Bayesian_Estimation_and_Simulation.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<style>\n",
       "@import url('http://fonts.googleapis.com/css?family=Source+Code+Pro');\n",
       "@import url('http://fonts.googleapis.com/css?family=Vollkorn');\n",
       "@import url('http://fonts.googleapis.com/css?family=Arimo');\n",
       "@import url('http://fonts.googleapis.com/css?family=Fira_sans');\n",
       "\n",
       "    div.cell{\n",
       "        width: 900px;\n",
       "        margin-left: 0% !important;\n",
       "        margin-right: auto;\n",
       "    }\n",
       "    div.text_cell code {\n",
       "        background: transparent;\n",
       "        color: #000000;\n",
       "        font-weight: 600;\n",
       "        font-size: 11pt;\n",
       "        font-style: bold;\n",
       "        font-family:  'Source Code Pro', Consolas, monocco, monospace;\n",
       "   }\n",
       "    h1 {\n",
       "        font-family: 'Open sans',verdana,arial,sans-serif;\n",
       "\t}\n",
       "\t\n",
       "    div.input_area {\n",
       "        background: #F6F6F9;\n",
       "        border: 1px solid #586e75;\n",
       "    }\n",
       "\n",
       "    .text_cell_render h1 {\n",
       "        font-weight: 200;\n",
       "        font-size: 30pt;\n",
       "        line-height: 100%;\n",
       "        color:#c76c0c;\n",
       "        margin-bottom: 0.5em;\n",
       "        margin-top: 1em;\n",
       "        display: block;\n",
       "        white-space: wrap;\n",
       "        text-align: left;\n",
       "    } \n",
       "    h2 {\n",
       "        font-family: 'Open sans',verdana,arial,sans-serif;\n",
       "        text-align: left;\n",
       "    }\n",
       "    .text_cell_render h2 {\n",
       "        font-weight: 200;\n",
       "        font-size: 16pt;\n",
       "        font-style: italic;\n",
       "        line-height: 100%;\n",
       "        color:#c76c0c;\n",
       "        margin-bottom: 0.5em;\n",
       "        margin-top: 1.5em;\n",
       "        display: block;\n",
       "        white-space: wrap;\n",
       "        text-align: left;\n",
       "    } \n",
       "    h3 {\n",
       "        font-family: 'Open sans',verdana,arial,sans-serif;\n",
       "    }\n",
       "    .text_cell_render h3 {\n",
       "        font-weight: 200;\n",
       "        font-size: 14pt;\n",
       "        line-height: 100%;\n",
       "        color:#d77c0c;\n",
       "        margin-bottom: 0.5em;\n",
       "        margin-top: 2em;\n",
       "        display: block;\n",
       "        white-space: wrap;\n",
       "        text-align: left;\n",
       "    }\n",
       "    h4 {\n",
       "        font-family: 'Open sans',verdana,arial,sans-serif;\n",
       "    }\n",
       "    .text_cell_render h4 {\n",
       "        font-weight: 100;\n",
       "        font-size: 14pt;\n",
       "        color:#d77c0c;\n",
       "        margin-bottom: 0.5em;\n",
       "        margin-top: 0.5em;\n",
       "        display: block;\n",
       "        white-space: nowrap;\n",
       "    }\n",
       "    h5 {\n",
       "        font-family: 'Open sans',verdana,arial,sans-serif;\n",
       "    }\n",
       "    .text_cell_render h5 {\n",
       "        font-weight: 200;\n",
       "        font-style: normal;\n",
       "        color: #1d3b84;\n",
       "        font-size: 16pt;\n",
       "        margin-bottom: 0em;\n",
       "        margin-top: 0.5em;\n",
       "        display: block;\n",
