notebooks/2M2.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Statistical Rethinking: Excercise 2M2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Like in the previous notebook for 2M1, we have the following globe tossing results:\n",
    "\n",
    "(1) W, W, W <br>\n",
    "(2) W, W, W, L <br>\n",
    "(3) L, W, W, L, W, W, W <br>\n",
    "\n",
    "<br>\n",
    "We want to estimate the probability of Water $p$ as parameter of the model. The parameter is at least 0 and at most 1.\n",
    "<br>\n",
    "<br>\n",
    "The key point for this exercise is that we employ a different prior:\n",
    "$$    \\text{prior}(p) =\n",
    "    \\begin{cases}\n",
    "      0, & \\text{if}\\ p < 0.5 \\\\\n",
    "      1, & \\text{otherwise}\n",
    "    \\end{cases}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[0.0, 0.02, 0.04, 0.06, 0.08, 0.1, 0.12, 0.14, 0.16, 0.18, 0.2, 0.22, 0.24, 0.26, 0.28, 0.3, 0.32, 0.34, 0.36, 0.38, 0.4, 0.42, 0.44, 0.46, 0.48, 0.5, 0.52, 0.54, 0.56, 0.58, 0.6, 0.62, 0.64, 0.66, 0.68, 0.7000000000000001, 0.72, 0.74, 0.76, 0.78, 0.8, 0.8200000000000001, 0.84, 0.86, 0.88, 0.9, 0.92, 0.9400000000000001, 0.96, 0.98, 1.0]"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "require \"rubythinking\"\n",
    "grid_size = 50\n",
    "step_size = 1.0 / grid_size.to_f\n",
    "grid = 0.step(by: step_size, to: 1).to_a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
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     "execution_count": 42,
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   "source": [
    "prior = grid.map do |x| \n",
    "  y =\n",
    "    if x < 0.5\n",
    "      0\n",
    "    else\n",
    "      1.to_f / (grid_size.to_f / 2.0)\n",
    "    end\n",
    "  \n",
    "  {x: x, y: y}\n",
    "end\n",
    "\n",
    "Vega.lite\n",
    "  .data(prior)\n",
    "  .mark(type: \"area\")\n",
    "  .encoding(\n",
    "    x: {field: \"x\", type: \"quantitative\"},\n",
    "    y: {field: \"y\", type: \"quantitative\", scale: {\"domain\": [0,0.1]}}\n",
    "  )"
   ]
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  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "#<Proc:0x00007f9f558c8d08@<main>:6 (lambda)>"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "factorial = ->(n) do\n",
    "  return 1 if n < 1\n",
    "  n.to_i.downto(1).inject(:*)\n",
    "end\n",
    "\n",
    "\n",
    "likelihood = ->(w, l, p) do\n",
    "  (factorial[w+l].to_f / (factorial[w] * factorial[l])).to_f * (p**w) * ((1-p)**l)\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, let's compute the grid aprroximation of the posterior for each of the cases. The difference is only the data input we give in terms of \"count of Water\" versus \"count of Land\" of our tossing result given in the exercise."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
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      ]
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     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# For case (1)\n",
    "w = 3\n",
    "l = 0\n",
    "\n",
    "print_posterior = ->(w, l) do\n",
    "  u_posterior = grid.map do |x| \n",
    "    prior_x = prior.detect { |p| p[:x] == x }[:y]\n",
    "    y = prior_x * likelihood[w, l, x]\n",
    "    {x: x, y: y}\n",
    "  end\n",
    "\n",
    "  posterior = u_posterior.map do |pair| \n",
    "    y = pair[:y].to_f / u_posterior.map { |pair| pair[:y] }.sum.to_f\n",
    "   {x: pair[:x], y: y}\n",
    "  end\n",
    "\n",
    "  Vega.lite\n",
    "    .data(posterior)\n",
    "    .mark(type: \"area\")\n",
    "    .encoding(\n",
