1 - Bayes for a proportion - Grid Estimate.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Introduction\n",
    "\n",
    "In this tutorial we'll use a grid of probabilities to estimate a population proportion using Bayesian statistics. We'll use [R](https://www.r-project.org/) to carry out the analysis.\n",
    "\n",
    "In statistical inference we are usually interested in estimating a population parameter using a sample from that population. We take the parameter from the sample and use that as representative for the whole population.\n",
    "\n",
    "The Bayesian approach to statistics gives us information about the probability distribution of the parameter of interest. \n",
    "\n",
    "At Stirling University I teach a module to third year students. Suppose I believe that 10% of students who take my module are capable of getting a first class mark. That 10% isn't set in stone though... I'll entertain anything from 5% to 20%. \n",
    "\n",
    "To what degree do I believe in these different percentages? Can I put my money where my mouth is?\n",
    "\n",
    "To set some probabilities for the different proportion values I think are credible I can arbitrarily set weights on those different proportions reflecting the degree to which I believe in them."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th scope=col>props</th><th scope=col>wgts</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><td>0.05</td><td>4.00</td></tr>\n",
       "\t<tr><td>0.1</td><td>8.0</td></tr>\n",
       "\t<tr><td>0.15</td><td>3.00</td></tr>\n",
       "\t<tr><td>0.2</td><td>0.5</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{ll}\n",
       " props & wgts\\\\\n",
       "\\hline\n",
       "\t 0.05 & 4.00\\\\\n",
       "\t 0.1 & 8.0\\\\\n",
       "\t 0.15 & 3.00\\\\\n",
       "\t 0.2 & 0.5\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "1. 0.05\n",
       "2. 0.1\n",
       "3. 0.15\n",
       "4. 0.2\n",
       "5. 4\n",
       "6. 8\n",
       "7. 3\n",
       "8. 0.5\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "     props wgts\n",
       "[1,]  0.05  4.0\n",
       "[2,]  0.10  8.0\n",
       "[3,]  0.15  3.0\n",
       "[4,]  0.20  0.5"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "props <- seq(0.05, 0.2, 0.05) # my proportions\n",
    "wgts <- c(4, 8, 3, 0.5) # my beliefs in each proportion\n",
    "cbind(props, wgts)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now I can create proabilities for each proportion by dividing through by the total weight thus scaling them to lie in the interval [0,1] and also making sure they sum to 1. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th scope=col>props</th><th scope=col>wgts</th><th scope=col>priors</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><td>0.05</td><td>4.00</td><td>0.26</td></tr>\n",
       "\t<tr><td>0.10</td><td>8.00</td><td>0.52</td></tr>\n",
       "\t<tr><td>0.15</td><td>3.00</td><td>0.19</td></tr>\n",
       "\t<tr><td>0.20</td><td>0.50</td><td>0.03</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{lll}\n",
       " props & wgts & priors\\\\\n",
       "\\hline\n",
       "\t 0.05 & 4.00 & 0.26\\\\\n",
       "\t 0.10 & 8.00 & 0.52\\\\\n",
       "\t 0.15 & 3.00 & 0.19\\\\\n",
       "\t 0.20 & 0.50 & 0.03\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "1. 0.05\n",
       "2. 0.1\n",
       "3. 0.15\n",
       "4. 0.2\n",
       "5. 4\n",
       "6. 8\n",
       "7. 3\n",
       "8. 0.5\n",
       "9. 0.26\n",
       "10. 0.52\n",
       "11. 0.19\n",
       "12. 0.03\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "     props wgts priors\n",
       "[1,]  0.05  4.0   0.26\n",
       "[2,]  0.10  8.0   0.52\n",
       "[3,]  0.15  3.0   0.19\n",
       "[4,]  0.20  0.5   0.03"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "priors <- round(wgts/sum(wgts),2) # scale to [0,1]\n",
    "cbind(props, wgts, priors)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The ```priors``` column in the above represents my 'degree of belief' in each proportion as a probability. Since these are discrete probabilities we can plot them as a probability mass."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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3bt2bNnCwoKTKX45ptvOi0tAAAO1s7K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\nMDDQ/KGws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   "source": [
