07_euro_soln.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Bite Size Bayes\n",
    "\n",
    "Copyright 2020 Allen B. Downey\n",
    "\n",
    "License: [Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0)](https://creativecommons.org/licenses/by-nc-sa/4.0/)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Review\n",
    "\n",
    "[In the previous notebook](https://colab.research.google.com/github/AllenDowney/BiteSizeBayes/blob/master/05_test.ipynb) ...\n",
    "\n",
    "So far just Bayes's Theorem, now on to Bayesian statistics\n",
    "\n",
    "and here's the function we defined"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Probability Mass Functions\n",
    "\n",
    "When we do more than one update, we don't always want to keep the whole Bayes table.  In this section we'll replace the Bayes table with a more compact representation, using a Pandas Series.\n",
    "\n",
    "Here's a function that takes a sequence of values, `xs`, and their corresponding probabilities, `ps`, and returns a Pandas Series:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "def make_pmf(xs, ps, **options):\n",
    "    \"\"\"Make a Series that represents a PMF.\n",
    "    \n",
    "    xs: sequence of values\n",
    "    ps: sequence of probabilities\n",
    "    options: keyword arguments passed to Series constructor\n",
    "    \n",
    "    returns: Pandas Series\n",
    "    \"\"\"\n",
    "    pmf = pd.Series(ps, index=xs, **options)\n",
    "    return pmf"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "def bayes_update(pmf, likelihood):\n",
    "    \"\"\"Do a Bayesian update.\n",
    "    \n",
    "    pmf: Series that represents the prior\n",
    "    likelihood: sequence of likelihoods\n",
    "    \n",
    "    returns: float probability of the data\n",
    "    \"\"\"\n",
    "    pmf *= likelihood\n",
    "    prob_data = pmf.sum()\n",
    "    pmf /= prob_data\n",
    "    return prob_data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "xs = np.arange(101)\n",
    "prior = 1/101\n",
    "\n",
    "pmf = make_pmf(xs, prior)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "likelihood_vanilla = xs / 100\n",
    "likelihood_chocolate = 1 - likelihood_vanilla"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "data = 'VVC'\n",
    "\n",
    "for cookie in data:\n",
    "    if cookie == 'V':\n",
    "        bayes_update(pmf, likelihood_vanilla)\n",
    "    else:\n",
    "        bayes_update(pmf, likelihood_chocolate)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "pmf.plot()\n",
    "\n",
    "plt.xlabel('Bowl #')\n",
    "plt.ylabel('Probability')\n",
    "plt.title('Three cookies');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The Euro problem\n",
    "\n",
    "In this notebook we'll work on a problem from David MacKay's book, [*Information Theory, Inference, and Learning Algorithms*](http://www.inference.org.uk/mackay/itila/p0.html), which is the book where I first learned about Bayesian statistics.  MacKay writes:\n",
    "\n",
    "> A statistical statement appeared in The Guardian on\n",
    "Friday January 4, 2002:\n",
    ">\n",
    "> >\"When spun on edge 250 times, a Belgian one-euro coin came\n",
    "up heads 140 times and tails 110. ‘It looks very suspicious\n",
    "to me’, said Barry Blight, a statistics lecturer at the London\n",
    "School of Economics. ‘If the coin were unbiased the chance of\n",
    "getting a result as extreme as that would be less than 7%’.\"\n",
    ">\n",
    "> But [asks MacKay] do these data give evidence that the coin is biased rather than fair?\n",
    "\n",
    "To answer this question, we have to make some modeling decisions.\n",
    "\n",
    "First, let's assume that if you spin a coin on edge, there is some probability that it will land heads up.  I'll call that probability $x$.\n",
    "\n",
    "Second, let's assume that $x$ varies from one coin to the next, depending on how the coin is balanced and maybe other factors.\n",
    "\n",
    "With these assumptions we can formulate MacKay's question as an inference problem: given the data, 140 heads and 110 tails, what do we think $x$ is for this coin.\n",
    "\n",
    "This formulation is similar to the 101 Bowls problem we saw in the previous notebook; in fact, we will use the same likelihoods.\n",