       "        white-space: nowrap;\n",
       "    }\n",
       "    div.text_cell_render{\n",
       "        font-family: 'Fira sans', verdana,arial,sans-serif;\n",
       "        line-height: 125%;\n",
       "        font-size: 115%;\n",
       "        text-align:justify;\n",
       "        text-justify:inter-word;\n",
       "    }\n",
       "    div.output_subarea.output_text.output_pyout {\n",
       "        overflow-x: auto;\n",
       "        overflow-y: scroll;\n",
       "        max-height: 50000px;\n",
       "    }\n",
       "    div.output_subarea.output_stream.output_stdout.output_text {\n",
       "        overflow-x: auto;\n",
       "        overflow-y: scroll;\n",
       "        max-height: 50000px;\n",
       "    }\n",
       "    div.output_wrapper{\n",
       "        margin-top:0.2em;\n",
       "        margin-bottom:0.2em;\n",
       "}\n",
       "\n",
       "    code{\n",
       "      font-size: 70%;\n",
       "    }\n",
       "    .rendered_html code{\n",
       "    background-color: transparent;\n",
       "    }\n",
       "    ul{\n",
       "        margin: 2em;\n",
       "    }\n",
       "    ul li{\n",
       "        padding-left: 0.5em; \n",
       "        margin-bottom: 0.5em; \n",
       "        margin-top: 0.5em; \n",
       "    }\n",
       "    ul li li{\n",
       "        padding-left: 0.2em; \n",
       "        margin-bottom: 0.2em; \n",
       "        margin-top: 0.2em; \n",
       "    }\n",
       "    ol{\n",
       "        margin: 2em;\n",
       "    }\n",
       "    ol li{\n",
       "        padding-left: 0.5em; \n",
       "        margin-bottom: 0.5em; \n",
       "        margin-top: 0.5em; \n",
       "    }\n",
       "    ul li{\n",
       "        padding-left: 0.5em; \n",
       "        margin-bottom: 0.5em; \n",
       "        margin-top: 0.2em; \n",
       "    }\n",
       "    a:link{\n",
       "       font-weight: bold;\n",
       "       color:#447adb;\n",
       "    }\n",
       "    a:visited{\n",
       "       font-weight: bold;\n",
       "       color: #1d3b84;\n",
       "    }\n",
       "    a:hover{\n",
       "       font-weight: bold;\n",
       "       color: #1d3b84;\n",
       "    }\n",
       "    a:focus{\n",
       "       font-weight: bold;\n",
       "       color:#447adb;\n",
       "    }\n",
       "    a:active{\n",
       "       font-weight: bold;\n",
       "       color:#447adb;\n",
       "    }\n",
       "    .rendered_html :link {\n",
       "       text-decoration: underline; \n",
       "    }\n",
       "    .rendered_html :hover {\n",
       "       text-decoration: none; \n",
       "    }\n",
       "    .rendered_html :visited {\n",
       "      text-decoration: none;\n",
       "    }\n",
       "    .rendered_html :focus {\n",
       "      text-decoration: none;\n",
       "    }\n",
       "    .rendered_html :active {\n",
       "      text-decoration: none;\n",
       "    }\n",
       "    .warning{\n",
       "        color: rgb( 240, 20, 20 )\n",
       "    } \n",
       "    hr {\n",
       "      color: #f3f3f3;\n",
       "      background-color: #f3f3f3;\n",
       "      height: 1px;\n",
       "    }\n",
       "    blockquote{\n",
       "      display:block;\n",
       "      background: #fcfcfc;\n",
       "      border-left: 5px solid #c76c0c;\n",
       "      font-family: 'Open sans',verdana,arial,sans-serif;\n",
       "      width:680px;\n",
       "      padding: 10px 10px 10px 10px;\n",
