soln/chap18.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Conjugate Priors"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "tags": [
     "remove-cell"
    ]
   },
   "source": [
    "Think Bayes, Second Edition\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",
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   "metadata": {
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   "source": [
    "# If we're running on Colab, install empiricaldist\n",
    "# https://pypi.org/project/empiricaldist/\n",
    "\n",
    "import sys\n",
    "IN_COLAB = 'google.colab' in sys.modules\n",
    "\n",
    "if IN_COLAB:\n",
    "    !pip install empiricaldist"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "execution": {
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     "remove-cell"
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   "outputs": [],
   "source": [
    "# Get utils.py\n",
    "\n",
    "from os.path import basename, exists\n",
    "\n",
    "def download(url):\n",
    "    filename = basename(url)\n",
    "    if not exists(filename):\n",
    "        from urllib.request import urlretrieve\n",
    "        local, _ = urlretrieve(url, filename)\n",
    "        print('Downloaded ' + local)\n",
    "    \n",
    "download('https://github.com/AllenDowney/ThinkBayes2/raw/master/soln/utils.py')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "execution": {
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    },
    "tags": [
     "remove-cell"
    ]
   },
   "outputs": [],
   "source": [
    "from utils import set_pyplot_params\n",
    "set_pyplot_params()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the previous chapters we have used grid approximations to solve a variety of problems.\n",
    "One of my goals has been to show that this approach is sufficient to solve many real-world problems.\n",
    "And I think it's a good place to start because it shows clearly how the methods work.\n",
    "\n",
    "However, as we saw in the previous chapter, grid methods will only get you so far.\n",
    "As we increase the number of parameters, the number of points in the grid grows (literally) exponentially.\n",
    "With more than 3-4 parameters, grid methods become impractical.\n",
    "\n",
    "So, in the remaining three chapters, I will present three alternatives:\n",
    "\n",
    "1. In this chapter we'll use **conjugate priors** to speed up some of the computations we've already done.\n",
    "\n",
    "2. In the next chapter, I'll present Markov chain Monte Carlo (MCMC) methods, which can solve problems with tens of parameters, or even hundreds, in a reasonable amount of time.\n",
    "\n",
    "3. And in the last chapter we'll use Approximate Bayesian Computation (ABC) for problems that are hard to model with simple distributions.\n",
    "\n",
    "We'll start with the World Cup problem."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The World Cup Problem Revisited\n",
    "\n",
    "In <<_PoissonProcesses>>, we solved the World Cup problem using a Poisson process to model goals in a soccer game as random events that are equally likely to occur at any point during a game.\n",
    "\n",
    "We used a gamma distribution to represent the prior distribution of $\\lambda$, the goal-scoring rate.  And we used a Poisson distribution to compute the probability of $k$, the number of goals scored.\n",
    "\n",
    "Here's a gamma object that represents the prior distribution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "execution": {
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   "outputs": [],
   "source": [
    "from scipy.stats import gamma\n",
    "\n",
    "alpha = 1.4\n",
    "dist = gamma(alpha)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And here's a grid approximation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "execution": {
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    }
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from utils import pmf_from_dist\n",
    "\n",
    "lams = np.linspace(0, 10, 101)\n",
    "prior = pmf_from_dist(dist, lams)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here's the likelihood of scoring 4 goals for each possible value of `lam`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "execution": {
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   "outputs": [],
   "source": [
    "from scipy.stats import poisson\n",
    "\n",
    "k = 4\n",
    "likelihood = poisson(lams).pmf(k)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And here's the update."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "execution": {
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.05015532557804499"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "posterior = prior * likelihood\n",