    "\n",
    "But in the 101 Bowls problem, we are told that we choose a bowl at random, which implies that all bowls have the same prior probability.\n",
    "\n",
    "For the Euro problem, we have to think harder.  What values of $x$ do you think are reasonable?\n",
    "\n",
    "I seems likely that many coins are \"fair\", meaning that the probability of heads is close to 50%.  Do you think there are coins where $x$ is 75%?  How about 90%?\n",
    "\n",
    "To be honest, I don't really know.  To get started, I will assume that all values of $x$, from 0% to 100%, are equally likely.  So that's the same prior we used for the 101 Bowls.\n",
    "\n",
    "So here's the prior:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "xs = np.arange(101)\n",
    "prior = 1/101\n",
    "pmf = make_pmf(xs, prior)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here are the likelihoods for heads and tails:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "likelihood_heads = xs / 100\n",
    "likelihood_tails = 1 - likelihood_heads"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And here are the updates for 140 heads and 110 tails."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "for i in range(140):\n",
    "    bayes_update(pmf, likelihood_heads)\n",
    "\n",
    "for i in range(110):\n",
    "    bayes_update(pmf, likelihood_tails)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here's what the results look like:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "pmf.plot()\n",
    "\n",
    "plt.xlabel('Possible values of x')\n",
    "plt.ylabel('Probability')\n",
    "plt.title('140 heads, 110 tails');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This curve shows the \"posterior distribution\" of $x$; a \"distribution\" is a set of possible values and their probabilities."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Put a function on it\n",
    "\n",
    "Before we go on, let's put that update in a function, because we are going to need it again."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "def bayes_update_euro(pmf, data):\n",
    "    \"\"\"Do a Bayesian update.\n",
    "    \n",
    "    pmf: Series that represents a prior PMF\n",
    "    data: tuple of number of heads, tails\n",
    "    \"\"\"\n",
    "    heads, tails = data\n",
    "    xs = pmf.index\n",
    "    \n",
    "    likelihood_heads = xs / 100\n",
    "    likelihood_tails = 1 - likelihood_heads\n",
    "\n",
    "    for i in range(heads):\n",
    "        bayes_update(pmf, likelihood_heads)\n",
    "\n",
    "    for i in range(tails):\n",
    "        bayes_update(pmf, likelihood_tails)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This function takes a PMF that represents the prior, and a tuple that contains the number of heads and tails.\n",
    "\n",
    "Here's the uniform prior again."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "xs = np.arange(101)\n",
    "prior = 1/101\n",
    "uniform = make_pmf(xs, prior)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here's the update."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "data = 140, 110\n",
    "bayes_update_euro(uniform, data)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And here are the results again."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "uniform.plot()\n",
    "\n",
    "plt.xlabel('Possible values of x')\n",
    "plt.ylabel('Probability')\n",
    "plt.title('140 heads, 110 tails');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## A better prior\n",
    "\n",
    "But remember that this result is based on a uniform prior, which assumes that any value of $x$ from 0 to 100 is equally likely.\n",
    "\n",
    "Given what we know about coins, that's probabily not true.  I can believe that if you spin a lop-sided coin on edge, it might be somewhat more likely to land on heads or tails.  \n",
    "\n",
    "But unless the coin is heavily weighted on one side, I would be surprised if $x$ were greater than 60% or less than 40%.\n",
    "\n",
    "Of course, I could be wrong, but in general I would expect to find $x$ closer to 50%, and I would be surprised to find it near 0% or 100%.\n",
    "\n",
    "I can represent that prior believe with a triangle-shaped prior.\n",
