1, \\beta_2 ) =  L(\\beta_1, \\beta_2)$ ($x_i$ は定数としている)\n",
    "\n",
    "* ベイズモデルでの事後分布は$(尤度) \\times (事前分布)$ なので、尤度からベイズモデルを作れる\n",
    "\n",
    "\\begin{equation}\n",
    "p(\\beta_1, \\beta_2 | Y) \\propto p(Y | \\beta_1, \\beta_2) p(\\beta_1) p(\\beta_2)\n",
    "\\end{equation}\n",
    "\n",
    "## 9.3 無情報事前分布\n",
    "\n",
    "* ベイズ統計モデルにおける事前分布 $p(\\beta_*)$\n",
    "  * 突然出てきた架空植物の切片 $\\beta_1$ , 体サイズ $x_i$ の効果を表す傾き $\\beta_2$ などの確率分布がわからない\n",
    "  * みどり本では「無情報っぽい」事前分布として「ひらべったい正規分布」を使う "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 601,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "beta1 = Normal('beta1', 0, 1e-4, value=0)\n",
    "beta2 = Normal('beta2', 0, 1e-4, value=0)\n",
    "\n",
    "@deterministic(plot=False)\n",
    "def  lambda_(beta1=beta1, beta2=beta2, x=df.x, mean=df.x.mean()):\n",
    "    return np.exp(beta1 + beta2 * (x - mean))\n",
    "\n",
    "y = Poisson('y', mu=lambda_, value=df.y, observed=True)\n",
    "\n",
    "model = Model([beta1, beta2])\n",
    "M = MCMC(model)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 602,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " [-----------------100%-----------------] 16000 of 16000 complete in 8.4 sec"
     ]
    }
   ],
   "source": [
    "M.sample(iter=16000, burn=100, thin=3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 603,
   "metadata": {
    "collapsed": false,
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Plotting beta1\n",
      "Plotting beta2\n"
     ]
    },
    {
     "data": {
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fc81e03f990>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
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       "A2cFyVNhmsO5AAAAAElFTkSuQmCC\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fc81e14d750>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Matplot.plot(M)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 604,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# M.sample(iter=16000, burn=100, thin=3)\n",
    "# M.sample(iter=16000, burn=100, thin=3)\n",
    "# M.trace('beta1', chain=1)[:]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 605,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2.05220624389\n",
      "0.074749000301\n"
     ]
    }
   ],
   "source": [
    "print M.trace('beta1')[:].mean()\n",
    "print M.trace('beta2')[:].mean()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 606,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "beta1:\n",
      " \n",
      "\tMean             SD               MC Error        95% HPD interval\n",
      "\t------------------------------------------------------------------\n",
      "\t[[ 2.052]]       [[ 0.035]]       [[ 0.001]]       [ 1.984  2.115]\n",
      "\t\n",
      "\t\n",
      "\tPosterior quantiles:\n",
      "\t\n",
      "\t2.5             25              50              75             97.5\n",
      "\t |---------------|===============|===============|---------------|\n",
      "\t[[ 1.987]]       [[ 2.028]]      [[ 2.052]]     [[ 2.076]]    [[ 2.12]]\n",
      "\t\n",
      "\n",
      "beta2:\n",
      " \n",
      "\tMean             SD               MC Error        95% HPD interval\n",
      "\t------------------------------------------------------------------\n",
      "\t[[ 0.075]]       [[ 0.035]]       [[ 0.001]]       [ 0.006  0.139]\n",
      "\t\n",
      "\t\n",
      "\tPosterior quantiles:\n",
      "\t\n",
      "\t2.5             25              50              75             97.5\n",
      "\t |---------------|===============|===============|---------------|\n",
      "\t[[ 0.009]]       [[ 0.049]]      [[ 0.073]]     [[ 0.099]]    [[ 0.144]]\n",
      "\t\n"
     ]
    }
   ],
   "source": [
    "M.summary()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 607,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "ename": "SyntaxError",
     "evalue": "invalid syntax (<ipython-input-607-e03c77ef4611>, line 1)",
     "output_type": "error",
     "traceback": [
      "\u001b[1;36m  File \u001b[1;32m\"<ipython-input-607-e03c77ef4611>\"\u001b[1;36m, line \u001b[1;32m1\u001b[0m\n\u001b[1;33m    M.\u001b[0m\n\u001b[1;37m      ^\u001b[0m\n\u001b[1;31mSyntaxError\u001b[0m\u001b[1;31m:\u001b[0m invalid syntax\n"
     ]
    }
   ],
   "source": [
    "M."
   ]
  }
 ],
 "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": "2.7.6"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}