    "options(repr.plot.width=5, repr.plot.height=5) # plot sizing\n",
    "plot(props, priors, type='h', xlab = 'Proportion', ylab='Probability', lwd=4)\n",
    "points(props, priors, pch=16, cex=2)"
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   "source": [
    "Clearly at the moment I believe most strongly (p=0.52) that 10% of the students will get a first class mark. \n",
    "\n",
    "The marks for 2015 show that 8 out of 104 students got a mark of 70% or more, a proportion of 7.7%. How should this change my belief about the proportion of 10%?\n",
    "\n",
    "I can use Bayes Theorem to calculate this. Bayes Theorem says:\n",
    "\n",
    "$$P(\\theta | D) = \\frac{P(D | \\theta)P(\\theta)}{P(D)}$$"
   ]
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   "source": [
    "In English $\\theta$ is my proportion (0.05, 0.1, 0.15, 0.2) and $D$ is my data (0.077). Bayes gives me a route to get from the probability of some data given a parameter - $P(D | \\theta)$) - to the probability of the parameter given some data - $P(\\theta | D)$.\n",
    "\n",
    "I will set up 4 hypotheses:\n",
    "\n",
    "* $H_1: \\theta = 0.05$. My belief  = 0.26\n",
    "* $H_2: \\theta = 0.10$. My belief  = 0.52\n",
    "* $H_3: \\theta = 0.15$. My belief  = 0.19\n",
    "* $H_3: \\theta = 0.20$. My belief  = 0.03\n",
    "\n",
    "To get the posterior probability mass I need to calculate $P(D|\\theta)$ - the likelihood - for each hypothesis i.e. I need to calculate the likelihood of seeing 8/104 people get a first class mark given the proportion in each hypothesis.\n",
    "\n",
    "The [binomial distribution](https://en.wikipedia.org/wiki/Binomial_distribution) allows me to calculate these likelihoods. Basically it gives me the probability of getting $s$ successes in $n$ trials. Here success is defined as a first class mark and there are 104 trials.\n",
    "\n",
    "The formula for the binomial probability distribution is:\n",
    "\n",
    "$P(s) = {n \\choose s}\\theta^s(1-\\theta)^{n-s}$\n",
    "\n",
    "The ${n\\choose s}$ part is called the binomial coefficient and is calculated by $\\frac{n!}{s!(n-s)!}$.\n",
    "\n",
    "Plugging in the numbers for the first hypothesis ($H_1: \\theta = 0.05$) we get:\n",
    "\n",
    "$P(8) = {104\\choose 8}0.05^8(1-0.05)^{96} = 0.07$\n",
    "\n",
    "So the likelihood of getting 8 successes out of 104 trials if the 'true' proportion of successes is 5% is 0.07 (or 7%).\n",
    "\n",
    "```R``` provides a density function for the binomial distribution that allows us to calculate this directly and we can use this to quickly calculate the probabilities for each hypothesis."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>0.0731</li>\n",
       "\t<li>0.1043</li>\n",
       "\t<li>0.0111</li>\n",
       "\t<li>3e-04</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 0.0731\n",
       "\\item 0.1043\n",
       "\\item 0.0111\n",
       "\\item 3e-04\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 0.0731\n",
       "2. 0.1043\n",
       "3. 0.0111\n",
       "4. 3e-04\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1] 0.0731 0.1043 0.0111 0.0003"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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KioqJycnN/+9rdOrgoAcFVqmhG67A71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\nL774QpGHB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      ],
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {
      "image/svg+xml": {
       "isolated": true
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     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# e.g h1_like <- dbinom(8,104,0.05), but we vectorise this below for each prop\n",
    "\n",
    "like_vec <- dbinom(8,104,props)\n",
    "round(like_vec,4)\n",
    "plot(props, like_vec, type='h', xlab = 'Proportion', ylab='Likelihood', lwd=4)\n",
    "points(props, like_vec, pch=16, cex=2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's start laying out the different components in a table."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th></th><th scope=col>theta</th><th scope=col>prior</th><th scope=col>likelihood</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>1</th><td>0.0500</td><td>0.2600</td><td>0.0731</td></tr>\n",