    "\n",
    "First I'll make an array that ramps up from 0 to 100 an array that ramps down from 100 to 0."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "ramp_up = xs\n",
    "ramp_down = 100 - ramp_up"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To construct a triangle prior I'll start with a copy of `ramp_up` and replace the second half with `ramp_down`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "prior = ramp_up.copy()\n",
    "\n",
    "high = (xs > 50)\n",
    "prior[high] = ramp_down[high]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Then I'll put the results into a PMF and normalize it so it adds up to 1."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "triangle = make_pmf(xs, prior)\n",
    "triangle /= triangle.sum()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here's what the triangle prior looks like."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "triangle.plot(color='C1')\n",
    "\n",
    "plt.xlabel('Possible values of x')\n",
    "plt.ylabel('Probability')\n",
    "plt.title('Triangle prior');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now let's update it with the data."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "data = 140, 110\n",
    "bayes_update_euro(triangle, data)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And plot the results, along with the posterior based on a uniform prior."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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xHCp6G2lo76ayODfxdlPFKL/pR2Vwae5+Dz/8MI8//jh/+ctfyM/P59hjj6WzM7gTLScnZ/d2yZQHH8jM+OEPf8gJJ5wwtsAH8RaEcxko3vg6b8QTjIHoJ9GVP5NZavKxEBNIS0sL5eXl5Ofns27dOtasWTPsPitWrOD+++8H4M4770y4zYknnsgPf/jD3Ullw4YN7No19n8XniCcy0DBGIhK5pblDbmNFc1mFk1sa/YEMVF86EMfoqOjg8MOO4yrrrqKlStXDrvPD37wA775zW+yYsUKamtrKS19e6vyggsuYPHixSxbtoyDDz6Yz33ucymZNMjLfTuXaXq7sKtn8f3ej3Dkud/i3YsTVwno+vn/pOZva3jsA7/jM8dOvbLfU6Xcd3t7OwUFBUjijjvu4P777+e+++4b1bFGWu7br0E4l2laahDGlvhMPlw6RBcTkDN9HrP1kHcxTXJr1qzh4osvJh6PM336dG699dZxO7cnCOcyTdPrAGGZjaG7mFQyh0J10tDUOF6RuTQ4/vjjdw/IG29+DcK5TBPe4tqSN5eCnL18hysORlN3N24dj6gy0mTpIh8Po/lbeYJwLtM0baaHaUwrnbv37cKxENa6fRyCyjx5eXk0NDR4kkiCmdHQ0EBe3tAt0kS8i8m5TNP8BjtiM5kzvWDv24UtiJyOHfT2xcnOmlrf96qqqqipqcHngklOXl4eVVVVI9rHE4RzmWbndmri5UOPgegXtiAqaWJHaxfzhtt+kpk2bRqLFk29u7fG09T6yuHcBBBvq+WtvuLhE0RuMb3ZhcxWE9t9LISLgCcI5zJNex0NVsqc0uH7i+NFs5mpJra1+MRBLvU8QTiXSbo7iPW0U2+lSXUZxUrmBOU2vAXhIuAJwrlM0h5ccK2nhDlJJIjs0rnMUTPbvQXhIuAJwrlMEiaIRkqZlUyF1uKwi6mpI+LA3FTkCcK5TBImCCuYkdxtq8VzyKGH1ub6iANzU5EnCOcySVswzWRWyazktg/npu5r2RZVRG4K8wThXCYJWxDTipNMEOFgufzOHfT5zHIuxTxBOJdJ2utoo4DSkuLktg8Hy81UE00d3REG5qYiTxDOZRBrq6M+XkxFYc7wGwMUhQmCZhraPEG41Io0QUg6SdIGSdWSLkuw/jhJz0nqlXTmoHXnStoY/pwbZZzOZYre1lrqKKWiKMkEMS2PnpwyZqmJhvauaINzU05kCUJSFnADcDKwBDhH0pJBm70JnAf8fNC+5cDlwEpgBXC5pOlRxepcpoi31dJgpVQUJXGLa/8+RbOYpSYa270F4VIryhbECqDazDaZWTdwJ3DawA3MbLOZvQjEB+17IvB7M2s0sybg98BJEcbqXEaIddRTbyXMSLaLiWDioFlq8i4ml3JRJoh5wJYBz2vCZSnbV9L5ktZKWuslf92E19dLdmcjDZRSnmwXE5BdNo+ZaqLBWxAuxaJMEEqwLNn78JLa18xuNLPlZra8sjLxxO7OTRi7GhFGvZVQUZh8F1OsaBaVaqGxzcttuNSKMkHUAPMHPK8Ckh3NM5Z9nZuYwkFyDZQyvWBa8vsVVDCNPna1NkUUmJuqokwQa4DFkhZJygHOBlYlue+DwAckTQ8vTn8gXObc5BUOkuvKKR/Z7HCFMwDoafVuVpdakSUIM+sFLiL4YF8P3G1m6yRdJelUAElHSaoBzgL+S9K6cN9G4OsESWYNcFW4zLnJK0wQfQUj7C4tCBKEtTekOiI3xUU65aiZrQZWD1r2tQGP1xB0HyXa9xbglijjcy6jhAkiVjTSBFEOQFanJwiXWj6S2rlM0VZLD9nkF1eMbL+wiym3q9HrMbmU8gThXKZor6eREspHMEgO2N3FNJ1Wr8fkUsoThHMZIt5eS228JPkyG/1yCujLyqNcrT6a2qWUJwjnMkTfzpGX2ejXm1dOhXZS3+b1mFzqeIJwLlO011FPafKVXAewghlMx1sQLrU8QTiXCczI6qgLR1GPPEHECiso105PEC6lPEE4lwm6WonFu6kfZRdTdnElFWr1gn0upTxBOJcJwjEQDVbCjJFepAZihTMoV6vPCeFSyhOEc5kgTBCNKqMkbwR1mPoVVFBIJztb21IcmJvKPEE4lwnCQn29+RXEYomKGQ/D6zG5CHiCcC4ThC0IG2kdpn4Fwehra69PVUTOeYJwLiOECWJa8WgTRNCCiO3yekwudTxBOJcJ2utooZjSooLR7R92MU3ravJ6TC5lPEE4lwnaaoMxEKO4gwnY3cVUzk6avR6TSxFPEM5lgHhbHXVWwoxRjIEAIK+MuLLCW109QbjU8AThXAboa6un3kooH8UoagBiMXpzyyjHB8u51PEE4VwGUEc9TVY8qjIb/Szfy2241PIE4Vy6xfvI6mqmkZJRldnoJx9N7VLME4Rz6barCWE0WPGoymz0yy6u9C4ml1KeIJxLt3BwW5MVj/4aBEFF1xkx72JyqeMJwrl06wgGt+3MKqUoN3v0xymYQSltNLZ1pCgwN9V5gnAu3TqCFkQ8rwJpFHWY+hVUEMPoam1MUWBuqos0QUg6SdIGSdWSLkuwPlfSXeH6pyUtDJdPk3SbpJckrZf05SjjdC6twi4mFVWM7TjhaGpr83pMLjUiSxCSsoAbgJOBJcA5kpYM2uyzQJOZ7Q9cB3wzXH4WkGtmhwBHAhf0Jw/nJp2O4Bt/VtEo6zD1C0dTy+sxuRSJsgWxAqg2s01m1g3cCZw2aJvTgNvCx/cCJyhoYxtQKCkbyAe6gZ0Rxupc+nTU00oB04sKx