Overdispersion_learning_to_play_darts.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Overdispersion in learning to play darts"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Suppose a player plays T=100 games of darts, each game consisting of ten throws.  Assume the player gets better at hitting the dartboard over time, learning in an S-curve over the 100 games, but holding success probability constant over the course of a game."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 183,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Warning message:\n",
      "In rbinom(1, n, ps[i]): NAs produced"
     ]
    },
    {
     "data": {
      "image/png": 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       "</svg>\n"
      ],
      "text/plain": [
       "Plot with title “prob of hit”"
      ]
     },
     "metadata": {
      "image/svg+xml": {
       "isolated": true
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "n=10\n",
    "T = 100\n",
    "x = rep(0,T);\n",
    "linear = (-50:50)/10\n",
    "ps=1/(1+exp(-linear))\n",
    "for(i in 0:T)\n",
    "{\n",
    "  x[i]<-rbinom(1,n,ps[i]);\n",
    "}\n",
    "par(mfrow=c(1, 2)); plot(x, main=\"no. of hits\", xlab=\"Game number\"); plot(ps,main=\"prob of hit\", xlab=\"Game number\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 184,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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       "0.48"
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       "0.48"
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       "[1] 0.48"
      ]
     },
     "execution_count": 184,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "phat = mean(x)/n; phat"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Assuming that the number of hits in each of the 100 games is binomial(n,.5), we'd expect a variance of"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 185,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "2.496"
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       "2.496"
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      "text/markdown": [
       "2.496"
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      "text/plain": [
       "[1] 2.496"
      ]
     },
     "execution_count": 185,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n*phat*(1-phat)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "But the observed actual variance is higher:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 186,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "15.7373737373737"
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       "15.7373737373737"
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       "15.7373737373737"
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      "text/plain": [
       "[1] 15.73737"
      ]
     },
     "execution_count": 186,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "var(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 187,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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      "text/plain": [
       "Plot with title “Histogram of x”"
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    "hist(x)"
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