       "      text-align:justify;\n",
       "      text-justify:inter-word;\n",
       "      }\n",
       "      blockquote p {\n",
       "        margin-bottom: 0;\n",
       "        line-height: 125%;\n",
       "        font-size: 100%;\n",
       "      }\n",
       "</style>\n",
       "<script>\n",
       "    MathJax.Hub.Config({\n",
       "                        TeX: {\n",
       "                           extensions: [\"AMSmath.js\", \"autobold.js\"]\n",
       "                           },\n",
       "                tex2jax: {\n",
       "                    inlineMath: [ ['$','$'], [\"\\\\(\",\"\\\\)\"] ],\n",
       "                    displayMath: [ ['$$','$$'], [\"\\\\[\",\"\\\\]\"] ]\n",
       "                },\n",
       "                displayAlign: 'center', // Change this to 'center' to center equations.\n",
       "                \"HTML-CSS\": {\n",
       "                    scale:100,\n",
       "                        availableFonts: [],\n",
       "                        preferredFont:null,\n",
       "                        webFont: \"TeX\",\n",
       "                    styles: {'.MathJax_Display': {\"margin\": 4}}\n",
       "                }\n",
       "        });\n",
       "</script>\n"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# style the notebook\n",
    "from IPython.core.display import HTML\n",
    "import urllib.request\n",
    "response = urllib.request.urlopen('http://bit.ly/1LC7EI7')\n",
    "HTML(response.read().decode(\"utf-8\"))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Bayesian Estimation\n",
    "\n",
    "\n",
    "This is from Candy, not me.\n",
    "\n",
    "\n",
    "\n",
    "Bayes. Variable or parameter X, and data Y.\n",
    "\n",
    "$$P(X|Y) = \\frac{P(Y|X)\\times P(X)}{P(Y)} = \\frac{P(Y|X)\\times P(X)}{\\int P(Y|X)\\times P(X) \\, \\mathrm{d}X}$$\n",
    "\n",
    "$P(Y|X)$ is the *likelihood*.\n",
    "\n",
    "$P(X)$ is the *prior*.\n",
    "\n",
    "$P(Y)$ is the *evidence*.\n",
    "\n",
    "\n",
    "# MAP\n",
    "\n",
    "Estimation requires finding most probable value of P(X|Y). Logical to find the most probably value - its maximum. *Maximu a posteriori* (MAP) estimate is the value of x that maximixes the posterior density:\n",
    "\n",
    "$$ \\max\\limits_x P(X|Y)$$\n",
    "\n",
    "Find this by differentiating, set to 0, and solve:\n",
    "\n",
    "$$\\nabla_x P(X|Y)\\bigr\\rvert_{X=\\hat{X}_{MAP}}=0$$\n",
    "\n",
    "where $\\rvert_x$ means *evaluated* at x.\n",
    "\n",
    "The *gradient vector* $\\nabla_x \\in R^{N_X \\times 1}$\n",
    "\n",
    "$$\\nabla_X := \\begin{bmatrix}\\frac{\\partial}{\\partial X_1}\\cdots\\frac{\\partial}{\\partial X_{N_X}}\\end{bmatrix}^\\mathsf{T}$$\n",
    "\n",
    "(not sure what $N_X$ is)\n",
    "\n",
    "\n",
    "$\\ln$ is useful since we often work with exponential densities. Logarithm is monotonically increasing so maximums hap[en at the same points. \n",
    "\n",
    "\n",
    "$$\\nabla_x \\ln P(X|Y)\\bigr\\rvert_{X=\\hat{X}_{MAP}}=0$$\n",
    "\n",
    "Apply Bayes rule:\n",
    "$$\\ln P(X|Y) = \\ln P(Y|X) + \\ln P(X) - \\ln P(Y)$$\n",
    "\n",
    "P(Y) is not a function of X, so map equation is\n",
    "\n",
    "$$\\nabla_x \\ln P(X|Y)\\bigr\\rvert_{X=\\hat{X}_{MAP}} = \\nabla_x  (\\ln P(Y|X) + \\ln P(X))\\bigr\\rvert_{X=\\hat{X}_{MAP}} = 0$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "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": "3.4.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}