3acMEFkdzV6PSaXElEmiHnAlgHPa8JlCbcxs16gBaggSBbtwHbgTeBaM3tbx6qk8yWtlbS2rs7r4LsJqr1+zHcwAbu7mMpppWVXTwoCc1NdUglC0n2SPiRpJAkl0dW2wV9rhtpmBdAHzAUWAV+UtO/bNjS70cyWm9nyysoxNs+dS5O+9nrqrZjyMYyBAPYo2Nfog+VcCiT7gf8j4OPARknXSDowiX1qgPkDnlcB24baJuxOKgUaw3M9YGY9ZlYL/AlYnmSszk0ofW31NFjJmMpsAJCdS292EeVqpbHdWxBu7JJKEGb2sJl9AjgC2Az8XtKfJX1a0lAT6K4BFktaJCkHOBtYNWibVcC54eMzgT+YmRF0K71PgULgaOBvI3lhzk0U2t3FNPoyG/3i+eVhPSZvQbixS7rLSFIFcB7wj8DzwPcJEsbvE20fXlO4CHgQWA/cbWbrJF0l6dRws5uBCknVwCVA/62wNwBFwMsEieZWM3txZC/NuQnAjKzORhopobxwqO9aI1A4g3K8BeFSI6lhm5J+CRwI3A6cYmbbw1V3SVo71H5mthpYPWjZ1wY87iS4pXXwfm2Jljs36XS3EYt305CiFkRW0QzK9SovegvCpUCy4/pvCj/sd5OUa2ZdZubXBpwbrbDMRhMpuIuJYCzFDD3rkwa5lEi2i+nqBMueSmUgzk1J7UGCaFEJJXljqMPUr6Cc6dpJU5u3INzY7fVfpKTZBGMV8iUdzt9vSy0BRjm7unNut7AOU0/uGOsw9SuYQS49tLe1jP1Ybsob7ivLiQQXpquA7w5Y3gr8R0QxOTd1hF1M/WMYxiwcLBdv93pMbuz2miDM7DbgNklnmNl94xSTc1NH+EGeVTwjNccrCI/T7vWY3NgN18X0STO7A1go6ZLB683suwl2c84lq6OebrIpKCpLzfHCFkRWZwNmlppuKzdlDdfF1F89rCjqQJybkjoaaBrjXNR7CLuqSuIt7OrpoyAnBRe+3ZQ1XBfTf4W/rxyfcJybWuJt9dTHU3OLK7BHPaaGtm4Kyj1BuNEbrovpB3tbb2afT204zk0tvW11NKaikmu/3GLisWmUq5Wmjm7ml/vNhm70hvt68ey4ROHcFGXtDTQyd+yF+vpJ9OZVUNG90wfLuTFL5i4m51xEsnY10GjvYEmqEgRgBRWUt7bS5AnCjdFwXUzfM7OLJf2at8/lgJmdmmA351wyervJ7mml0YqpGOtcEAPEiiopr63hdU8QboyG62K6Pfx9bdSBODfl7AomSQwquaboLiYgu7iSCr3iXUxuzIbrYno2/P1YOKfDgQQtiQ3hPNPOudEKB8k1UUJZfgpKfYdUMCO4SO0Jwo1RsuW+PwT8GHiNoB7TIkkXmNnvogzOuUltdx2m6cRiKRzQVlhBEbtoaWtL3THdlJTsTdLfAd5rZtUAkvYDfgt4gnButMI6TPH8FNVh6heOhehrrUvtcd2Uk2y579r+5BDaBNRGEI9zU0dYL0mFKarD1M/rMbkUGe4uptPDh+skrQbuJrgGcRbBVKDOudHqqCeOyC1JcQtiQD0m58ZiuC6mUwY83gG8J3xcB0yPJCLnpoqOBlooYnpRfmqPG7Ygcroa6e2Lk52V9NTzzu1huLuYPj1egTg31cTb62mIp2Yu6j2ELYgKtdLU0UNlcYqP76aMZO9iygM+CywF8vqXm9lnIorLuUmvr7WORopTV2ajX14ZcWUxPazH5AnCjVaybc/bgdkEM8w9RjDDXGtUQTk3FcTb6mm0EqanOkHEYvTmllERVnR1brSSTRD7m9lXgfawPtOHgEOG20nSSZI2SKqWdFmC9bmS7grXPy1p4YB1h0p6StI6SS+FrRjnJo1YRx31VpL6FgTBrbP9FV2dG61kE0RP+LtZ0sFAKbBwbztIygJuAE4GlgDnSFoyaLPPAk1mtj9wHfDNcN9s4A7gQjNbChw/IAbnJr6+XrK7mqinNHWlvgeIFc6gXF7R1Y1NsgniRknTga8Cq4BXCD/M92IFUG1mm8KyHHcCpw3a5jSgv2LsvcAJCuZI/ADwopn9FcDMGsysL8lYnct8HfUIo95KI2lBZBdXUsFOL7fhxiSpi9RmdlP48DFg3ySPPQ/YMuB5DbByqG3MrFdSC1ABHACYpAeBSuBOM/vW4BNIOh84H2DBggVJhuVcBmgLxpnWWWnqr0EQtCAqYq00eoJwY5BUC0JShaT/lPScpGclfU/ScKN7EhWXGVwyfKhtsoFjgU+Evz8i6YS3bWh2o5ktN7PllZWVSbwS5zJEe5AgduWUMy2KcQqFMyijjaa2jtQf200Zyf7LvJOgtMYZwJlAPXDXMPvUAPMHPK8Ctg21TXjdoRRoDJc/Zmb1ZtYBrAaOSDJW5zJfW1Anqa9gZjTHDwfL9bT6aGo3eskmiHIz+7qZvR7+XA2UDbPPGmCxpEVhqfCzCa5fDLQKODd8fCbwBzMz4EHgUEkFYeJ4D8F1D+cmh7Ydwe/CiBJEYViwLywp7txoJJsg/ijpbEmx8OejBNVch2RmvcBFBB/264G7zWydpKsk9c9EdzNQIakauAS4LNy3CfguQZJ5AXjOzPZ6PucmlPY6OsmloLg0muOHLYisDm9BuNEbrlhfK8E1ARF8gN8RrooBbcDle9vfzFYTdA8NXPa1AY87CQr/Jdr3jgHnc25yaaulnlJmRjXKOSy3kd3ZiJkR3Bzo3MgMV4upeLwCcW4qibfVUhsvia4MRjgnRIm10NrVS0le6masc1NHshMGEXYLHRc+fdTMfhNNSM5Nfn2tO6i30sgTRAU7qW/t8gThRiXZ21yvAf6V4ELxK8C/hsucc6PRVhckiKKIEkTWNHpzSijXTupau6I5h5v0km1BfBBYZmZxAEm3Ac8TXlR2zo1AvI/srkbqKGVJSXQlxuL5FVTsaqWuzROEG52RjNAZeFtrRLdeODcFdDQgi0fbxUQwlWk53oJwo5dsC+IbwPOS/khwR9NxwJcji8q5ySwss1FvpcwoSn2ZjX7ZxZVU6C1PEG7Uhk0QYfG8J4GjgaMIEsSXzOytiGNzbnIKB8l15laQm50V2WkU1mPyBOFGa9gEYWYm6VdmdiRvHwntnBup9qDMhkVVZqNfwQzKaKWutTPa87hJK9lrEH+RdFSkkTg3VYRdTFklESeIwkqy6aOr1cttuNFJ9hrEe4ELJW0G2gm6mczMDo0qMOcmrfZausihqGR6tOcpnhX8bt0R7XncpJVsgjg50iicm0KsrZZ6K2VmhLe4AlA0G4Bpu2qJx41YzMttuJEZrhZTHnAhsD/wEnBzWITPOTdKfa211FmEZTb6FQcJotKaaOropiyjN3MAABaASURBVCKqQXlu0hruGsRtwHKC5HAy8J3II3JukuvbuYO6iMdAALsTxEw1+2A5NyrDdTEtMbNDACTdDDwTfUjOTW7qqKPeDmZ+UcRdTDmF9E0rYmZvE3WtXRw4O9rTuclnuBZET/8D71pyLgXifWR3Ngalvkui7/KJF85ippp8LIQbleFaEIdJ2hk+FpAfPu+/i6kk0uicm2w6GolZX7SF+gZQyRxmNtbynCcINwrDzQcR3TBP56ai9mAMRJNKKSuIvgR3VslsZmmjtyDcqIykWJ9zbqzCQXK9+ZXjMsubimeHXUw+mtqNnCcI58ZTWGaDwohHUfcrnk0e3bTvbByf87lJxROEc+Npd5mNWeNzvnCwnO302ppu5DxBODee2mvpJpui0vLxOV9YbiPW4eU23Mh5gnBuHMVba4NBciX543PC4jkAFHTV09XbNz7ndJNGpAlC0kmSNkiqlvS26Ukl5Uq6K1z/tKSFg9YvkNQm6dIo43RuvPTs3BH5THJ7KApaELPURENb9/ic000akSUISVnADQQlOpYA50haMmizzwJNZrY/cB3wzUHrrwN+F1WMzo23eGvtuI2BACC3mL6s/KDcht/q6kYoyhbECqDazDaZWTdwJ3DaoG1OI6j3BHAvcEI4gx2S/gHYBKyLMEbnxlVW23ZqrWz8WhASvYWzPEG4UYkyQcwDtgx4XhMuS7hNWMqjBaiQVAh8CbhybyeQdL6ktZLW1tXVpSxw5yLRs4ucrga22gxmjleCACiezSw1Ue8F+9wIRZkgEo0CsiS3uRK4zsza9nYCM7vRzJab2fLKyspRhuncONm5DYBtVjF+LQggu3Q2lXgLwo1cshMGjUYNMH/A8ypg2xDb1EjKBkqBRmAlcKakbwFlQFxSp5ldH2G8zkWrJWhQN+XMIm/a+FWxySqZy6yYl/x2IxdlglgDLJa0CNgKnA18fNA2q4BzgaeAM4E/mJkB7+7fQNIVQJsnBzfhtdQA0FM4d3zPWzSLQjrZ2dI0vud1E15kCcLMeiVdBDwIZAG3mNk6SVcBa81sFXAzcLukaoKWw9lRxeNc2rXUEEdYyTgniHDioL6W7eN7XjfhRdmCwMxWA6sHLfvagMedwFnDHOOKSIJzbrw1b6FB0ykvKR7f84ZjIdTuo6ndyPhIaufGibVsoaavgqrp4zSKul84mnpaR+34ntdNeJ4gnBsnfU1b2GrpSBBBC6Ksr4G2Lp8Y0iXPE4Rz48GM2M4atloF86cXjO+588roi+UwU81sa941vud2E5onCOfGQ3s9sXg322wG88vHOUFI9BYEo6lrmjrG99xuQvME4dx4aHkTgO1UMLcsb9xPHyuZzUya2dLoLQiXPE8Qzo2HcAzErvx55GaP/1Tv2aVzmB1rYkujtyBc8jxBODcewgSRXT5/mA2joeI5zFILW7yLyY2AJwjnxkNLDR3kUVY+TnNRD1Y8iyLaqWv00dQueZ4gnBsH8eYtbI1XjP8F6n6lQculr+nN9JzfTUieIJwbBz2Nb7DNKqhKV4Io3xeAiu5ttOzqSU8MbsLxBOHcOFBLmsZA9Ju+CICF2uEXql3SPEE4F7WeXeR0NbLVKsd/FHW/gnL6ckrYR29R0+S3urrkeIJwLmotWwF4SzOYUzr+YyAAkLDpi1ioHT5YziXNE4RzUQsnCuounEN2Vvr+y2XN2I+FsVrvYnJJ8wThXNTCBKHSBWkNQ+WLmKc6tje2pjUON3F4gnAuauFEQYUzqtIbR/m+ZNNHV8Mb6Y3DTRieIJyLWG/Tm9RaGXMrStMbSHir67SWzQQz+zq3d54gnItYd8ObbLUZVJWn6Q6mfmGCmB1/i4b27vTG4iYETxDORa1lC9vSOQaiX9Es+rLyWei3urokeYJwLkq9XeS1b+UNm5W+Mhv9JHpK92EfHyznkuQJwrko1W8kZn1s0nwqi3LTHQ3ZM/YLRlP7WAiXBE8QzkWp7m8A7CxaTCymNAcTJIgFsVq2NranOxQ3AUSaICSdJGmDpGpJlyVYnyvprnD905IWhsv/h6RnJb0U/n5flHE6F5na9fQRIz5j/3RHEihfRC49tNV5VVc3vMgShKQs4AbgZGAJcI6kJYM2+yzQZGb7A9cB3wyX1wOnmNkhwLnA7VHF6VyUrG49bzI7/be49gvvZIo1vZ7mQNxEEGULYgVQbWabzKwbuBM4bdA2pwG3hY/vBU6QJDN73sy2hcvXAXmS0t+B69wI9b61nr/1zWPp3MxKEIVtbxKP+1gIt3dRJoh5wJYBz2vCZQm3MbNeoAWoGLTNGcDzZtY1+ASSzpe0VtLaurq6lAXuXEr0dJLdvJlXrYqDMyVBlMyjT9nM4y227+xMdzQuw0WZIBJdkRv8lWWv20haStDtdEGiE5jZjWa23MyWV1ZWjjpQ5yJR/yoizmvM54DZRemOJhDLoqdkAftoB69s25nuaFyGizJB1AADZ2ivArYNtY2kbKAUaAyfVwH3A58ys9cijNO5aIR3MHWXv4Pc7Kw0B/N30yr3Z6F28NLWlnSH4jJclAliDbBY0iJJOcDZwKpB26wiuAgNcCbwBzMzSWXAb4Evm9mfIozRucjYjlfoIYvpVQemO5Q9ZFXsx6LYDl7a0pTuUFyGiyxBhNcULgIeBNYDd5vZOklXSTo13OxmoEJSNXAJ0H8r7EXA/sBXJb0Q/syMKlbnotC1fR2vx2dz0PwZ6Q5lT+X7kk8n27a+6UX73F5lR3lwM1sNrB607GsDHncCZyXY72rg6ihjcy5q8R3redWqMucOpn6zDwFg/q71vLWzkzmlaS4i6DKWj6R2LgrdHeS311BtVSyZU5LuaPY0dxnx2DSOjL3KSzV+HcINzROEc1Go34AwWkv2Jz8ncy5QAzAtH2YfypGxjX6h2u2VJwjnolAb3ME0bfbg4gGZIbbPMRwW28QrNfXpDsVlME8QzkWgfevLdFk2MxdmZoJg/gpy6aan5q9+odoNyROEcxHYtfVlNtkcllYNLgyQIapWALB/1ytsb/ER1S4xTxDORSCnYQMbrYolczPsAnW/kjl0FVVxROxVXvQL1W4IniCcS7WmzZR0befN/CUU501LdzRDyt7naJbHXuXlmuZ0h+IylCcI51LMNvwOgPq5701zJHuXtc/RzFYTW9+sTncoLkN5gnAuxdpf/i3V8bnsd+Ch6Q5l7+YH1yFyt6/xC9UuIU8QzqVS507ytz7FI/EjOPng2emOZu9mLqUnK5939KznzUafo9q9nScI51LIXvsDWdZL7ZzjmVGU4XNcZWXTN+dIjoy9ym9f2p7uaFwG8gThXAo1v/Brmq2QA448Id2hJCVv32NYGnuD3z37mnczubfxBOFcqsT7yH39YR6LL+PEQ6vSHU1yFh1HFnH2bXiUdT6BkBvEE4RzKWI1aynobWbbrPdQVpCT7nCSs8+x9JXvz3nZv+eXz21NdzQuw3iCcC5Fdqz9Fb0WY+7yU9IdSvJiMbJWns/hsY1sfOFxevvi6Y7IZRBPEM6lghl69QHW2oG8d9nidEczMoedQ292Aad2/ZYnqr14n/s7TxDOpUD3ul8zq3MTGys/QEkGj55OKK8EHXYOp2Y/xUPPrEt3NC6DeIJwbqy6O+hY9W/8LT6ffU/8XLqjGZWsleeTSw8VG+9iZ2dPusNxGcIThHNjtH31NyjrfovH9/833nVAhg+OG8rMA2mb+07O0e+58ldeAtwFPEE4NwZdtdVUvPAjHoq9m7M/+vF0hzMmRcf9C/NUz4Evf4d71takOxyXATxBODdafb3U/Pwiui2LolO+MfGuPQz2jpOJr7iAf8pezeu/voZXd7SmOyKXZp4gnBuFXQ01bL7uBPZrfoqH55zPOw8/JN0hjZ1E7KRr6DzgVL4Uu4N7f3odta0+mdBUFmmCkHSSpA2SqiVdlmB9rqS7wvVPS1o4YN2Xw+UbJJ0YZZzOJc2M1/50H53Xv5OZreu5f+HXOOmzV6Q7qtSJxcg76ye0zFzJpR3f46Frz+Vnqx+js6cv3ZG5NFBUF6MkZQGvAv8DqAHWAOeY2SsDtvlfwKFmdqGks4GPmNnHJC0BfgGsAOYCDwMHmNmQ/0qXL19ua9eujeS1uKnJ4nGaG96icfsbtG7fSO/Gh1nY8AQzrJHXtICdH/4Jhx95dLrDjMauZlpX/Tv56+9D1sejWe+kfe4xlC44hPmLD6Ni5hyK83OJxZTuSN0YSXrWzJYnWpcd4XlXANVmtikM4k7gNOCVAducBlwRPr4XuF6SwuV3mlkX8Lqk6vB4T6U6yJaGHTTdMDEKq7mRG/rjyxDBlyNZ8FjEiVmcHLrJtW7y6GK64kwP92i3PF4pPIr1i97PoSd+hv1KMnQ60VTIL6P4YzfCzivY+sB3Wfm3Oymq+VPwVe/PwSbtlkuH8ukjmz5lYcSII1Dw1xzqr+/3R6VeQ+Fijvzi/Sk/bpQJYh6wZcDzGmDlUNuYWa+kFqAiXP6XQfvOG3wCSecD5wMsWLBgVEEqK5vGgkWj2tdNFEN9UAUfZgCmLECYYlhWLvHsPCw7HxXNYtr0eRRUzmfR0mM4Kr9gHOPOACVzmffRa8G+TV9zDW9VP0/DG+voaW/GOndCdxvEeyAeR9YLFg9GlQ+ZBjw9RKG7eH4kx40yQST6Xzn4X8dQ2ySzL2Z2I3AjBF1MIw0QoKSsgiMu/fVodnVu6pDImj6feUfNZ95Rp6Y7GjdOorxIXQMMTGtVwLahtpGUDZQCjUnu65xzLkJRJog1wGJJiyTlAGcDqwZtswo4N3x8JvAHC66arwLODu9yWgQsBp6JMFbnnHODRNbFFF5TuAh4EMgCbjGzdZKuAtaa2SrgZuD28CJ0I0ESIdzuboIL2r3AP+/tDibnnHOpF9ltruPNb3N1zrmR29ttrj6S2jnnXEKeIJxzziXkCcI551xCniCcc84lNGkuUkuqA94YwyFmAFNtQt6p9pqn2usFf81TxVhe8z5mVploxaRJEGMlae1QV/Inq6n2mqfa6wV/zVNFVK/Zu5icc84l5AnCOedcQp4g/u7GdAeQBlPtNU+11wv+mqeKSF6zX4NwzjmXkLcgnHPOJeQJwjnnXEJTPkFIOknSBknVki5LdzxRkDRf0h8lrZe0TtK/hsvLJf1e0sbw9/ThjjXRSMqS9Lyk34TPF0l6OnzNd4Wl6CcNSWWS7pX0t/D9Pmayv8+SvhD+u35Z0i8k5U2291nSLZJqJb08YFnC91WBH4SfaS9KOmK0