BIC

BIC.ipynb

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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### BIC (Bayesian information criterion)\n",
    "- [ベイズ情報量基準 (朱鷺の杜Wiki)](http://ibisforest.org/index.php?BIC)\n",
    "\n",
    "[ここ](https://gist.github.com/shotahorii/99342b1b4d63b2e743301b7ab8e56f19)で、AICについて書いた。そこで書いたように、AICはモデル選択基準の一つで、モデルの予測の良さによってモデルを選択する基準である。  \n",
    "\n",
    "AICの他の選択基準でモデル選択においてよく使われるものの一つにBIC(ベイズ情報量基準)がある。Schwarz情報量基準とも呼ばれる。この記事ではこのBICについて見ていく。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "BICは、データ$$\\{x\\}^N$$が与えられた時の、モデル$$M$$が発生する条件付確率$$P(M|\\{x\\}^N)$$\n",
       "を最大化するモデルが良いモデルであると考える。<br><br>\n",
       "\n",
       "別の言い方をすると、AICが予測の良いモデルを良いモデルと考えるのに対して、\n",
       "BICは真のモデルである確率の大きいモデルを良いモデルと考える。\n",
       "この思想の違いによって、モデル選択基準にAICを使うのかBICを使うのかを決めると良い。<br><br>\n",
       "\n",
       "BICは式で表すと以下である。<br><br>\n",
       "$$BIC = -2log(L^*) + k log(N)$$<br>\n",
       "$$= D + k log(N)$$<br><br>\n",
       "\n",
       "ここで、$$log(L^*)$$は最大対数尤度、$$k$$は推定したパラメータ数、$$N$$は標本数(=観測データ数)、\n",
       "$$D$$は逸脱度である。"
      ],
      "text/plain": [
       "<IPython.core.display.Latex object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%%latex\n",
    "BICは、データ$$\\{x\\}^N$$が与えられた時の、モデル$$M$$が発生する条件付確率$$P(M|\\{x\\}^N)$$\n",
    "を最大化するモデルが良いモデルであると考える。<br><br>\n",
    "\n",
    "別の言い方をすると、AICが予測の良いモデルを良いモデルと考えるのに対して、\n",
    "BICは真のモデルである確率の大きいモデルを良いモデルと考える。\n",
    "この思想の違いによって、モデル選択基準にAICを使うのかBICを使うのかを決めると良い。<br><br>\n",
    "\n",
    "BICは式で表すと以下である。<br><br>\n",
    "$$BIC = -2log(L^*) + k log(N)$$<br>\n",
    "$$= D + k log(N)$$<br><br>\n",
    "\n",
    "ここで、$$log(L^*)$$は最大対数尤度、$$k$$は推定したパラメータ数、$$N$$は標本数(=観測データ数)、\n",
    "$$D$$は逸脱度である。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 試してみる\n",
    "[ここ](https://gist.github.com/shotahorii/99342b1b4d63b2e743301b7ab8e56f19)と同じ例を使ってAICとBICを比較してみる。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import math\n",
    "import numpy as np\n",
    "from functools import reduce\n",
    "\n",
    "from matplotlib import pyplot as plt\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# create matrix of X \n",
    "# for example, if fitting with 3 degrees, return [[1, x, x**2, x**3],[1, x, x**2, x**3],...]\n",
    "def create_matrix(x,d):\n",
    "    \"\"\"\n",
    "    x: predictor variable data(array)\n",
    "    d: dimension (int>0) including intercept\n",
    "    \"\"\"\n",
    "    X = np.zeros((len(x), d), float) # make a matrix size of (data size - dimension)\n",
    "    for i in range(d):\n",
    "        X[:,i] = x**i\n",
    "    return X\n",
    "\n",
    "def fitting(x,y,d):\n",
    "    d = d+1 # +1 means adding intercept\n",
    "    X = create_matrix(x,d) \n",
    "    (beta, residuals, rank, s) = np.linalg.lstsq(X, y) # numpy function for least square \n",
    "    return lambda x: reduce(lambda a,b:a+b, [beta[i]*(x**i) for i in range(d)])\n",
    "\n",
    "def log_likelihood(model,x,y):\n",
    "    n = len(x)\n",
    "    b = reduce(lambda a,b: a+b, [(y[i]-model(x[i]))**2 for i in range(n)])/n\n",
    "    l = -n*math.log(2*math.pi*b)/2 - \\\n",
    "        reduce(lambda a,b: a+b, [(y[i]-model(x[i]))**2 for i in range(n)])/(2*b)\n",
    "    return l"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# calc AIC\n",
    "def aic(l,k): return -2*l+2*k\n",
    "# calc BIC\n",
    "def bic(l,k,n): return -2*l+k*math.log(n)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# generate random dataset\n",
    "x = np.linspace(0, 1, 100)\n",
    "y = 0.5 + 0.4*np.sin(2*np.pi*x) + np.random.normal(0.0, 0.1, len(x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# models with 1~10 degrees\n",
    "models = [fitting(x,y,i) for i in range(1,11)]\n",
    "\n",
    "ks = [i+1 for i in range(1,11)] # num beta + intercept\n",
    "llhs = [log_likelihood(m,x,y) for m in models]\n",
    "\n",
    "n = len(x) # num samples\n",
    "\n",
    "aics = [aic(llhs[i],ks[i]) for i in range(len(models))]\n",
    "bics = [bic(llhs[i],ks[i],n) for i in range(len(models))]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
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ELyLiJWXP3R0ROyJia0TcWq0alBkQxFavhi1bssZ9SZJUmGrOiH0SuDel1A7cC/xHgIhY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\n3RtrFQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x109248c50>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(1,1,figsize=(10,7))\n",
    "aic_plot = ax.plot(ks,aics,'b')\n",
    "bic_plot = ax.plot(ks,bics,'r')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "AICではk=7が最良、BICではk=3が最良となった。  \n",
    "一般に、BICの方が小さいkのモデル(=シンプルなモデル)を選択しやすいとされる。"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.5.2"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}