       "\t<tr><th scope=row>2</th><td>0.1000</td><td>0.5200</td><td>0.1043</td></tr>\n",
       "\t<tr><th scope=row>3</th><td>0.1500</td><td>0.1900</td><td>0.0111</td></tr>\n",
       "\t<tr><th scope=row>4</th><td>2e-01</td><td>3e-02</td><td>3e-04</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|lll}\n",
       "  & theta & prior & likelihood\\\\\n",
       "\\hline\n",
       "\t1 & 0.0500 & 0.2600 & 0.0731\\\\\n",
       "\t2 & 0.1000 & 0.5200 & 0.1043\\\\\n",
       "\t3 & 0.1500 & 0.1900 & 0.0111\\\\\n",
       "\t4 & 2e-01 & 3e-02 & 3e-04\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/plain": [
       "  theta prior likelihood\n",
       "1  0.05  0.26     0.0731\n",
       "2  0.10  0.52     0.1043\n",
       "3  0.15  0.19     0.0111\n",
       "4  0.20  0.03     0.0003"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bayes_tab <- data.frame(theta = props, prior = priors, likelihood = round(like_vec,4))\n",
    "bayes_tab"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we have to calculate the product of the likelihood and the prior - $P(D|\\theta)P(\\theta)$. This is $P(D|\\theta)$ multiplied by how much I believe in each value of $\\theta$ i.e. my priors."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th></th><th scope=col>theta</th><th scope=col>prior</th><th scope=col>likelihood</th><th scope=col>like_x_prior</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>1</th><td>0.0500</td><td>0.2600</td><td>0.0731</td><td>0.0190</td></tr>\n",
       "\t<tr><th scope=row>2</th><td>0.1000</td><td>0.5200</td><td>0.1043</td><td>0.0542</td></tr>\n",
       "\t<tr><th scope=row>3</th><td>0.1500</td><td>0.1900</td><td>0.0111</td><td>0.0021</td></tr>\n",
       "\t<tr><th scope=row>4</th><td>2e-01</td><td>3e-02</td><td>3e-04</td><td>0e+00</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|llll}\n",
       "  & theta & prior & likelihood & like\\_x\\_prior\\\\\n",
       "\\hline\n",
       "\t1 & 0.0500 & 0.2600 & 0.0731 & 0.0190\\\\\n",
       "\t2 & 0.1000 & 0.5200 & 0.1043 & 0.0542\\\\\n",
       "\t3 & 0.1500 & 0.1900 & 0.0111 & 0.0021\\\\\n",
       "\t4 & 2e-01 & 3e-02 & 3e-04 & 0e+00\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/plain": [
       "  theta prior likelihood like_x_prior\n",
       "1  0.05  0.26     0.0731       0.0190\n",
       "2  0.10  0.52     0.1043       0.0542\n",
       "3  0.15  0.19     0.0111       0.0021\n",
       "4  0.20  0.03     0.0003       0.0000"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "like_x_prior <- like_vec * priors\n",
    "bayes_tab$like_x_prior <- round(like_x_prior,4)\n",
    "bayes_tab"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally we need to calculate the denominator for Bayes rule i.e. $P(D)$. To do this we sum the likelihood and prior products we have calculated i.e.\n",
    "\n",
    "$P(\\theta) = \\Sigma\\ {P(D|\\theta) P(\\theta)}$ "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "0.075"
      ],
      "text/latex": [
       "0.075"
      ],
      "text/markdown": [
       "0.075"
      ],
      "text/plain": [
       "[1] 0.075"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "denom <- sum(bayes_tab$like_x_prior)\n",
    "round(denom,3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "With all these parts in place we can calculate the posterior distribution for our proportions i.e. the products divided by the evidence.\n",
    "\n",
    "$$P(\\theta | D) = \\frac{P(D | \\theta)P(\\theta)}{P(D)}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>0.2523</li>\n",
       "\t<li>0.7198</li>\n",
       "\t<li>0.0279</li>\n",
       "\t<li>0</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 0.2523\n",
       "\\item 0.7198\n",
       "\\item 0.0279\n",
       "\\item 0\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 0.2523\n",
       "2. 0.7198\n",
       "3. 0.0279\n",
       "4. 0\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1] 0.2523 0.7198 0.0279 0.0000"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "post <- bayes_tab$like_x_prior/denom\n",
    "round(post,4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th></th><th scope=col>theta</th><th scope=col>prior</th><th scope=col>likelihood</th><th scope=col>like_x_prior</th><th scope=col>post</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>1</th><td>0.0500</td><td>0.2600</td><td>0.0731</td><td>0.0190</td><td>0.2523</td></tr>\n",