553SCUJSFnADcDKwBDhH0pL0RhWJXuCLZnYQcDTwz+HrvAx4xMwWA4+EzyebfwXWD3j+TeC68DU3AZ9NS1TR+T7wgJkdCBxG8Non7fssaR7weWC5mR1MMLXA2Uy+9/mnwEmDlg31vp5MMIfOYoIpmX802pNO6QQBrACqzWyTmXUDdwKnpTmmlDOz7Wb2XPi4leBDYx7Ba70t3Ow24B/SE2E0JFUBHwJuCp8LeB9wb7jJpHrNkkqA4wjmWcHMus2smUn+PhPMa5MfzkpZAGxnkr3PZvY4wZw5Aw31vp4G/MwCfwHKJM0ZzXmneoKYB2wZ8LwmXDZpSVoIHA48Dcwys+0QJBFgZvoii8T3gH8H4uHzCqDZzHrD55Pt/d4XqANuDbvVbpJUyCR+n81sK3At8CZBYmgBnmVyv8/9hnpfU/a5NtUThBIsm7T3/UoqAu4DLjaznemOJ0qSPgzUmtmzAxcn2HQyvd/ZwBHAj8zscKCdSdSdlEjY734asAiYCxQSdLEMNpne5+Gk7N/5VE8QNcD8Ac+rgG1piiVSkqYRJIf/NrNfhot39Dc9w9+16YovAu8CTpW0maDr8H0ELYqysCsCJt/7XQPUmNnT4fN7CRLGZH6f3w+8bmZ1ZtYD/BJ4J5P7fe431Puass+1qZ4g1gCLwzsecggubq1Kc0wpF/a93wysN7PvDli1Cjg3fHwu8P/GO7aomNmXzazKzBYSvK9/MLNPAH8Ezgw3m2yv+S1gi6R3hItOIJjXfdK+zwRdS0dLKgj/nfe/5kn7Pg8w1Pu6CvhUeDfT0UBLf1fUSE35kdSSPkjwzTILuMXM/k+aQ0o5SccCTwAv8ff++P8guA5xN7CA4D/aWWY2+ELYhCfpeOBSM/uwpH0JWhTlwPPAJ82sK53xpZKkZQQX5XOATcCnCb4ITtr3WdKVwMcI7tZ7HvhHgj73SfM+S/oFcDxBWe8dwOXAr0jwvoaJ8nqCu546gE+b2dpRnXeqJwjnnHOJTfUuJuecc0PwBOGccy4hTxDOOecS8gThnHMuIU8QzjnnEvIE4SYESX2SXggrdt4jqSBFx71Q0qfCx49KetvE75LOk3R9Ks43RAzH91ebHU+SKsOKp89Levd4n99lPk8QbqLYZWbLwoqd3cCFqTiomf3YzH6WimNNQCcAfzOzw83siXQH4zKPJwg3ET0B7A8g6ZKwVfGypIvDZYWSfivpr+Hyj4XLr5H0Slgj/9pw2RWSLh1w7E9K+nO434rBJw6/dd8naU34864E2zwtaemA549KOlLSivDYz4e/35Fg3z3iCeNYGD7+pKRnwpbUfymY6yJL0k/D7V6S9IUEx9xH0iPh635E0oJwQN23gA+Gx8sfsH2pgjlS3hE+/4Wkf9r7W+Imo+zhN3Euc4T1dU4GHpB0JMFI4ZUEBcqelvQYQVXTbWb2oXCfUknlwEeAA83MJJUNcYpCM3unpOOAW4CDB63/PsE8A09KWgA8CBw0aJs7gY8Cl4c1cuaa2bP95bjNrFfS+4H/C5yR5Os+iGC08LvMrEfSD4FPAOuAeWHLiiFe1/UE5Z9vk/QZ4Adm9g+SvkYwj8JFAzc2sxZJFwE/lfR9YLqZ/SSZON3k4i0IN1HkS3oBWEtQVuBm4FjgfjNrN7M2gkJt7yYoKfJ+Sd+U9G4zawF2Ap3ATZJOJyhBkMgvYHf9/ZIEH7jvB64PY1kVblM8aJu7gbPCxx8F7gkflwL3KJgV7DpgKck7ATgSWBOe+wSCRLgJ2FfSf0o6KXydgx0D/Dx8fDvB322vzOz3BH/HGwhKV7gpyFsQbqLYZWbLBi4Ia868jZm9GrYuPgh8Q9JDZnZV2GV0AkHxvosIKry+bfdhnseAY8xs11CBmtlWSQ2SDiX41n9BuOrrwB/N7CNht9GjCXbvZc8vbnnhbwG3mdmXB+8g6TDgROCfCRLSZ4aKbYjX9DaSYgQto10E9YxqhtvHTT7egnAT2ePAPyio5FlI0IX0hKS5QIeZ3UEwmcwRCubCKDWz1cDFwLIhjtl/veJYgiqYLYPWP0SQXAi3G+o4dxJMVlRqZi+Fy0qBreHj84bYbzNBiW4UzCW8KFz+CHCmpJnhuvLw2sIMIGZm9wFf7d93kD8TJEUIuqWeHOLcA32BYObBc4BbFJSLd1OMtyDchGVmz0n6KfBMuOgmM3te0onAtyXFgR7gc0Ax8P8k5RF8G3/bxdxQk6Q/AyUk/ib+eeAGSS8S/P95nMR3VN1LcL3i6wOWfQu4TdIlwB+GOP99BKWaXyAoR/9q+FpfkfQV4KHw230PQYthF8EMcv1f9t7WwghjvkXSvxHMOPfpIc4NgKQDCLqVVphZq6THga8QVBB1U4hXc3XOOZeQdzE555xLyBOEc865hDxBOOecS8gThHPOuYQ8QTjnnEvIE4RzzrmEPEE455xL6P8DvFItfflgo/8AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "uniform.plot(label='Uniform')\n",
    "triangle.plot(label='Triangle')\n",
    "\n",
    "plt.xlabel('Possible values of x')\n",
    "plt.ylabel('Probability')\n",
    "plt.title('140 heads, 110 tails')\n",
    "plt.legend();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The posterior distributions are almost identical because, in this case, we have enough data to \"swamp the prior\"; that is, the posteriors depend strongly on the data and only weakly on the priors.\n",
    "\n",
    "This is good news, because it suggests that we can use data to resolve arguments.  Suppose two people disagree about the correct prior.  If neither can persuade the other, they might have to agree to disagree.\n",
    "\n",
    "But if they get new data, and each of them does a Bayesian update, they will usually find their beliefs converging.\n",
    "\n",
    "And with enough data, the remaining difference can be so small that it makes no difference in practice."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Summarizing the posterior distribution\n",
    "\n",
    "The posterior distribution contains all of the information we have about the value of $x$.  But sometimes we want to summarize this information.\n",
    "\n",
    "We have already seen one way to summarize a posterior distribution, the Maximum Aposteori Probability, or MAP:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "56"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "uniform.idxmax()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`idxmax` returns the value of $x$ with the highest probability.\n",
    "\n",
    "In this example, we get the same MAP with the triangle prior:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "56"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "triangle.idxmax()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Another way to summarize the posterior distribution is the posterior mean.\n",
