{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\newcommand{\\si}{\\sigma}\n",
    "\\newcommand{\\al}{\\alpha}\n",
    "\\newcommand{\\tta}{\\theta}\n",
    "\\newcommand{\\Tta}{\\Theta}\n",
    "\\newcommand{\\Si}{\\Sigma}\n",
    "\\newcommand{\\ld}{\\ldots}\n",
    "\\newcommand{\\cd}{\\cdots}\n",
    "\\newcommand{\\Ga}{\\Gamma} \n",
    "\\newcommand{\\bet}{\\beta}\n",
    "\\newcommand{\\cU}{\\mathcal{U}}\n",
    "\\newcommand{\\cN}{\\mathcal{N}}\n",
    "\\newcommand{\\R}{\\mathbb{R}}\n",
    "\\newcommand{\\p}{\\mathbb{P}}\n",
    "\\newcommand{\\f}{\\frac}\n",
    "\\newcommand{\\ff}{\\frac{1}}\n",
    "\\newcommand{\\ds}{\\displaystyle}\n",
    "\\newcommand{\\bE}{\\mathbf{E}}\n",
    "\\newcommand{\\E}{\\mathbb{E}}\n",
    "\\newcommand{\\bF}{\\mathbf{F}}\n",
    "\\newcommand{\\ii}{\\mathrm{i}}\n",
    "\\newcommand{\\me}{\\mathrm{e}}\n",
    "\\newcommand{\\hsi}{\\hat{\\sigma}}\n",
    "\\newcommand{\\hmu}{\\hat{\\mu}}\n",
    "\\newcommand{\\ste}{\\, ;\\, }\n",
    "\\newcommand{\\op}{\\operatorname} \n",
    "\\newcommand{\\argmax}{\\op{argmax}}\n",
    "\\newcommand{\\lfl}{\\lfloor}\n",
    "\\newcommand{\\ri}{\\right}\n",
    "\\newcommand{\\supp}{\\operatorname{supp}}$\n",
    "\n",
    "<a href=\"https://colab.research.google.com/github/judelo/algosto/blob/master/python/TP_importance_sampling_enonce.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>\n",
    "\n",
    "# Echantillonnage préférentiel (*importance sampling*)\n",
    "\n",
    "\n",
    "## Exercice 1\n",
    "\n",
    "On souhaite calculer numériquement l'intégrale \n",
    "$$\\int_0^{10} e^{-2|x-5|}dx.$$\n",
    "Une première manière de faire est d'écrire cette intégrale comme \n",
    "$$10\\E[g(X)]$$\n",
    "avec $g(x) = e^{-2|x-5|}$ et avec $X$ de loi uniforme sur $[0,10]$. \n",
    "\n",
    "1. Utilisez la méthode de Monte-Carlo pour approcher l'intégrale précédente : tirez $N$ réalisations indépendantes $x_1\\dots x_N$ d'une loi uniforme sur $[0,10]$ et l'approximation est donnée par \n",
    "$$10 \\frac 1 N \\sum_{i=1}^N e^{-2|x_i-5|}.$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x114110630>]"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.linspace(1,10,100)\n",
    "def g(x):\n",
    "    return np.exp(-2*np.abs(x-5))\n",
    "plt.plot(x,g(x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Cependant, la fonction $g$ atteint son maximum en $x=5$ et décroît\n",
    "rapidement après, il est\n",
    "donc sans doute plus malin d'utiliser une fonction d'échantillonnage\n",
    "préférentiel gaussienne $f_Y$ centrée en 5 et de variance 1 (par exemple). On réécrit donc  \n",
    "$$\\int_0^{10} e^{-2|x-5|}dx = \\int \\mathbf{1}_{[0,10]}(x)\n",
    "\\frac{e^{-2|x-5|}}{\\frac{1}{\\sqrt{2\\pi}}e^{-(x-5)^2/2}}\\frac{1}{\\sqrt{2\\pi}}e^{-(x-5)^2/2}dx\n",
    "= \\E[\\frac{g(Y)}{f_Y(Y)}  \\mathbf{1}_{[0,10]}(Y)],$$\n",
    "avec $f_Y(x) = \\frac 1 {\\sqrt{2\\pi} }e^{-(x-5)^2/2} $ et $Y$ de loi gaussienne $\\mathcal{N}(5,1)$.\n",
    "\n",
    "Pour calculer l'intégrale précédente, on tire donc $N$ réalisations indépendantes $x_1\\dots x_N$ d'une loi $\\mathcal{N}(5,1)$ et l'approximation est donnée par \n",
    "$$ \\frac 1 N \\sum_{i=1}^N \\sqrt{2\\pi}e^{+(y_i-5)^2/2}e^{-2|y_i-5|} \\times \\mathbf{1}_{[0,10]}(y_i).$$\n",
    "\n",
    "2. Utilisez cette nouvelle approche pour calculer l'intégrale et comparer les variances et vitesses de convergence des deux approches. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercice 2\n",
    "\n",
    "On souhaite calculer \n",
    "$$\\int_{-\\infty}^{+\\infty} x^2 \\frac 1 2 e^{-|x|}dx,$$\n",
    "mais on ne sait pas échantillonner suivant la densité $p(x) = \\frac 1 2 e^{-|x|}dx$.\n",
    "\n",
    "Réécrire cette intégrale à l'aide d'une fonction d'échantillonnage préférentiel gaussienne de $\\cN(0,4)$ et estimer sa valeur par une méthode de MOnte-Carlo. "
   ]
  }
 ],
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