    "posterior.normalize()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So far, this should be familiar.\n",
    "Now we'll solve the same problem using the conjugate prior."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The Conjugate Prior\n",
    "\n",
    "In <<_TheGammaDistribution>>, I presented three reasons to use a gamma distribution for the prior and said there was a fourth reason I would reveal later.\n",
    "Well, now is the time.\n",
    "\n",
    "The other reason I chose the gamma distribution is that it is the \"conjugate prior\" of the Poisson distribution, so-called because the two distributions are connected or coupled, which is what \"conjugate\" means.\n",
    "\n",
    "In the next section I'll explain *how* they are connected, but first I'll show you the consequence of this connection, which is that there is a remarkably simple way to compute the posterior distribution.\n",
    "\n",
    "However, in order to demonstrate it, we have to switch from the one-parameter version of the gamma distribution to the two-parameter version.  Since the first parameter is called `alpha`, you might guess that the second parameter is called `beta`.\n",
    "\n",
    "The following function takes `alpha` and `beta` and makes an object that represents a gamma distribution with those parameters."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "execution": {
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   "source": [
    "def make_gamma_dist(alpha, beta):\n",
    "    \"\"\"Makes a gamma object.\"\"\"\n",
    "    dist = gamma(alpha, scale=1/beta)\n",
    "    dist.alpha = alpha\n",
    "    dist.beta = beta\n",
    "    return dist"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here's the prior distribution with `alpha=1.4` again and `beta=1`. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "execution": {
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     "shell.execute_reply": "2021-04-16T19:40:54.622194Z"
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   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.4"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "alpha = 1.4\n",
    "beta = 1\n",
    "\n",
    "prior_gamma = make_gamma_dist(alpha, beta)\n",
    "prior_gamma.mean()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now I claim without proof that we can do a Bayesian update with `k` goals just by making a gamma distribution with parameters `alpha+k` and `beta+1`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "execution": {
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   "outputs": [],
   "source": [
    "def update_gamma(prior, data):\n",
    "    \"\"\"Update a gamma prior.\"\"\"\n",
    "    k, t = data\n",
    "    alpha = prior.alpha + k\n",
    "    beta = prior.beta + t\n",
    "    return make_gamma_dist(alpha, beta)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here's how we update it with `k=4` goals in `t=1` game."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-04-16T19:40:54.632759Z",
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     "shell.execute_reply": "2021-04-16T19:40:54.634216Z"
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   },
   "outputs": [],
   "source": [
    "data = 4, 1\n",
    "posterior_gamma = update_gamma(prior_gamma, data)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "After all the work we did with the grid, it might seem absurd that we can do a Bayesian update by adding two pairs of numbers.\n",
    "So let's confirm that it works.\n",
    "\n",
    "I'll make a `Pmf` with a discrete approximation of the posterior distribution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "execution": {
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   "outputs": [],
   "source": [
    "posterior_conjugate = pmf_from_dist(posterior_gamma, lams)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The following figure shows the result along with the posterior we computed using the grid algorithm."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "execution": {
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     "hide-cell"
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   "outputs": [],
   "source": [
    "from utils import decorate\n",
    "\n",
    "def decorate_rate(title=''):\n",
    "    decorate(xlabel='Goal scoring rate (lam)',\n",
    "             ylabel='PMF',\n",
    "             title=title)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
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    "tags": [
     "hide-input"
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   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "posterior.plot(label='grid posterior', color='C1')\n",
