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@hagino3000
Last active October 19, 2016 17:51
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データ解析のための統計モデリング入門 8章
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{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# 8 マルコフ連鎖モンテカルロ(MCMC)法とベイズ統計モデル\n",
"\n",
"[データ解析のための統計モデリング入門8章](http://hosho.ees.hokudai.ac.jp/~kubo/ce/IwanamiBook.html) \n",
"担当 hagino3000\n",
"\n",
"前半\n",
"- (効率のよくない)試行錯誤による最尤推定法を紹介する\n",
"- それをMCMCアルゴリズムに改造し、MCMCサンプリングをする"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 8. 1 例題:種子の生存確率(個体差なし)\n",
"\n",
"20個体の生存種子数は"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"y = [4,3,4,5,5,2,3,1,4,0,1,5,5,6,5,4,4,5,3,4]"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"とする。このデータをヒストグラムで図示すると"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"collapsed": false
},
"outputs": [
{
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],
"text/plain": [
"<matplotlib.figure.Figure at 0x1051dcb10>"
]
},
"metadata": {
"image/png": {
"height": 310,
"width": 405
}
},
"output_type": "display_data"
}
],
"source": [
"plt.hist(y, bins=range(9))\n",
"_ = plt.title(u'図8.1(B)')"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"過分散ではないようなので、生存種子数$yi$が二項分布に従うと仮定する。"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"import scipy.misc"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"def p_y_given_q(q, y):\n",
" return scipy.misc.comb(8, y) * (q**y) * (1-q)**(8-y)"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"<matplotlib.text.Text at 0x106bc0750>"
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
},
{
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],
"text/plain": [
"<matplotlib.figure.Figure at 0x10595c210>"
]
},
"metadata": {
"image/png": {
"height": 312,
"width": 429
}
},
"output_type": "display_data"
}
],
"source": [
"q = np.linspace(0, 1, 100)\n",
"plt.plot(q, p_y_given_q(q, 1))\n",
"plt.plot(q, p_y_given_q(q, 2))\n",
"plt.plot(q, p_y_given_q(q, 3))\n",
"plt.plot(q, p_y_given_q(q, 4))\n",
"plt.plot(q, p_y_given_q(q, 5))\n",
"plt.plot(q, p_y_given_q(q, 6))\n",
"plt.plot(q, p_y_given_q(q, 7))\n",
"plt.legend(['$yi=1$','$yi=2$','$yi=3$','$yi=4$','$yi=5$','$yi=6$','$yi=7$'])\n",
"plt.title(u'ある個体$i$の生存種子数が$yi$である確率')"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"尤度 $L(q)$ は20個体ぶんの「データが得られる確率」の積\n",
"\n",
"$\\displaystyle\n",
"L(q) = p(Y|q) = \\Pi_{i}p(yi|q)$"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"def likelifood(q, y):\n",
" return np.prod([scipy.misc.comb(8, yi) * (q**yi) * (1-q)**(8-yi) for yi in y])\n",
"\n",
"def calc_likelifood(X, y):\n",
" return [likelifood(x, y) for x in X]\n",
"\n",
"def log_likelifood(q, y):\n",
" return sum([yi * np.log(q) + (8-yi)*np.log(1-q) for yi in y]) + 1\n",
"\n",
"def calc_log_likelifood(X, y):\n",
" return [log_likelifood(x, y) for x in X]"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"X = np.linspace(0.01, 0.99, 100)\n",
"l = calc_likelifood(X, y)\n",
"log_l = calc_log_likelifood(X, y)"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"<matplotlib.text.Text at 0x10562b190>"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
},
{
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],
"text/plain": [
"<matplotlib.figure.Figure at 0x106b67390>"
