BDA 3.10.7.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Problem 3.10.7\n",
    "\n",
    "Poisson and binomial distributions: a student sits on a street corner for an hour and records the number of bicycles $ b $ and the number of other vehicles $v$ that go by. Two models are considered: \n",
    "\n",
    "* The outcomes $b$ and $v$ have independent Poisson distributions, with unknown means $\\theta_b$ and $\\theta_v$ . \n",
    "\n",
    "* The outcome $b$ has a binomial distribution, with unknown probability $p$ and sample size $b + v$. \n",
    "\n",
    "Show that the two models have the same likelihood if we define $p = \\theta_b/( \\theta_b +\\theta_v)$.\n",
    "\n",
    "Gelman, Andrew; Carlin, John B.; Stern, Hal S.; Dunson, David B.; Vehtari, Aki; Rubin, Donald B.. Bayesian Data Analysis, Third Edition (Chapman & Hall/CRC Texts in Statistical Science) (Page 81). CRC Press. Kindle Edition. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This problem has no computational element, it's a fact about poisson distributions.  In the first case we have\n",
    "$$\n",
    "P(b=b_0)=\\frac{\\theta_b^{b_0}e^{-\\theta_b)}}{b_0!}\n",
    "$$\n",
    "and\n",
    "$$\n",
    "P(v=v_0)=\\frac{\\theta_v^{v_0}e^{-\\theta_v)}}{v_0!}\n",
    "$$\n",
    "It's also a fact that the sum of two poisson variables with rates $\\theta_v$ and $\\theta_b$ is poisson with \n",
    "rate $\\theta_v+\\theta_b$.  \n",
    "\n",
    "A direct calculation gives \n",
    "$$\n",
    "P(b=b_0,v=v_0|b_0+v_0=N)=\\binom{N}{b_0}\\frac{\\theta_b^{b_0}\\theta_v^{N-b_0}}{(\\theta_b+\\theta_v)^{N}}\n",
    "$$\n",
    "which is what we're supposed to show."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "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": "3.6.4"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}