    "posterior_conjugate.plot(label='conjugate posterior', \n",
    "                         color='C4', ls=':')\n",
    "\n",
    "decorate_rate('Posterior distribution')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "They are the same other than small differences due to floating-point approximations."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-04-16T19:40:54.860075Z",
     "iopub.status.busy": "2021-04-16T19:40:54.859461Z",
     "iopub.status.idle": "2021-04-16T19:40:54.862369Z",
     "shell.execute_reply": "2021-04-16T19:40:54.861978Z"
    },
    "tags": [
     "hide-cell"
    ]
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.allclose(posterior, posterior_conjugate)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## What the Actual?\n",
    "\n",
    "To understand how that works, we'll write the PDF of the gamma prior and the PMF of the Poisson likelihood, then multiply them together, because that's what the Bayesian update does.\n",
    "We'll see that the result is a gamma distribution, and we'll derive its parameters.\n",
    "\n",
    "Here's the PDF of the gamma prior, which is the probability density for each value of $\\lambda$, given parameters $\\alpha$ and $\\beta$:\n",
    "\n",
    "$$\\lambda^{\\alpha-1} e^{-\\lambda \\beta}$$\n",
    "\n",
    "I have omitted the normalizing factor; since we are planning to normalize the posterior distribution anyway, we don't really need it.\n",
    "\n",
    "Now suppose a team scores $k$ goals in $t$ games.\n",
    "The probability of this data is given by the PMF of the Poisson distribution, which is a function of $k$ with $\\lambda$ and $t$ as parameters.\n",
    "\n",
    "$$\\lambda^k e^{-\\lambda t}$$\n",
    "\n",
    "Again, I have omitted the normalizing factor, which makes it clearer that the gamma and Poisson distributions have the same functional form.\n",
    "When we multiply them together, we can pair up the factors and add up the exponents.\n",
    "The result is the unnormalized posterior distribution,\n",
    "\n",
    "$$\\lambda^{\\alpha-1+k} e^{-\\lambda(\\beta + t)}$$\n",
    "\n",
    "which we can recognize as an unnormalized gamma distribution with parameters $\\alpha + k$ and $\\beta + t$.\n",
    "\n",
    "This derivation provides insight into what the parameters of the posterior distribution mean: $\\alpha$ reflects the number of events that have occurred; $\\beta$ reflects the elapsed time."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Binomial Likelihood\n",
    "\n",
    "As a second example, let's look again at the Euro problem.\n",
    "When we solved it with a grid algorithm, we started with a uniform prior:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-04-16T19:40:54.867138Z",
     "iopub.status.busy": "2021-04-16T19:40:54.866520Z",
     "iopub.status.idle": "2021-04-16T19:40:54.868867Z",
     "shell.execute_reply": "2021-04-16T19:40:54.868399Z"
    }
   },
   "outputs": [],
   "source": [
    "from utils import make_uniform\n",
    "\n",
    "xs = np.linspace(0, 1, 101)\n",
    "uniform = make_uniform(xs, 'uniform')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We used the binomial distribution to compute the likelihood of the data, which was 140 heads out of 250 attempts."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-04-16T19:40:54.872896Z",
     "iopub.status.busy": "2021-04-16T19:40:54.872260Z",
     "iopub.status.idle": "2021-04-16T19:40:54.874560Z",
     "shell.execute_reply": "2021-04-16T19:40:54.874118Z"
    }
   },
   "outputs": [],
   "source": [
    "from scipy.stats import binom\n",
    "\n",
    "k, n = 140, 250\n",
    "xs = uniform.qs\n",
    "likelihood = binom.pmf(k, n, xs)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Then we computed the posterior distribution in the usual way."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-04-16T19:40:54.879347Z",
     "iopub.status.busy": "2021-04-16T19:40:54.878723Z",
     "iopub.status.idle": "2021-04-16T19:40:54.882457Z",
     "shell.execute_reply": "2021-04-16T19:40:54.881808Z"
    },
    "tags": [
     "hide-output"
    ]
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.003944617569326651"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "posterior = uniform * likelihood\n",
    "posterior.normalize()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can solve this problem more efficiently using the conjugate prior of the binomial distribution, which is the beta distribution.\n",
    "\n",
    "The beta distribution is bounded between 0 and 1, so it works well for representing the distribution of a probability like `x`.\n",