    "      x: {field: \"x\", type: \"quantitative\"},\n",
    "      y: {field: \"y\", type: \"quantitative\", scale: {\"domain\": [0,0.1]}}\n",
    "    )\n",
    "end\n",
    "\n",
    "print_posterior[3, 0]"
   ]
  },
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       "#<Vega::LiteChart:0x00007f9f54198e08 @schema=\"https://vega.github.io/schema/vega-lite/v4.json\", @spec={\"$schema\"=>\"https://vega.github.io/schema/vega-lite/v4.json\", \"width\"=>\"container\", \"height\"=>\"container\", \"data\"=>{:values=>[{:x=>0.0, :y=>0.0}, {:x=>0.02, :y=>0.0}, {:x=>0.04, :y=>0.0}, {:x=>0.06, :y=>0.0}, {:x=>0.08, :y=>0.0}, {:x=>0.1, :y=>0.0}, {:x=>0.12, :y=>0.0}, {:x=>0.14, :y=>0.0}, {:x=>0.16, :y=>0.0}, {:x=>0.18, :y=>0.0}, {:x=>0.2, :y=>0.0}, {:x=>0.22, :y=>0.0}, {:x=>0.24, :y=>0.0}, {:x=>0.26, :y=>0.0}, {:x=>0.28, :y=>0.0}, {:x=>0.3, :y=>0.0}, {:x=>0.32, :y=>0.0}, {:x=>0.34, :y=>0.0}, {:x=>0.36, :y=>0.0}, {:x=>0.38, :y=>0.0}, {:x=>0.4, :y=>0.0}, {:x=>0.42, :y=>0.0}, {:x=>0.44, :y=>0.0}, {:x=>0.46, :y=>0.0}, {:x=>0.48, :y=>0.0}, {:x=>0.5, :y=>0.030333668411168073}, {:x=>0.52, :y=>0.032756401520313755}, {:x=>0.54, :y=>0.03515475121344381}, {:x=>0.56, :y=>0.03750262568233768}, {:x=>0.58, :y=>0.03977206941818762}, {:x=>0.6, :y=>0.041933263211598744}, {:x=>0.62, :y=>0.043954524152588936}, {:x=>0.64, :y=>0.04580230563058892}, {:x=>0.66, :y=>0.047441197334442235}, {:x=>0.68, :y=>0.04883392525240524}, {:x=>0.7000000000000001, :y=>0.04994135167214711}, {:x=>0.72, :y=>0.05072247518074984}, {:x=>0.74, :y=>0.05113443066470823}, {:x=>0.76, :y=>0.05113248930992992}, {:x=>0.78, :y=>0.05067005860173534}, {:x=>0.8, :y=>0.04969868232485778}, {:x=>0.8200000000000001, :y=>0.04816804056344327}, {:x=>0.84, :y=>0.04602594970105078}, {:x=>0.86, :y=>0.04321836242065198}, {:x=>0.88, :y=>0.03968936770463141}, {:x=>0.9, :y=>0.035381190834786434}, {:x=>0.92, :y=>0.030234193392327212}, {:x=>0.9400000000000001, :y=>0.024186873257876735}, {:x=>0.96, :y=>0.01717586461147086}, {:x=>0.98, :y=>0.00913593793255812}, {:x=>1.0, :y=>0.0}]}, \"mark\"=>{:type=>\"area\"}, \"encoding\"=>{:x=>{:field=>\"x\", :type=>\"quantitative\"}, :y=>{:field=>\"y\", :type=>\"quantitative\", :scale=>{:domain=>[0, 0.1]}}}}>"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# For case (2)\n",
    "w = 3\n",
    "l = 1\n",
    "print_posterior[3, 1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "<div id=\"chart-a2d2734370bc12e92a998a5380446da5\" style=\"width: 100%; height: 300px;\"></div>\n",
       "<script>\n",
       "  require.config({\n",
       "    paths: {\n",
       "      'vega': 'https://cdn.jsdelivr.net/npm/vega@5.16.1?noext',\n",
       "      'vega-lite': 'https://cdn.jsdelivr.net/npm/vega-lite@4.16.2?noext',\n",
       "      'vega-embed': 'https://cdn.jsdelivr.net/npm/vega-embed@6.12.2?noext'\n",
       "    }\n",
       "  });\n",
       "  require(['vega', 'vega-lite', 'vega-embed'], function(vega, vegaLite, vegaEmbed) {\n",
       "    vegaEmbed(\"#chart-a2d2734370bc12e92a998a5380446da5\", {\"$schema\":\"https://vega.github.io/schema/vega-lite/v4.json\",\"width\":\"container\",\"height\":\"container\",\"data\":{\"values\":[{\"x\":0.0,\"y\":0.0},{\"x\":0.02,\"y\":0.0},{\"x\":0.04,\"y\":0.0},{\"x\":0.06,\"y\":0.0},{\"x\":0.08,\"y\":0.0},{\"x\":0.1,\"y\":0.0},{\"x\":0.12,\"y\":0.0},{\"x\":0.14,\"y\":0.0},{\"x\":0.16,\"y\":0.0},{\"x\":0.18,\"y\":0.0},{\"x\":0.2,\"y\":0.0},{\"x\":0.22,\"y\":0.0},{\"x\":0.24,\"y\":0.0},{\"x\":0.26,\"y\":0.0},{\"x\":0.28,\"y\":0.0},{\"x\":0.3,\"y\":0.0},{\"x\":0.32,\"y\":0.0},{\"x\":0.34,\"y\":0.0},{\"x\":0.36,\"y\":0.