]
},
"metadata": {
"image/png": {
"height": 329,
"width": 669
}
},
"output_type": "display_data"
}
],
"source": [
"fig, ax = plt.subplots(1, 2, figsize=(10, 4))\n",
"ax[0].plot(X, l)\n",
"ax[0].set_title(u'尤度')\n",
"ax[0].set_xlabel(u'$q$')\n",
"ax[1].plot(X, log_l)\n",
"ax[1].set_title(u'対数尤度')\n",
"ax[1].set_xlabel(u'$q$')"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$\\displaystyle\n",
"\\frac{dlogL(q)}{dq} = 0$ となる$\\hat{q}$ を(解析的に)もとめると\n",
"\n",
"0.45625 (式の展開はホワイトボードで)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 8.2 ふらふら試行錯誤による最尤推定"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"<matplotlib.text.Text at 0x106e5e750>"
]
},
"execution_count": 11,
"metadata": {},
"output_type": "execute_result"
},
{
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],
"text/plain": [
"<matplotlib.figure.Figure at 0x105980410>"
]
},
"metadata": {
"image/png": {
"height": 334,
"width": 458
}
},
"output_type": "display_data"
}
],
"source": [
"X = np.linspace(0.23, 0.64, 30)\n",
"l = calc_log_likelifood(X, y)\n",
"plt.plot(X, l, 'o')\n",
"plt.xlabel(u'生存確率$q$')\n",
"plt.ylabel(u'対数尤度')\n",
"plt.title(u'$q$を離散化して、それぞれの尤度を見て進めていく')"
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"def naive_hill_climb(X, likelifood):\n",
" history = []\n",
" def next():\n",
" return 1 if np.random.random() > 0.5 else -1\n",
" \n",
" def update_judge(new_likelifood, current_likelifood):\n",
" # 尤度が上がるなら更新\n",
" return new_likelifood > current_likelifood\n",
" \n",
" pos = 0\n",
" current = likelifood[0]\n",
" while pos <= len(likelifood):\n",
" tmp = pos + next() # +1 or -1 ランダムで動かしてみる\n",
" new = likelifood[tmp]\n",
" history.append(X[tmp])\n",
" if update_judge(new, current):\n",
" pos = tmp\n",
" if np.abs(new - current) < 0.05:\n",
" break # 変化しなくなったら終了\n",
" current = new\n",
" return X[pos], history\n",
"\n",
"X = np.linspace(0.1, 0.9, 100)\n",
"l = calc_log_likelifood(X, y)\n",
"m, history = naive_hill_climb(X, l)"
]
},
{
"cell_type": "code",
"execution_count": 16,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"<matplotlib.text.Text at 0x108812710>"
]
},
"execution_count": 16,
"metadata": {},
"output_type": "execute_result"
},
{
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],
"text/plain": [
"<matplotlib.figure.Figure at 0x1087f3bd0>"
]
},
"metadata": {
"image/png": {
"height": 312,
"width": 428
}
},
"output_type": "display_data"
}
],
"source": [
"plt.plot(history)\n",
"plt.title(u'$q$の推移')"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 8.3 MCMCアルゴリズムのひとつ:メトロポリス法"
]
},
{
"cell_type": "code",
"execution_count": 17,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"def metropolis(X, likelifoods, step):\n",
" history = []\n",
" def next():\n",
" return 1 if np.random.random() > 0.5 else -1\n",
" \n",
" def update_judge(new_likelifood, current_likelifood):\n",
" # 尤度比を使って動かす\n",
" l_rate = np.exp(new_likelifood - current_likelifood)\n",
" return np.random.random() < l_rate\n",
" \n",
" pos = 10\n",
" current = likelifoods[pos]\n",
" t = 0\n",
" while t < step:\n",
" tmp = np.abs(pos + next()) # +1 or -1 ランダムで動かしてみる\n",
" new = likelifoods[tmp]\n",
" history.append(X[pos])\n",
" if update_judge(new, current):\n",
" pos = tmp\n",
" current = new\n",
" t += 1\n",
" return history"
]
},