    "It has two parameters, called `alpha` and `beta`, that determine the shape of the distribution.\n",
    "\n",
    "SciPy provides an object called `beta` that represents a beta distribution.\n",
    "The following function takes `alpha` and `beta` and returns a new `beta` object."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-04-16T19:40:54.887666Z",
     "iopub.status.busy": "2021-04-16T19:40:54.886655Z",
     "iopub.status.idle": "2021-04-16T19:40:54.889122Z",
     "shell.execute_reply": "2021-04-16T19:40:54.889678Z"
    }
   },
   "outputs": [],
   "source": [
    "import scipy.stats\n",
    "\n",
    "def make_beta(alpha, beta):\n",
    "    \"\"\"Makes a beta object.\"\"\"\n",
    "    dist = scipy.stats.beta(alpha, beta)\n",
    "    dist.alpha = alpha\n",
    "    dist.beta = beta\n",
    "    return dist"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It turns out that the uniform distribution, which we used as a prior, is the beta distribution with parameters `alpha=1` and `beta=1`.\n",
    "So we can make a `beta` object that represents a uniform distribution, like this:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-04-16T19:40:54.895074Z",
     "iopub.status.busy": "2021-04-16T19:40:54.894403Z",
     "iopub.status.idle": "2021-04-16T19:40:54.897275Z",
     "shell.execute_reply": "2021-04-16T19:40:54.896661Z"
    }
   },
   "outputs": [],
   "source": [
    "alpha = 1\n",
    "beta = 1\n",
    "\n",
    "prior_beta = make_beta(alpha, beta)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now let's figure out how to do the update.  As in the previous example, we'll write the PDF of the prior distribution and the PMF of the likelihood function, and multiply them together.  We'll see that the product has the same form as the prior, and we'll derive its parameters.\n",
    "\n",
    "Here is the PDF of the beta distribution, which is a function of $x$ with $\\alpha$ and $\\beta$ as parameters.\n",
    "\n",
    "$$x^{\\alpha-1} (1-x)^{\\beta-1}$$\n",
    "\n",
    "Again, I have omitted the normalizing factor, which we don't need because we are going to normalize the distribution after the update.\n",
    "\n",
    "And here's the PMF of the binomial distribution, which is a function of $k$ with $n$ and $x$ as parameters.\n",
    "\n",
    "$$x^{k} (1-x)^{n-k}$$\n",
    "\n",
    "Again, I have omitted the normalizing factor.\n",
    "Now when we multiply the beta prior and the binomial likelihood, the result is\n",
    "\n",
    "$$x^{\\alpha-1+k} (1-x)^{\\beta-1+n-k}$$\n",
    "\n",
    "which we recognize as an unnormalized beta distribution with parameters $\\alpha+k$ and $\\beta+n-k$.\n",
    "\n",
    "So if we observe `k` successes in `n` trials, we can do the update by making a beta distribution with parameters `alpha+k` and `beta+n-k`.\n",
    "That's what this function does:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-04-16T19:40:54.900940Z",
     "iopub.status.busy": "2021-04-16T19:40:54.900479Z",
     "iopub.status.idle": "2021-04-16T19:40:54.902177Z",
     "shell.execute_reply": "2021-04-16T19:40:54.902524Z"
    }
   },
   "outputs": [],
   "source": [
    "def update_beta(prior, data):\n",
    "    \"\"\"Update a beta distribution.\"\"\"\n",
    "    k, n = data\n",
    "    alpha = prior.alpha + k\n",
    "    beta = prior.beta + n - k\n",
    "    return make_beta(alpha, beta)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Again, the conjugate prior gives us insight into the meaning of the parameters; $\\alpha$ is related to the number of observed successes; $\\beta$ is related to the number of failures.\n",
    "\n",
    "Here's how we do the update with the observed data."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-04-16T19:40:54.906405Z",
     "iopub.status.busy": "2021-04-16T19:40:54.905881Z",
     "iopub.status.idle": "2021-04-16T19:40:54.907966Z",
     "shell.execute_reply": "2021-04-16T19:40:54.907527Z"
    }
   },
   "outputs": [],
   "source": [
    "data = 140, 250\n",
    "posterior_beta = update_beta(prior_beta, data)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To confirm that it works, I'll evaluate the posterior distribution for the possible values of `xs` and put the results in a `Pmf`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-04-16T19:40:54.912183Z",
     "iopub.status.busy": "2021-04-16T19:40:54.911680Z",
     "iopub.status.idle": "2021-04-16T19:40:54.914222Z",
     "shell.execute_reply": "2021-04-16T19:40:54.913751Z"
    }
   },
   "outputs": [],
   "source": [
    "posterior_conjugate = pmf_from_dist(posterior_beta, xs)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And we can compare the posterior distribution we just computed with the results from the grid algorithm."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-04-16T19:40:54.917775Z",
