{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "#Statistical Inference for Everyone: Technical Supplement\n", "\n", "\n", "\n", "This document is the technical supplement, for instructors, for [Statistical Inference for Everyone], the introductory statistical inference textbook from the perspective of \"probability theory as logic\".\n", "\n", "\n", "\n", "[Statistical Inference for Everyone]: http://web.bryant.edu/~bblais/statistical-inference-for-everyone-sie.html\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Estimating the Ratio of Two Variances $\\kappa\\equiv \\sigma_x^2/\\sigma_y^2$\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$\\newcommand{\\bvec}[1]{\\mathbf{#1}}$\n", "\n", "From *Estimating the Deviation* we have\n", "\\begin{eqnarray}\n", "p(\\sigma_x|\\bvec{x},I)&\\propto& \\frac{1}{\\sigma_x^{n}}e^{-V_x/2\\sigma_x^2} \n", "\\end{eqnarray}\n", "For independent $\\bvec{x}$ and $\\bvec{y}$ we have\n", "\\begin{eqnarray}\n", "p(\\sigma_x,\\sigma_y|\\bvec{x},\\bvec{y},I)&\\propto& \n", "\\frac{1}{\\sigma_x^{n}}\\frac{1}{\\sigma_y^{m}}\n", "e^{-V_x/2\\sigma_x^2}e^{-V_y/2\\sigma_y^2}\n", "\\end{eqnarray}\n", "Changing variables to $\\kappa\\equiv \\sigma_x^2/\\sigma_y^2$, we have the\n", "following definitions\n", "\\begin{eqnarray}\n", "\\kappa&\\equiv& \\sigma_x^2/\\sigma_y^2 \\\\\\\\\n", "\\sigma_x&=&\\sigma_y \\kappa^{1/2} \\\\\\\\\n", "\\sigma_y&=&\\sigma_x \\kappa^{-1/2}\n", "\\end{eqnarray}\n", "and then transform the posterior\n", "\\begin{eqnarray}\n", "p(\\kappa,\\sigma_x|\\bvec{x},\\bvec{y},I)&=&\n", " p(\\sigma_x,\\sigma_y|\\bvec{x},\\bvec{y},I)\\times\n", " \\left|\\frac{\\partial(\\sigma_x,\\sigma_y)}{\\partial(\\kappa,\\sigma_x)}\\right| \\\\\\\\\n", "&=& p(\\sigma_x,\\sigma_y|\\bvec{x},\\bvec{y},I)\\times\n", "\\left| \\begin{array}{cc}\n", " \\frac{\\partial\\sigma_x}{\\partial \\kappa} & \\frac{\\partial\n", "\\sigma_x}{\\partial \\sigma_x}\\\\\\\\ \n", " \\frac{\\partial\\sigma_y}{\\partial \\kappa} & \\frac{\\partial\n", "\\sigma_y}{\\partial \\sigma_x} \\\\\\\\\n", " \\end{array}\n", " \\right|\\\\\\\\\n", "&=& p(\\sigma_x,\\sigma_y|\\bvec{x},\\bvec{y},I)\\times\n", "\\left| \\begin{array}{cc}\n", " \\frac{1}{2}\\sigma_y \\kappa^{-1/2} & 1 \\\\\\\\\n", " -\\frac{1}{2}\\sigma_x \\kappa^{-3/2} & 0 \n", " \\end{array}\n", " \\right|\\\\\\\\\n", "&\\propto&\\frac{1}{\\sigma_x^{n}}\\frac{\\kappa^{m/2}}{\\sigma_x^{m}}\n", "e^{-V_x/2\\sigma_x^2}e^{-V_y\\kappa/2\\sigma_x^2} \\sigma_x \\kappa^{-3/2}\n", "\\end{eqnarray}\n", "Now we integrate out the nuisance parameter, $\\sigma_x$, to get\n", "\n", "\\begin{eqnarray}\n", "p(\\kappa|\\bvec{x},\\bvec{y},I)&=&\\int d\\sigma_x\n", "p(\\kappa,\\sigma_x|\\bvec{x},\\bvec{y},I) \\\\\\\\\n", "&\\propto& \\kappa^{(m-3)/2} \\int d\\sigma_x \\frac{1}{\\sigma_x^{n+m-1}}\n", "e^{-(V_x+V_y\\kappa)/2\\sigma_x^2}\n", "\\end{eqnarray}\n", "from the gaussian integral trick we get\n", "\n", "\\begin{eqnarray}\n", "p(\\kappa|\\bvec{x},\\bvec{y},I)&\\propto&\\kappa^{(m-3)/2}\n", "(V_x+V_y\\kappa)^{(n+m-2)/2}\n", "\\end{eqnarray}\n", "\n", "A more common form is found with the substitutions\n", "\\begin{eqnarray}\n", "\\eta &\\equiv&\\kappa\\times \\frac{(V_y/f_y)}{(V_x/f_x)} \\\\\\\\\n", "f_x&\\equiv&n-1 \\\\\\\\\n", "f_y&\\equiv&m-1\n", "\\end{eqnarray}\n", "\n", "from which it follows\n", "\\begin{eqnarray}\n", "p(\\eta|\\bvec{x},\\bvec{y},I)&\\propto&\n", "\\eta^{\\frac{f_y}{2}-1} (f_x+f_y\\eta)^{(f_x+f_y)/2}\n", "\\end{eqnarray}\n", "which is the commonly used F distribution.\n", "\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "---------------------" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "\n", "\n" ], "text/plain": [ "" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "from IPython.core.display import HTML\n", "\n", "\n", "def css_styling():\n", " styles = open(\"../styles/custom.css\", \"r\").read()\n", " return HTML(styles)\n", "css_styling()" ] } ], "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": "2.7.9" } }, "nbformat": 4, "nbformat_minor": 0 }