    "\n",
    "Given a set of values, $x_i$, and the corresponding probabilities, $p_i$, the mean of the distribution is:\n",
    "\n",
    "$\\sum_i x_i p_i$\n",
    "\n",
    "Using the posterior PMF, we can compute it like this:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "While we're at it, I'll make a function to compute the posterior mean, too:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [],
   "source": [
    "def pmf_mean(pmf):\n",
    "    \"\"\"Compute the mean of a PMF.\n",
    "    \n",
    "    pmf: Series representing a PMF\n",
    "    \n",
    "    return: float\n",
    "    \"\"\"\n",
    "    return np.sum(pmf.index * pmf)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here's the posterior mean based on the uniform prior:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "55.95238095238095"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pmf_mean(uniform)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And here's the posterior mean with the triangle prior:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "55.74349943859505"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pmf_mean(triangle)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The posterior means are not identical, but they are close enough that the difference probably doesn't matter.\n",
    "\n",
    "In this example, the posterior mean is very close to the MAP.  That's true when the posterior distribution is symmetric, but it is not always true."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If someone asks what we think $x$ is, the MAP or the posterior mean might be a good answer.\n",
    "\n",
    "But MacKay asked a different question: do these data give evidence that the coin is biased rather than fair?\n",
    "\n",
    "We have more work to do before we can really answer this question.  But first, I want to rule out an approach that is tempting, but incorrect."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Posterior probability\n",
    "\n",
    "If the coin is \"fair\", that means that $x$ is 50%.  So it might be tempting to use the posterior PMF to compute the probability that $x$ is 50%:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.020976526129544655"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "uniform[50]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The result is the posterior probability that $x$ is 50%, but it is not the probability that the coin is fair.\n",
    "\n",
    "The problem is that $x$ is really a continuous quantity, which means it could have any value between 0 and 1.  \n",
    "\n",
    "For purposes of computation, I broke this interval into 101 discrete values, but that was an arbitrary choice.  I could have done the computation with 201 hypotheses, like this:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "201"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xs2 = np.linspace(0, 100, 201)\n",
    "prior2 = 1/201\n",
    "\n",
    "uniform2 = make_pmf(xs2, prior2)\n",
    "len(uniform2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here's the update."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [],
   "source": [
    "bayes_update_euro(uniform2, data)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And here's what the results look like."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "uniform2.plot(color='C2')\n",
    "\n",
    "plt.xlabel('201 possible values of x')\n",
    "plt.ylabel('Probability')\n",
    "plt.title('140 heads, 110 tails');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The results are visually similar, but you might notice that the curve is a little smoother at the peak.\n",
    "\n",
    "The MAPs are the same and the posterior means are almost the same:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(56, 56.0)"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "uniform.idxmax(), uniform2.idxmax()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(55.95238095238095, 55.95238095238094)"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pmf_mean(uniform), pmf_mean(uniform2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "But the total probability is spread out over twice as many hypotheses, so the proability of any single hypothesis is smaller.\n",