     "iopub.status.busy": "2021-04-16T19:40:54.917270Z",
     "iopub.status.idle": "2021-04-16T19:40:54.918951Z",
     "shell.execute_reply": "2021-04-16T19:40:54.919307Z"
    },
    "tags": [
     "hide-cell"
    ]
   },
   "outputs": [],
   "source": [
    "def decorate_euro(title):\n",
    "    decorate(xlabel='Proportion of heads (x)',\n",
    "             ylabel='Probability',\n",
    "             title=title)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-04-16T19:40:54.954377Z",
     "iopub.status.busy": "2021-04-16T19:40:54.936044Z",
     "iopub.status.idle": "2021-04-16T19:40:55.108007Z",
     "shell.execute_reply": "2021-04-16T19:40:55.107548Z"
    },
    "tags": [
     "hide-input"
    ]
   },
   "outputs": [
    {
     "data": {
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uF3xwuqp/9NFHrb7fcNxzzz2Ul5e3eLvbb7+dd955JwIRNS5i96BU1S8iNwBvAl7gYVVdLSJz3PK5ItIZWApkAgERuRkYDAwHrgRWisgKd5c/UdXXIxWvMfFI3aST4IPkLh0PKS+sTGPhB8kUlnjI7LKd805s6whjX9euXXnuuedafb/z588nPT2diRMntvq+m3PPPfdwxRVXkJqaGvY2tbW1/PKXv2zRcWpra/F6vS0Nr15E+5Sq6uuqOkBV+6nqb9z35qrqXHf5a1XtrqqZqprtLher6gJVFVUdrqoj3X+WnIxpIS3ZW78c3JxXp8qTycJlyaxen8iXa1v+F3Wse+yxxxg+fDgjRozgyiuvBGDLli2cdtppDB8+nNNOO42tW7cCzqgON910ExMnTqRv3771SSl40sJ58+Zxww031O//7LPPZv78+QA89NBDDBgwgKlTp/Ktb32rfr1XX32V8ePHM2rUKE4//XR27drF5s2bmTt3Ln/6058YOXIkH374IXv27OHCCy9k7NixjB07loULFx7yeebNm8fMmTOZPn06AwcO5Be/+EV92d13383QoUMZOnRo/WjmZWVlzJgxgxEjRjB06FCefvpp7r33Xnbs2MEpp5zCKaecAsBbb73FiSeeyOjRo7n44ospLS0FoHfv3vzyl79k0qRJPPvss8yePbv+vLz77ruMGjWKYcOGce2111JVVdXoNkfDhjoyJp4FNdtJ+qE99HJ6datfLtxfFdFnod554nPeffJzAE67bDinzxoeUv7aQ8tY8NJaAM66djQnnz84pPyFvyxiyZtfAnDed8Yzfnr/wx5v9erV/OY3v2HhwoXk5+ezf/9+AG644Qauuuoqrr76ah5++GFuuukmXnrpJQB27tzJggUL+OKLLzj33HPDbtrbsWMHv/rVr1i+fDkZGRmceuqpjBgxAnDGBPzkk08QEf7+97/z+9//nj/+8Y/MmTOnfk4ngFmzZvG9732PSZMmsXXrVr7xjW+wdu3aQ461ePFiVq1aRWpqKmPHjmXGjBmICI888giLFi1CVRk/fjxTpkxh48aNdO3alddeew1wRj/Pysri7rvv5r333iM/P5+9e/fy61//mnfeeYe0tDR+97vfcffdd3P77bcDkJyczIIFCwB44403AKisrGT27Nm8++67DBgwgKuuuooHHniAm2+++ZBtjoYlKGPi2JZ1+8lMFDLStNEElZKdybSpAdKTnWehqCiCtJwoRNr6/vOf/3DRRReRn+/UHHNznfttH3/8MS+84PS7uvLKK/nhD39Yv815552Hx+Nh8ODB7Nq1K+xjLV68mClTptQf4+KLL2b9+vUAFBQUcMkll7Bz506qq6vp06dPo/t45513WLPm4KOexcXFlJSUkJGREbLeGWecQV6e811ecMEFLFiwABHh/PPPrx+9/IILLuDDDz9k+vTp3Hrrrdx2222cffbZnHzyyYcc95NPPmHNmjWcdNJJAFRXV3PiiQfbehubsmPdunX06dOHAQMGAM40H/fff399gmqtaT4sQRkTp6rKyvnbYx4gi8QE+MWVWY0++3HyqVnoAfcpjvIDcZOgwp06I3idpKSkkO0b8vl89QO+glOTaGrdOjfeeCO33HIL5557LvPnzw+ZbDBYIBDg448/rp+/Kpx46143dfwBAwawbNkyXn/9dX784x8zbdq0+ppRHVXljDPO4Mknn2x0H01NS3I4rTXNR3yNa2KMqXdg0/b65fSsBDxN3KwOrllFsqv56bOGc+erV3Dnq1cc0rwHMOObJ9SXN2zeA7jghvH15c017wGcdtppPPPMM+zb53ymuia+iRMn8tRTTwHw+OOPNzmte2N69+7NihUrCAQCbNu2jcWLFwPO5IXvv/8+Bw4cwO/38/zzz9dvU1RURLduTlPqo48+Wv9+8FQbANOmTeMvf/lL/esVK1Y0GsPbb7/N/v37qaio4KWXXuKkk05i8uTJvPTSS5SXl1NWVsaLL77IySefzI4dO0hNTeWKK67g1ltvZfny5Ycce8KECSxcuJANG5zZlcvLy+trf00ZNGgQmzdvrt8mUtN8WA3KmDjlL95Ht05+Cos9ZOc3/Vd5SNNfHD0LNWTIEH76058yZcoUvF4vo0aNYt68edx7771ce+21/OEPf6BDhw5hTU9RV2s56aST6NOnD8OGDWPo0KH1g89269aNn/zkJ4wfP56uXbsyePDg+qk97rjjDi6++GK6devGhAkT2LRpE+DM8HvRRRfx8ssvc99993Hvvffyne98h+HDh+P3+5k8eTJz5849JJZJkyZx5ZVXsmHDBmbNmsWYMc4g4LNnz2bcuHEAXHfddYwaNYo333yTH/zgB3g8HhISEnjggQcAZxDdM888ky5duvDee+8xb948LrvssvqODr/+9a/rm+8ak5yczCOPPMLFF1+M3+9n7NixzJkzJ6zvpSVsug1j4lTtuvn4F7nNNr0nkjT56sbXW/su/iXPAOAdMBnfhMtb5fixMN1Ga1i2bBm33HIL77///mHXq5vaw+/3c/7553Pttddy/vnnt2os8+bNY+nSpSE1rfamJdNtWA3KmDilpfvrl71ZTY+xt6cohXdeT+VAkYf8XgVcNqEtomsfli5dyqxZs/jtb3/b7Lp33HEH77zzDpWVlUybNo3zzjsv8gHGOUtQxsQpLT38M1B1/ImZrFyXCEAVZRGPqz0ZM2ZMs/dj6tx1110RjsZpxqubLv5YYJ0kjIlToQmq6RpUbp/u9csH9lWH9FI76hji6BaCOXot/XmwGpQxcWr1skLSE71kZwbICx5zr4HUnGwunOEnK7WC7MwAVJVCSuZRHz85OZl9+/aRl5cXVndvE99UlX379pGcnBz2NpagjIlDVSVlPPWiABl4vfCr67MOu/7IcVlooTO8DWX7WiVBde/enYKCAvbs2XPU+zLxITk5me7duze/ossSlDFx6MCWgzPbZOUk4PUd/ldd0vPRQnfuqNJ9kN/4aActkZCQ0OSoCcaEwxKUMfGofD+Dj6vmQJGX3O6Hrz0BSHpQE2AcPQtl2jdLUMbEoQ5ZFVx6jjM6ufe4Q0dtaCi4E0WgZC9HPkGCMa3HEpQxcShkyKLD9OCrU7A7iX89nU5hsYeuxxVwjc0LZWKAJShj4lBwgpK05hOUJy2HrTucy0HyLnsWysQGew7KmDgU7kO6dXL7dK1fLtrfus9CGXOkrAZlTBz6+P0i0hMTyM4M0Du1+ekzUnJzmH1JFdmpVWRlBJCackhKb4NIjWmaJShj4kxVSRmvvw2Q5jwDdUvzvfg8Hg/9BueghU73dC3dh1iCMlFmTXzGxJmSnbvrlzOymn8Gqo4EjzZRdqC1wzKmxawGZUyc8daWMXZ4FaXlQlpe+LUgST44eoRWFkciNGNaxBKUMXEmM7Wac06rAMDTvfmZZ+ulZACgCtXFhRx+4nFjIi+iTXwiMl1E1onIBhH5USPlg0TkYxGpEpFbW7KtMaYJQbUfcZNOOD5bqfzhwUzu+HMWbzyzLRKRGdMiEUtQIuIF7gfOBAYDl4nI4Aar7QduAu46gm2NMY3QiqDmuZTmO0jU8SSnUVLmQVUoLSyPQGTGtEwka1DjgA2qulFVq4GngJnBK6jqblVdAtS0dFtjTBMqiuoXpQWjkqfnH0xm1eVVrRqSMUcikvegugHB7QQFwPjW3lZErgeuB+jZs2fLozQmzixbuJfygiTSUwMMGJVCuHWonsN6cMs3i0hPVRJyEiMaozHhiGSCamyGsnCnUwx7W1V9EHgQYMyYMTZ9pznmLfmohG2bnC4Oc87yhp2gErOyyc50f4WsF5+JAZFs4isAegS97g7saGLd1tzWmGNaSVF1/XJGx+bH4auXmAbiXBK0