{
"cell_type": "code",
"execution_count": 23,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"<matplotlib.text.Text at 0x10a61a210>"
]
},
"execution_count": 23,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
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"GtHR0Tf1mXKmOIlLREREREREREREJDBubEZEREREREREREQkME7iEhEREREREREREQmMk7hERERE\n",
"REREREREAuMkLhEREREREREREZHAOIlLREREREREREREJDBO4hIREREREREREREJjJO4RERERERE\n",
"RERERALjJC4RERERERERERGRwDiJS0RERERERERERCQwTuISERERERERERERCYyTuERERERERERE\n",
"REQC4yQuERERERERERERkcA4iUtEREREREREREQkME7iEhEREREREREREQmMk7hERERERERERERE\n",
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"H5nWtBflp9vlAAAAAElFTkSuQmCC\n"
],
"text/plain": [
"<matplotlib.figure.Figure at 0x10ab9d690>"
]
},
"metadata": {
"image/png": {
"height": 311,
"width": 696
}
},
"output_type": "display_data"
}
],
"source": [
"X = np.linspace(0.1, 0.9, 100)\n",
"l = calc_log_likelifood(X, y)\n",
"ret = metropolis(X, l, 100)\n",
"fig, ax = plt.subplots(1, 2, figsize=(10, 4))\n",
"ax[0].plot(ret)\n",
"ax[0].set_title(u'MCMC Step 100')\n",
"ax[0].set_ylabel('$q$')\n",
"ax[1].hist(ret, bins=100)\n",
"ax[1].set_xlim(0.1, 0.9)\n",
"ax[1].set_title(u'Histgram of $q$')\n"
]
},
{
"cell_type": "code",
"execution_count": 22,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"<matplotlib.text.Text at 0x106f28b90>"
]
},
"execution_count": 22,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
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],
"text/plain": [
"<matplotlib.figure.Figure at 0x10a2fb050>"
]
},
"metadata": {
"image/png": {
"height": 311,
"width": 689
}
},
"output_type": "display_data"
}
],
"source": [
"X = np.linspace(0.1, 0.9, 100)\n",
"l = calc_log_likelifood(X, y)\n",
"ret = metropolis(X, l, 100000)\n",
"fig, ax = plt.subplots(1, 2, figsize=(10, 4))\n",
"ax[0].plot(ret)\n",
"ax[0].set_title('MCMC Step 100000')\n",
"ax[0].set_ylabel('$q$')\n",
"ax[1].hist(ret, bins=100)\n",
"ax[1].set_xlim(0.1, 0.9)\n",
"ax[1].set_title(u'Histgram of $q$')\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 定常分布\n",
"\n",
"これは何? → $p(q|Y)$は尤度$L(q)$に比例する確率分布"
]
},
{
"cell_type": "code",
"execution_count": 37,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"<matplotlib.text.Text at 0x1079b6d50>"
]
},
"execution_count": 37,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
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],
"text/plain": [
"<matplotlib.figure.Figure at 0x10dc33f90>"
]
},
"metadata": {
"image/png": {
"height": 330,
"width": 326
}
},
"output_type": "display_data"
}
],
"source": [
"fig = plt.figure(figsize=(4, 4))\n",
"plot = fig.add_subplot(1, 1, 1)\n",
"X = np.linspace(0.01, 0.99, 100)\n",
"l = calc_likelifood(X, y)\n",
"plot.plot(X, l/sum(l))\n",
"plt.ylabel('$p(q|Y)$')\n",
"plt.title('$L(q)/\\Sigma(L(q))$')\n",
"plt.xlabel('$q$')"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"MCMCサンプリングとは何だったのか\n",
"\n",
"- ランダムサンプルを得るのが目的、最も尤度が高い点を求めるのでは無い\n",
"- 定常分布を求めて、データ$Y$に統計モデルをあてはめた時のパラメータの確率分布を得る事\n",
"- 複雑な統計モデルのあてはめで威力を発揮するらしい\n",
" - 例として簡単なモデルが使われたので、この章ではわかりにくい気がした\n",
" - 10章でやるっぽい"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 8.4 MCMCサンプリングとベイズ統計モデル\n",
"\n",
"\n",
"ベイズの話はPRMLで散々やってるのでさらっと"
]
}
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