0},{\"x\":0.38,\"y\":0.0},{\"x\":0.4,\"y\":0.0},{\"x\":0.42,\"y\":0.0},{\"x\":0.44,\"y\":0.0},{\"x\":0.46,\"y\":0.0},{\"x\":0.48,\"y\":0.0},{\"x\":0.5,\"y\":0.030230440239998296},{\"x\":0.52,\"y\":0.03389640455469191},{\"x\":0.54,\"y\":0.03759576306122878},{\"x\":0.56,\"y\":0.041257217006775745},{\"x\":0.58,\"y\":0.0448015448911555},{\"x\":0.6,\"y\":0.04814272579711524},{\"x\":0.62,\"y\":0.051189405068867475},{\"x\":0.64,\"y\":0.05384672730179831},{\"x\":0.66,\"y\":0.05601856160623748},{\"x\":0.68,\"y\":0.05761014410818551},{\"x\":0.7000000000000001,\"y\":0.058531162649892644},{\"x\":0.72,\"y\":0.05869930865318412},{\"x\":0.74,\"y\":0.05804432110842732},{\"x\":0.76,\"y\":0.05651254765203498},{\"x\":0.78,\"y\":0.054072047695399646},{\"x\":0.8,\"y\":0.05071826256815431},{\"x\":0.8200000000000001,\"y\":0.04648027763865389},{\"x\":0.84,\"y\":0.04142770137457235},{\"x\":0.86,\"y\":0.03567818630651054},{\"x\":0.88,\"y\":0.02940561685750976},{\"x\":0.9,\"y\":0.022848989001365234},{\"x\":0.92,\"y\":0.016322006712635093},{\"x\":0.9400000000000001,\"y\":0.010223420171239313},{\"x\":0.96,\"y\":0.005048130684543599},{\"x\":0.98,\"y\":0.0013990872898229294},{\"x\":1.0,\"y\":0.0}]},\"mark\":{\"type\":\"area\"},\"encoding\":{\"x\":{\"field\":\"x\",\"type\":\"quantitative\"},\"y\":{\"field\":\"y\",\"type\":\"quantitative\",\"scale\":{\"domain\":[0,0.1]}}}}, {\"actions\":false});\n",
       "  });\n",
       "</script>\n"
      ],
      "text/plain": [
       "#<Vega::LiteChart:0x00007f9f540bb738 @schema=\"https://vega.github.io/schema/vega-lite/v4.json\", @spec={\"$schema\"=>\"https://vega.github.io/schema/vega-lite/v4.json\", \"width\"=>\"container\", \"height\"=>\"container\", \"data\"=>{:values=>[{:x=>0.0, :y=>0.0}, {:x=>0.02, :y=>0.0}, {:x=>0.04, :y=>0.0}, {:x=>0.06, :y=>0.0}, {:x=>0.08, :y=>0.0}, {:x=>0.1, :y=>0.0}, {:x=>0.12, :y=>0.0}, {:x=>0.14, :y=>0.0}, {:x=>0.16, :y=>0.0}, {:x=>0.18, :y=>0.0}, {:x=>0.2, :y=>0.0}, {:x=>0.22, :y=>0.0}, {:x=>0.24, :y=>0.0}, {:x=>0.26, :y=>0.0}, {:x=>0.28, :y=>0.0}, {:x=>0.3, :y=>0.0}, {:x=>0.32, :y=>0.0}, {:x=>0.34, :y=>0.0}, {:x=>0.36, :y=>0.0}, {:x=>0.38, :y=>0.0}, {:x=>0.4, :y=>0.0}, {:x=>0.42, :y=>0.0}, {:x=>0.44, :y=>0.0}, {:x=>0.46, :y=>0.0}, {:x=>0.48, :y=>0.0}, {:x=>0.5, :y=>0.030230440239998296}, {:x=>0.52, :y=>0.03389640455469191}, {:x=>0.54, :y=>0.03759576306122878}, {:x=>0.56, :y=>0.041257217006775745}, {:x=>0.58, :y=>0.0448015448911555}, {:x=>0.6, :y=>0.04814272579711524}, {:x=>0.62, :y=>0.051189405068867475}, {:x=>0.64, :y=>0.05384672730179831}, {:x=>0.66, :y=>0.05601856160623748}, {:x=>0.68, :y=>0.05761014410818551}, {:x=>0.7000000000000001, :y=>0.058531162649892644}, {:x=>0.72, :y=>0.05869930865318412}, {:x=>0.74, :y=>0.05804432110842732}, {:x=>0.76, :y=>0.05651254765203498}, {:x=>0.78, :y=>0.054072047695399646}, {:x=>0.8, :y=>0.05071826256815431}, {:x=>0.8200000000000001, :y=>0.04648027763865389}, {:x=>0.84, :y=>0.04142770137457235}, {:x=>0.86, :y=>0.03567818630651054}, {:x=>0.88, :y=>0.02940561685750976}, {:x=>0.9, :y=>0.022848989001365234}, {:x=>0.92, :y=>0.016322006712635093}, {:x=>0.9400000000000001, :y=>0.010223420171239313}, {:x=>0.96, :y=>0.005048130684543599}, {:x=>0.98, :y=>0.0013990872898229294}, {:x=>1.0, :y=>0.0}]}, \"mark\"=>{:type=>\"area\"}, \"encoding\"=>{:x=>{:field=>\"x\", :type=>\"quantitative\"}, :y=>{:field=>\"y\", :type=>\"quantitative\", :scale=>{:domain=>[0, 0.1]}}}}>"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# For case (3)\n",
    "w = 5\n",
    "l = 2\n",
    "print_posterior[5, 2]"
   ]
  }
 ],
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