    "\n",
    "If use both posteriors to compute the probability that $x$ is 50%, we get very different results."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.020976526129544655, 0.01048826306477235)"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "uniform[50], uniform2[50]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Because $x$ is continuous, we divided the interval into discrete values.  But the number of values was an arbitrary choice, so the probability of any single value is not meaningful. \n",
    "\n",
    "However, we can meaningfully compute the probability that $x$ falls in an interval."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Credible intervals\n",
    "\n",
    "We can use a Booleans series to select values from the posterior distribution and add up their probabilities.\n",
    "\n",
    "Here's a function that computes the total probability of all values less than or equal to a given value of $x$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [],
   "source": [
    "def prob_le(pmf, threshold):\n",
    "    le = (pmf.index <= threshold)\n",
    "    total = pmf[le].sum()\n",
    "    return total"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For example, here's the probability that $x$ is less than or equal to 60%, based on the uniform prior with 101 values."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.928955163489148"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "prob_le(uniform, 60)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here's what we get with 201 values."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.9164519507463521"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "prob_le(uniform2, 60)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And with the triangle prior"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.9384439012832195"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "prob_le(triangle, 60)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The results are not identical, but they are close enough that the differences might not matter.\n",
    "\n",
    "So let's say that the probability is 92% that $x$ is less than or equal to 61.\n",
    "\n",
    "I'll also compute the probability that $x$ is less than or equal to 51:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.04056797774604359, 0.034538255921946155, 0.04535722804399313)"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "prob_le(uniform, 50), prob_le(uniform2, 50), prob_le(triangle, 50)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It looks like the probability is about 4% that $x$ is less than 50.\n",
    "\n",
    "Putting these results together, we can estimate the probability that $x$ is between 50 and 60; it's about 92% - 4% = 88%.\n",
    "\n",
    "And interval like this is called a \"credible interval\" because it tells us how credible it is that $x$ falls in the interval.\n",
    "\n",
    "In this case the interval from 50 to 60 is an 88% credible interval."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Summary\n",
    "\n",
    "In this notebook, we used data from a coin-spinning experiment to estimate the probability that a given coin lands on heads.\n",
    "\n",
    "We tried three different priors: uniform distributions with 101 and 201 values, and a triangle distribution.  The results are similar, which indicates that we have enough data to \"swamp the priors\".\n",
    "\n",
    "And we summarized the posterior distributions three ways, computing the value with Maximum Aposteori Probability (MAP), the posterior mean, and a credible interval.\n",
    "\n",
    "Although we have made progress, we have not yet answered the question I started with, \"Do these data give evidence that the coin is biased rather than fair?\"\n",
    "\n",
    "We'll come back to this in a future notebook, but [in the next notebook](https://colab.research.google.com/github/AllenDowney/BiteSizeBayes/blob/master/08_soccer.ipynb), we'll take on a different question, which I call the World Cup problem."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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