       "\t<tr><th scope=row>2</th><td>0.1000</td><td>0.5200</td><td>0.1043</td><td>0.0542</td><td>0.7198</td></tr>\n",
       "\t<tr><th scope=row>3</th><td>0.1500</td><td>0.1900</td><td>0.0111</td><td>0.0021</td><td>0.0279</td></tr>\n",
       "\t<tr><th scope=row>4</th><td>2e-01</td><td>3e-02</td><td>3e-04</td><td>0e+00</td><td>0e+00</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|lllll}\n",
       "  & theta & prior & likelihood & like\\_x\\_prior & post\\\\\n",
       "\\hline\n",
       "\t1 & 0.0500 & 0.2600 & 0.0731 & 0.0190 & 0.2523\\\\\n",
       "\t2 & 0.1000 & 0.5200 & 0.1043 & 0.0542 & 0.7198\\\\\n",
       "\t3 & 0.1500 & 0.1900 & 0.0111 & 0.0021 & 0.0279\\\\\n",
       "\t4 & 2e-01 & 3e-02 & 3e-04 & 0e+00 & 0e+00\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/plain": [
       "  theta prior likelihood like_x_prior   post\n",
       "1  0.05  0.26     0.0731       0.0190 0.2523\n",
       "2  0.10  0.52     0.1043       0.0542 0.7198\n",
       "3  0.15  0.19     0.0111       0.0021 0.0279\n",
       "4  0.20  0.03     0.0003       0.0000 0.0000"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bayes_tab$post <- round(post,4)\n",
    "bayes_tab"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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Vq1qhkTGBgohPjjjz8sERQAAEswtgiz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\nGXPkyBGbJAQASI6hIiwpKRH/vaaMzpNPPlmz3PpiYmKEEMuXL09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       "plot without title"
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   ],
   "source": [
    "plot(bayes_tab$theta, bayes_tab$post, type='h', xlab='Proportion', ylab='Posterior Probability', lwd=4)\n",
    "points(bayes_tab$theta, bayes_tab$post, pch=16, cex=2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So that's fine. These results strengthen my belief that 10% of students will get 70% or more."
   ]
  }
 ],
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    "43097/FE38EPWA": {
     "DOI": "10.1016/j.jmp.2015.10.003",
     "URL": "//www.sciencedirect.com/science/article/pii/S0022249615000607",
     "abstract": "Bayes factors provide a symmetrical measure of evidence for one model versus another (e.g. H1 versus H0) in order to relate theory to data. These properties help solve some (but not all) of the problems underlying the credibility crisis in psychology. The symmetry of the measure of evidence means that there can be evidence for H0 just as much as for H1; or the Bayes factor may indicate insufficient evidence either way. P -values cannot make this three-way distinction. Thus, Bayes factors indicate when the data count against a theory (and when they count for nothing); and thus they indicate when replications actually support H0 or H1 (in ways that power cannot). There is every reason to publish evidence supporting the null as going against it, because the evidence can be measured to be just as strong either way (thus the published record can be more balanced). Bayes factors can be B -hacked but they mitigate the problem because a) they allow evidence in either direction so people will be less tempted to hack in just one direction; b) as a measure of evidence they are insensitive to the stopping rule; c) families of tests cannot be arbitrarily defined; and d) falsely implying a contrast is planned rather than post hoc becomes irrelevant (though the value of pre-registration is not mitigated).",
     "accessed": {
      "day": "22",
      "month": "1",
      "year": "2017"
     },
     "author": [
      {
       "family": "Dienes",
       "given": "Zoltan"
      }
     ],
     "collection-title": "Bayes Factors for Testing Hypotheses in Psychological Research: Practical Relevance and New Developments",
     "container-title": "Journal of Mathematical Psychology",
     "container-title-short": "Journal of Mathematical Psychology",
     "id": "43097/FE38EPWA",
     "issued": {
      "month": "6",
      "year": "2016"
     },
     "journalAbbreviation": "Journal of Mathematical Psychology",
     "page": "78-89",
     "page-first": "78",
     "title": "How Bayes factors change scientific practice",
     "type": "article-journal",
     "volume": "72"
    }
   }
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