phL1VzezgTGRFcka1BKgv4j0AbYDlwKz2mBbY45Zqsr4YWUUFSkl5UJGl45hbysiSHIGWncPq7IkrJHQjYmUiCUoVfWLyA3Am4AXeFhVV4vIHLd8roh0BpYCmUBARG4GBqtqcWPbRipWY+JGTQUnjXZ74PmSSMpIa9n2KVnUlhVRVi5kFO4n2RKUiaKIPqirqq8Drzd4b27Q8tc4zXdhbWuMaUZQF3NJDv8ZqDrPvuJl5ZJsFLiqRwFDWvKgrzGtzMbiMyaOBD8DJS14BqqOLyWlvjdSye7C1gnKmCNkQx0ZE0dCxtBrwSgSdTJy0xAgNSVAoMoe1jXRZQnKmDjy5bLtrHs/mfQ0pW9SMn1auP2pM3sytdcSvB7wDvRGJEZjwmUJypg4smnNXj5angzAaV09LU5QCZnZ+N2GfxvR3ESb3YMyJo6U7i+rX87Ib3kTX8jgspagTJRZDcqYODLs+Fpy/BWUlnnoNiD8Z6DqBc8JVWEJykSXJShj4kjfbhX0SXYGek04vmuLt5eULPx+KC0XyveVtLiJ0JjWZAnKmHgSMpJ5y7uZa2Iqv74/m0DAef2rb1eQmGpTF5rosHtQxsQJVUUrSw6+cQQP6no8HtLSD/7dWrZrX2uEZswRsRqUMfGiugwCtQBIQjLiO7IpMzJzEqC2mvRUpbq0iCYGezEm4ixBGRMn9m/dxWuvpZKRqnTskclJR7ifOd/JRnfuBCAhp7b1AjSmhSxBGRMn9m/dzer1Tq2pd1niEScoT0oWdWlJK0oOu64xkWT3oIyJE6V7CuuX03OSj3xHKcFdzYsOs6IxkWU1KGPiRM/uykVnllFa5iFveO4R7ydkFHR7WNdEkSUoY+JEVkolwwfVAOAd2fmI91ObkMG+Qg9l5YLPu48+41orQmNaxhKUMfHiKJ+BqrN5c4CHHnGa+foMOMCcy486MmOOiN2DMiZOBA/uKkcw1Uad9E4HmwdLi6qPKiZjjobVoIyJEyFj5wWNqddSGZ07kJURID01QF6+Nr+BMRFiCcqYOPHPf5bjDaSSlqqcfVYqR9qPL71DLt+/rgTcuXW11o947VJh2p791BkTB2r9ftavD6A4z0HNTD/ye1Di8SDJ6QeHTaoshrQj7xVozJGye1DGxIHyvQeoa4xLSfGQkJx0dDsMnnaj0h7WNdFhNShj4kCSp4JrLy6hpMxDbVLeUe9PUjLRwu3OC5sXykRJRGtQIjJdRNaJyAYR+VEj5SIi97rln4vI6KCy74nIahFZJSJPishRPBpvTHzz+Uvp3b2WYQNrGD3hyJv36hTXpLG5wMuq9Qns2rirFSI0puUilqBExAvcD5wJDAYuE5HBDVY7E+jv/rseeMDdthtwEzBGVYcCXuDSSMVqTHvXWj346nz0ifLwsxk881oaa5d8fdT7M+ZIRLIGNQ7YoKobVbUaeAqY2WCdmcBj6vgEyBaRLm6ZD0gRER+QCuyIYKzGtG8hD+kefYLKzEuvXy7Zb/egTHREMkF1A7YFvS5w32t2HVXdDtwFbAV2AkWq+lYEYzWmXQvuyCCtUIPK655Lz65+BvWroWOu/6j3Z8yRiGQnCWnkvYZP/TW6jojk4NSu+gCFwLMicoWq/vOQg4hcj9M8SM+ePY8qYGPaq38/v51NK9JJT1WmdBD6DTu6/Q2e0IP+ZaUAeDrXtEKExrRcWDUoETlbRFpa2yoAegS97s6hzXRNrXM6sElV96hqDfACMLGxg6jqg6o6RlXHdOjQoYUhGhMfdhWUs/1rH+s2JlBVe5RdzAGChkqybuYmWsJNOpcCX4rI70Xk+DC3WQL0F5E+IpLo7uOVBuu8Alzl9uabgNOUtxOnaW+CiKSKiACnAWvDPK4xx5zSoqr65fQOOUe9v5BmQktQJkrCauJT1StEJBO4DHhERBR4BHhSVRv96VVVv4jcALyJ0wvvYVVdLSJz3PK5wOvAWcAGoBy4xi1bJCLPAcsBP/Ap8OCRf0xj4ttl55RTvK+UkjIPHfoe+VQb9ZIOdpLQylI0EEA89ly/aVth34NS1WIReR5IAW4Gzgd+ICL3qup9TWzzOk4SCn5vbtCyAt9pYtufAz8PNz5jjlWqSnZyCdldA0AtiTmtUIPy+tiyK4PiA5WUlnsYf3YhKbk23JFpW2ElKBE5F6d20w/4BzBOVXeLSCpO01ujCcoY0waqykADAEhCCuJLbJXdvvJuKnt2eQEYsG23JSjT5sKtQV0E/ElVPwh+U1XLReTa1g/LGBOu4HmgaIVnoOqkZyWyZ5dzb6tsz/5W268x4Qo3Qe1smJxE5HeqepuqvhuBuIwx4QoaRUKSj3yiwob6DEwnlX2kpShpydbV3LS9cBPUGcBtDd47s5H3jDFtbOWCLfzrb5mkpQYYOsHH6dNbZ7+nzehE7XFO51lfnj2sa9reYROUiPw38D9APxH5PKgoA1gYycCMMeEp3l1IcamH4lIPPSta5/4TAMn2LJSJruZqUE8A/wbuBIJHIy9RVWuUNiYGlOw7mDwygsbQO1ohY/rZlBsmCppLUKqqm0XkkK7gIpJrScqY6DtlSgIn5BZRUuYhe3LD4S6PgtWgTJSFU4M6G1iGM45e8Nh5CvSNUFzGmDD5/CVkZyrZmbX4uh39ZIV1KmuTWbk6kbIKwbfhAFNObbVdGxOWwyYoVT3b/b9P24RjjGmp4G7mrdmLr6ImmRffSgUgO7eSKa22Z2PC01wnidGHK1fV5a0bjjGmxVp5qo06aR0P1sbKSqwXn2l7zTXx/fEwZQpYpd+YKAoEAtSWFR8c9Tnl6Kd7r5OUmc7IwX5SkmpJTwtQW1WJNym51fZvTHOaa+I7pa0CMca0XFVxKb/4UzqpKQFycuCGK1uvm7nH4+HCCxLRMqcvlKe6BCxBmTbUXBPfqar6HxG5oLFyVX0hMmEZY8JRumsvCpRVeEhIScCZnaYVJWeAm6C0sgTJsDnXTNtprolvCvAf4JxGyhRnIkFjTJSU7TlQv5yR1YoP6bokOePgNNjW1dy0seaa+H7u/n9N24RjjGmJHj2En99USFmFEOjYvdX3H9zpQu1hXdPGwp1uIw9nbqZJODWnBcAvVXVfBGMzxjSnshivFzLTFW/X1usgUadgTyLrFyVRVuGhn2cHwwe0+iGMaVK4g8U+BXwAXOi+vhx4Gjg9EkEZY8KjFUHNbq041UadLdvg3Y9SAJBO+xne6kcwpmnhJqhcVf1V0Otfi8h5EYjHGNMSIQ/ptn6CSs87WCsrPVDe6vs35nDCTVDvicilwDPu64uA1yITkjEmXOUHCvHVgtdLRGpQXY7rwEknVJKeqnQ5vrbV92/M4TTXzbyEg2Pw3QL80y3yAKU496WMMVHyz4f2s+WrbFKSlNl9ld69W3f/nY/rzDcmVwIgWVaDMm2ruV58rTewlzGm1ZUUVwNQUSUkZ2e3+v5DptywbuamjYXbxIeI5AD9gfpHyRtOA2+MaWP+GgSnmSOjc27r7z/p4PxSWlWGBmoRj7f1j2NMI8LtZn4d8F2gO7ACmAB8jI3FZ0zUqL+am2cfIBCAiiovKbk5rX4M8XiRxDS0ugxQqCqLyL0uYxrjaX4VwElOY4Et7vh8o4A9zW0kItNFZJ2IbBCRHzVSLiJyr1v+efDo6SKSLSLPicgXIrJWRE4MM1Zjjg1uk5vHA+m56Xg84f46t8zi1Rm8+m4KT72ayt7NX0fkGMY0Jtyf6EpVrQQQkSRV/QIYeLgNRMQL3A+cCQwGLhORwQ1WOxOn2bA/cD3wQFDZn4E3VHUQMAJYG2asxhwTgueBIgJdzOus+jKRJZ8nsWZDIge2Nft3qTGtJtx7UAUikg28BLwtIgeAHc1sMw7YoKobAUTkKWAmsCZonZnAY6qqwCdurakLUAZMBmYDqGo1UB1mrMYcG4KGHpIINrulZSXVL5fsLYrYcYxpKKwEparnu4t3iMh7QBbwRjObdQO2Bb0uAMaHsU43wI/ThPiIiIzAmXL+u6pa1vAgInI9Tu2Lnj17hvNxjIkLpXsPUFEspKUqyRFMUKNOzKNn6hbSUpXe9itm2lBLevGN5uBYfAvdWs1hN2nkPQ1zHR8wGrhRVReJyJ+BHwE/O2Rl1QeBBwHGjBnTcP/GxK3F72zj7SeckR6mnl/DmSdF5jiDRnek1uP8untTKyNzEGMaEdY9KBG5HXgUyAPycWo2/6+ZzQqAHkGvu3Nos2BT6xQABaq6yH3/OZyEZYxxle4vrV9OzUqL2HEkOehxyMrSplc0ppWF20niMmCsqv7cnYJjAs6AsYezBOgvIn1EJBG4FHilwTqvAFe5vfkmAEWqulNVvwa2iUhdR4zTCL13ZcwxL0GqyEwP4PUoaXmtP5J5HQmaRt6m3DBtKdwmvs04D+jW1e+TgK8Ot4Gq+kXkBuBNwAs8rKqrRWSOWz4XeB04C9gAlAPB807dCDzuJreNDcqMOeZNm1LL6YOKUQXf1N6RO1Dww7qVlqBM22luLL77cO4JVQGrReRt9/UZOHNCHZaqvo6ThILfmxu0rMB3mth2BTCmuWMYc6zSCqdHnQh40lv/Id06lf4k/vV2CmXlHiSpjNkzInYoY0I0V4Na6v6/DHgx6P35EYnGGBMWVUXLC+tfBzfDtTZvRg7LVjldzX0+P4FAIGIPBRsTrLnBYh+tW3ab2urm01ynqjWRDMwYcxg1leCvcpa9CZCYGrFDJWWkkZQkVFUpfj9UHigiNS9yNTZj6oQ7Ft9UnF58m3G6hvcQkattsFhjoqN8zx4KCrxkpCuZnbNJksae2Gg9M85KJKFmPxlpARK0DLAEZSIv3E4SfwSmqeo6ABEZADwJnBCpwIwxTdvxRQEPP+t0/+4zIJk5/xXZ442ekE3g610AeKuto4RpG+E2JCfUJScAVV0PJEQmJGNMc4p2HqhfzshNifwBg7uaB937MiaSwq1BLRORh4B/uK8vx+k4YYyJgiTK6dHFT0mZh5xOkZ9XVFKzD76osPH4TNsIN0HNwekOfhPOPagPgL9GKihjzOEdP6CWAZc6ozr4Tugd8eOFPKxbbgnKtI1mE5SIeIBlqjoUuDvyIRljmhPSzJYa+Q4LBbt8vPNSGiWlHroOLOC/Gg77bEwENJugVDUgIp+JSE9V3doWQRljmtFGz0DVqfWms36Tc9vZl14e8eMZA+E38XXBGUliMc5cTQCo6rkRicoYc1gadB8o5P5QhGR1za9fLi60qdlM2wg3Qf0iolEYY8IWCAT4bHkZGak+MtICdG2LBNW9M5eeU0ZGWoDMDGckC4nws1fGNDcWXzJOB4njgJXAQ6rqb4vAjDGNK9uzn+ded7qWJyd7+MXNiRE/pi85mSGDE9Bqt3mvqhSSI9970BzbmnsO6lGcAVtXAmfiPLBrjImiou1f1y9nZLfh44ghPfkOHGZFY1pHc018g1V1GID7HNTiyIdkjDkcn7+UIQOqKSn1kNej7YYcktRstGin86K8CHLb7NDmGNVcgqofENad3ynC4RhjmtMhu4ZLZjhNbd5+w9rsuMGdMQLlhWEPQ2PMkWouQY0QkbqBtwRIcV8LznROmRGNzhhziNBnoLLb7LjLV3pY+EIGJaXCiTs3csZtJ7fZsc2xqbnpNrxtFYgxJkwVhfWLbfEMVJ1qTeHrPc4loXhPSZsd1xy7rJZuTDsTMtRQG9agMjsdPFbxvrKmVzSmlYT7HJQxJkYs+2g/Up5ARprSz5dBWzVz9B3Vk29fVkJGeoD07ta6byLPEpQx7cxbb1VSVpoGwI+vTCGpjY6b1qED3TrXAiBVNmCsiTxr4jOmHfFXVVNW6iYJIL1rx7Y7eMrBWpNWFKOB2rY7tjkmWQ3KmHaktqSQiaMrKSnzUEMyvoS2e1BXvD4kOQOtLAEUKkva9B6YOfZENEGJyHTgz4AX+Luq/rZBubjlZwHlwGxVXR5U7gWWAttV9exIxmpMe5CgpUyfUgmA5OY3s3YEpGRBZQl+P3iK95NgCcpEUMQSlJtc7gfOAAqAJSLyiqquCVrtTKC/+2888ID7f53vAmsBuyNrDKHPQLXFKOYN/eudRD5fnElFpYer8rcw5Ny+bR6DOXZE8h7UOGCDqm5U1WrgKWBmg3VmAo+p4xMgW0S6AIhId2AG8PcIxmhM+9LG80A1VOtJoaLSuWwU7drf5sc3x5ZIJqhuwLag1wXue+Gucw/wQyBwuIOIyPUislRElu7Zs+eoAjYm1mnQQ7rRuP+T2cEZwVxEqSy0h3VNZEXyHlRjA/dpOOuIyNnAblVdJiJTD3cQVX0QeBBgzJgxDfdvTFz5dMFOCtcnkZGmDBiU1ubjtZ74jZ6ckLOItBQlYYB1AjaRFckEVQD0CHrdHdgR5joXAeeKyFlAMpApIv9U1SsiGK8xMW/FoiI2rHHmgrpmqrfNE1R6x3xq0py/A0PGBDQmAiL5J9ASoL+I9BGRROBS4JUG67wCXCWOCUCRqu5U1R+randV7e1u9x9LTsZA0f7K+uWMLlHqxecKnnbemEiIWA3KnZ7jBuBNnG7mD6vqahGZ45bPBV7H6WK+Aaeb+TWRiseYeDBheBn7u/kpLvWQ071zmx8/pOeg1aBMhEX0OShVfR0nCQW/NzdoWYHvNLOP+cD8CIRnTLuiNVWMHVLqvPB4Scxru8kK6yVnAEKNXykpLKdjTTXehMhPOW+OTTaShDHthJYe7KUqablEYwJR8Xi57/Fc9ux2hjm67Zxd5Pbp0cxWxhwZ64ZjTHtRsrd+UTI6RC2MhJSDw9MW77BHO0zkWIIypp3Qkt31y5IevQSVlZuCiJKbVUtN8YGoxWHinzXxGdNOLHx9E19/nkJudoChPbJo+y4Sjguu6oHv5K/wesHbrSpKUZhjgSUoY9qJdSv2sWGN07zW5YzUqCWotC5d8G91X5RYE5+JHGviM6ad2LervH45r2/3qMUR3LwY3OxoTGuzGpQx7YDW+jn75EL2FQr7C73kRjNBZQQnKKtBmcixBGVMe1C2j/59augPSGoOiakp0YslLZfagIfCIti/qYyBFRUkpEQxHhO3LEEZ0w4E11QkIwpDHAURr4/7n8xn7+5qAG6+YAtdhg2KakwmPtk9KGPagdAE1TGKkTiyO6TWL+/f2HAMaGNahyUoY9qB0AQVvWeg6uR3zSQzPUDv7n68NYXRDsfEKWviM6YdePkfmzmwLY3c7AAnDcygU5TjOefq/pw19HMAvN0rohyNiVeWoIxpBzZ9WcqerxMAGOvNjm4wgCezE7XusvXkM5FiTXzGxLhAIMCBvQdHbMjrF70u5vUy7FkoE3lWgzIm1pUf4PpLi9lf5KW4MpXUnOxoR4SkH+xJqKX70Vo/4rXLiWld9hNlTKwr3UvnDgE6dwjgyW/rSd4bJ75Eiv3Z7N1Ryr5CDyNO2UFGj57RDsvEGUtQxsS4kHs8MdCDr87zb2awab2znH/aZgZZgjKtzO5BGRPjQqbZiKEEldc1s35535ZdUYzExCurQRkT42LtId06XY/Lp8e6LeRmB8hNK29+A2NayBKUMTHuwT/vpLI4g9ysAOeekEZetANyTZgxgDEZ7wPg6VoS5WhMPLImPmNiWCAQYNeOSnbv8/LFxgQS87tEO6R6wbU5Ld17mDWNOTKWoIyJYeV791NVpQAkJglpHWKjFx+EDlqrJXtQ1ShGY+JRRBOUiEwXkXUiskFEftRIuYjIvW755yIy2n2/h4i8JyJrRWS1iHw3knEaE6tSKeLH/13Ety8r4ZJLUvF4YudvSklMRZLSnRcBP5QfiG5AJu5E7B6UiHiB+4EzgAJgiYi8oqprglY7E+jv/hsPPOD+7we+r6rLRSQDWCYibzfY1pi4pyV7SElWunWuxdMzdjpI1Pm6LI8d66o5UORh5PGb6DI2dmp4pv2L5J9j44ANqrpRVauBp4CZDdaZCTymjk+AbBHpoqo7VXU5gKqWAGuBbhGM1ZiYpPu31i9LVucoRtK4BctSePGtVOYvSmbrqm3RDsfEmUgmqG5A8E9sAYcmmWbXEZHewChgUeuHaExs072b65c9+X2iF0gT8rrn1C/v22odJUzrimSCkkbea3gX9bDriEg68Dxws6oWN3oQketFZKmILN2zx0ZVNvHDX13Nri+3Udf3QGIwQXXp35n+vWs4ZUIlx/eyruamdUXyOagCoEfQ6+5Aw6k3m1xHRBJwktPjqvpCUwdR1QeBBwHGjBlj3YhM3Ni5cj1/eSSV5KQUBg1K4LKrMpvfqI0NO2MkA8tfBEASt6GqiDT2d6cxLRfJGtQSoL+I9BGRROBS4JUG67wCXOX25psAFKnqTnF+wh8C1qrq3RGM0ZiYtW35OgAqqwR/Uk4za0dJZick0Zn+XavLwKbeMK0oYglKVf3ADcCbOJ0cnlHV1SIyR0TmuKu9DmwENgB/A/7Hff8k4ErgVBFZ4f47K1KxGhOL/Pt3kZYSAKDn4K5RjqZxIhLS9BjYsymK0Zh4E9GhjlT1dZwkFPze3KBlBb7TyHYLaPz+lDHHjAmDdjOuczFFJULKN0ZEO5wmSYe+sGM1ALp3E/SbEOWITLywsfiMiUFaXYEWfY0IZGcJib2Pi3ZITapK68ni5UkU7PTif/tLrhkf7YhMvLAEZUwM0n1bqOvQKtldkYSk6AZ0GJLXkzfeTwHA4ymjqqycpLTUKEdl4kHsjJtijKkX+vxT76jFEY7UnGw6dHYSaCAA25fZgC+mdVgNypgYtOqDdSQXeenasZbU/L7RDqdZk6Z1pqZgLd0719Ita1+0wzFxwhKUMTHo1Wd2U1KcgQDfOzGfTtEOqBljZwzF/8lnAMgB68lnWoc18RkTYwq3bqek2A9AQqKQP7BflCNqXnBXc91rCcq0DktQxsQY/56tjBpSRce8Wrr3Tcfri/2GDsnuCj7nPpSWF6Jl+6MckYkHsf+Tb8wxJpudnD+twnkxqH08UyQeL568XgR2rScQgMqCDaQMHBftsEw7ZwnKmBgT+PqL+mVfp9jvIFFnw75OvP/cdrZ/7WPcxhWc80tLUOboWBOfMTFEi752n4ECPF48nQZEN6AWCKR3Y9O2BKprhG1rd0U7HBMHLEEZE0P8X31cv+zpNgxJTo9iNC3TY8zg+uXE2iICFTb9hjk61sRnTIwIBAL85adL6ZafwsjB1fSb0j7uP9XJ7NqJGedmcFzedvKyA+imRTD49GiHZdoxq0EZEyO+em8RX3/tZ9mqJJ78VxbaeUi0Q2qxk66YSl62MwJ77ZcLULUp2syRswRlTIz44p0l9csjTu6JLykxitEcGU/vMQe7mxftRPd8FeWITHtmCcqYGKA1VZwxdAPXXVLCmGFVnHDhSdEO6YhIQjLe3mPqXwe+XBDFaEx7ZwnKmBgQ2PopUltFz661zLwwm+5jhkU7pCPmGXAyfj+sWp/Ao39aRfmBwmiHZNopS1DGxIDAxk/ql719JyDSfufrlLze/PPfnXnmtTTWf+VlxbPvRDsk005ZgjImyvwFqwnsXOu+Ejx92/cDriLC0NOG1r9e8q9V1lnCHBFLUMZEUcmuvdxz7T9Z+UUCAJ6eI5G03ChHdfRG/dfp5OUqUydUcum0rwlsWxHtkEw7ZAnKmCgJBAI8e9tD7NkT4Nl/p/Hmx9n4Jlwe7bBaRWpuDt/75RBOPbGS7EzFv+ARAoU7oh2WaWcsQRkTJcWL32b35j31rwecOwNJzohiRK3LN+ZCJD3PeeGvwv/eX9Gq0ugGZdoVS1DGtDFVpXb9B6RseIX/vryEEcdXc9JZfRh05uRoh9aqJCkd3yn/c/C5qJI9rP7bXPzVNVGOzLQXEk83L8eMGaNLly6NdhjGNClQuJPaT/5JYPeG+vckpzve6bfhTWh/D+aGo3bLcqr+83+8+UEKn6xIokPnJGZ+71SOO2NKu+6taFqPiCxT1TEN34/oWHwiMh34M+AF/q6qv21QLm75WUA5MFtVl4ezrTGxTlWhpoKavTtY8OQnbFi2lSzvXs47o6x+HcnoQMKUbyNxmpwAvL1Gs7LqZD5ZsRKAPV9XUbP4WWpqPsLbZxyS2xPJ7Y4kpkY5UhNrIpagRMQL3A+cARQAS0TkFVVdE7TamUB/99944AFgfJjbtppA4U5qV7zMf94pZ9euWgBOOS2Fzp1DT8+bb5Szf59TPm16Knl53pDy114to7jYGYdsxtlpZGaFtqC+9GIZFeVO+czz00hNDS1/7plSamqcGu2FF6eTmBj61+VTT5RQV+G9dFZ6yF+fNTXKc8847fs+n3DxJaGjYFdUBHjpBefCmJwsnH9haHlxcYDXXnXK0zM8nHNuWkj5gf21vPHvcgBycrxMPyv0YrJ7dy3vvu2Ud+jo5fQzQsu3b/fzwXxnEr6uXX1MOSUlpHzL5ho+WlgJQK9ePiZOCi3fsKGGJYuc8r79Ehg/ITmkfN0X1SxfVgXAwEGJjD4hKaR81coqVn5eDcCQoYkMHxFa/unyKr5Y65SPHJXE8YNDE8aSxZVs+LIGVRg7Non+AxJwvgwFVT54v4INX1ZTWxPg1CnQt1sFWlEMtc42C1/IpLTcAyQwbICPfr0DeIdMwzt8BuKL3+RUZ+z1sygre4L3nllFt07V9OvpR/dtwe9OLbJnv4eHns0mOc1H/wEJnHNOKnh8IB7weFm/3s+SxVWIR+g/IIGx40K//zVrqvnsU+f7P35wIiNHhX6/n39WxepVzvc7bHgiQ4eFli9fVsW6L5zy0SckMXBQ6Hey6JNKNn7lNE2OG59Mv+MSQsoXLqhg6xY/ABMnJdOrV2j5++9VsGOHUz5lagpdu4VeW95+q5y9e5xry2lnpNKxY+i15Y3XyzlwwCmffmYqObmh5a++XEZpqXNtOfvcNDIyQq8tLz5fSmWlc/E474I0UlJCy595qpTaWqf84kvS8fkOXlsCAeXpJ51ri3jg0ssOvU/q7T8JT7ehh7x/tCJZgxoHbFDVjQAi8hQwEwhOMjOBx9RpZ/xERLJFpAvQO4xtW09lCYGtn7Lp8zQ2bXN+sMb2KiVQ7Q9ZbeOKdLbvck7ZpP4l5JTVhpSv/zSDfQecH5xTh2wkPTcQWr48k+JS5wdj+sivSM4IbV5duyyLqmrnB2PmuA11Tff1Vi/Jpm6LwKQNBLeO1FbD6qXZACT4lMCJX4ZsW10mrF6aBUB6aoDA2NDyqkIPq5dmApCbVUtgZOhUCeW7vaxe6vxgdulQS2BoaHlpwcHyXt38BAaG3gwv3uhj9VInKdb0riHQryykvHB9AquXOklRiquZ0LM8pPzAukRWL3WSXlJFFYGuFSHle75IYvVSJ6ll1lYR6BBavnvtwfIO3koCOZUh5V+vSWb1Muei1y2lgoHpVSHl21elsHql84X0yyonkFIdUr5zfSob1zkXtdIBZWjWwfssItCnh5+VbvlnW7sw6IbZeHK6c6zweDyc+v0rGHflPsqWvwV7F0Dg4O/Xzt1eyssDlJdXU55bSmDX9pDt93+ZyNpPne8/raaKQOcG33/Q95utlQzPC/1+d61JZvVS5/vtlFBBICv0+92xOoXVnzrfb4+0CvqnNvL9r3bKj8spp09i6Pe/dWUqq9c73+/gjmX0kND7bJs/T+PLzc61ZVT3UjrXhl5bNn2WztYdzrVlfJ8S8itDry0bVmTw9R7n2jJ5YAlZpYdeew4UOeWnD9tIWnboteeLZZmUVTjXnhmjvyIpLfTas2ZZFn6/c0E5f8IGPEH5NRA4eG0RIHDSBhrydB54yHutIZIJqhuwLeh1AU4tqbl1uoW5LQAicj1wPUDPnj2PLmJjjlDwHws1tUEvvAlIajbjzurOwLO60X/KaLJ7dWv7AGNEesc80qdfhpZPJ7B1Bbp/G4F9W9m5e1/9OslJh94Xj6Nb5aYFIpmgGrv72fDHrKl1wtnWeVP1QeBBcDpJtCTA+iCyu+Cbcj2ndy6mrMT5y6frgEx8maHV9Okdiqgod/7y6TAoE196aPk5eYVUu3/55AzJwpcSenrPyz6Av8b5yyZjWDa+pNBq+sUZ+wkEnI+QMjIHry+0Gn5Z2sFfYt8JuSFNfFKrzMrZ7yyL4Dsh9GHP9OoAszoecLb1efCNzAkpz6msZVa3QgASk7z4hmWHlOeX+ZnVpwiA5BQfviFZIeWdi2uYNbAYgNT0BHyDMkPKe4yoZtYwp9aVnpWAr39oed9hVcw6wal1ZeUk4usX2ozQf3AlsyY4ta7svCR8fUKbKI8fWEHuyU6tK69TMr4eoU2Uw/tX0Pk0twmySwq+bqFNkCf0LafXmc5f5Z26p+LrHNzEKEzoXcbAvc5f5V16pOHr5JSLeECEqceVM648gC85kbxuWSTmZkBqNviSEBHaz7y4bUNSc/AOOqX+9VnTqjmlsJiK/YUkeGpIyBTQAARqQQMMHlxB9smloEpOh2R8Peu+X+f3ZejACjpMdb7f/E4p+HqEfr8jjyun2zTn++3YNRVf19Am5DF9yug7w/l+O/dIrf9+60zoWcqgfU6tqmuvNHwdQpsYJ3cvYcQBp1bVo286vtzQ5o9TuhYztsi5tnQ/LgNfdmgT4rROxZSXOuWdG7n2nJVfRGWFc+3JH5yFLzX02jIzt5DqKvfaMzQbX3LoteXCrAP4/c61J214Dr7E0GvLJen760f7SBqVi9d78NqiqszK2F//2jcmj4Y8OT0Oea81RKwXn4icCNyhqt9wX/8YQFXvDFrn/4D5qvqk+3odMBWnie+w2zbGevEZY0z701Qvvkg+B7UE6C8ifUQkEbgUeKXBOq8AV4ljAlCkqjvD3NYYY0wci1gTn6r6ReQG4E2cruIPq+pqEZnjls8FXsfpYr4Bp5v5NYfbNlKxGmOMiT32oK4xxpioikYTnzHGGHPELEEZY4yJSZagjDHGxCRLUMYYY2JSXHWSEJE9wJaj2EU+sLeVwmnv7FwcZOfCYefhIDsXB7XGueilqh0avhlXCepoicjSxnqSHIvsXBxk58Jh5+EgOxcHRfJcWBOfMcaYmGQJyhhjTEyyBBXqwWgHEEPsXBxk58Jh5+EgOxcHRexc2D0oY4wxMclqUMYYY2KSJShjjDEx6ZhMUCIyXUTWicgGEflRI+UiIve65Z+LyOhoxNkWwjgXl7vn4HMR+UhERkQjzkhr7jwErTdWRGpF5KK2jK8thXMuRGSqiKwQkdUi8n5bx9hWwvj9yBKRV0XkM/dcXBONOCNNRB4Wkd0isqqJ8shcM1X1mPqHM33HV0BfIBH4DBjcYJ2zgH/jzOw7AVgU7bijeC4mAjnu8pnxeC7COQ9B6/0HZ5qYi6IddxR/JrKBNUBP93XHaMcdxXPxE+B37nIHYD+QGO3YI3AuJgOjgVVNlEfkmnks1qDGARtUdaOqVgNPATMbrDMTeEwdnwDZItKlrQNtA82eC1X9SFUPuC8/Abq3cYxtIZyfCYAbgeeB3W0ZXBsL51zMAl5Q1a0Aqhqv5yOcc6FAhogIkI6ToPxtG2bkqeoHOJ+tKRG5Zh6LCaobsC3odYH7XkvXiQct/ZzfxPkrKd40ex5EpBtwPjC3DeOKhnB+JgYAOSIyX0SWichVbRZd2wrnXPwFOB7YAawEvquqgbYJL6ZE5JoZsRl1Y5g08l7DvvbhrBMPwv6cInIKToKaFNGIoiOc83APcJuq1jp/LMetcM6FDzgBOA1IAT4WkU9UdX2kg2tj4ZyLbwArgFOBfsDbIvKhqhZHOLZYE5Fr5rGYoAqAHkGvu+P89dPSdeJBWJ9TRIYDfwfOVNV9bRRbWwrnPIwBnnKTUz5wloj4VfWlNomw7YT7+7FXVcuAMhH5ABgBxFuCCudcXAP8Vp0bMRtEZBMwCFjcNiHGjIhcM4/FJr4lQH8R6SMiicClwCsN1nkFuMrtmTIBKFLVnW0daBto9lyISE/gBeDKOPwLuU6z50FV+6hqb1XtDTwH/E8cJicI7/fjZeBkEfGJSCowHljbxnG2hXDOxVacmiQi0gkYCGxs0yhjQ0SumcdcDUpV/SJyA/AmTi+dh1V1tYjMccvn4vTSOgvYAJTj/JUUd8I8F7cDecBf3dqDX+NsFOcwz8MxIZxzoaprReQN4HMgAPxdVRvtftyehflz8StgnoisxGnmuk1V424aDhF5EpgK5ItIAfBzIAEie820oY6MMcbEpGOxic8YY0w7YAnKGGNMTLIEZYwxJiZZgjLGGBOTLEEZY4yJSZagTFxwRxhfISKrRORZ9/mctjz+Txq8/ijCxxvkft5PRaRfg7LSCB1zvoiE/YiBiNzc3DBIIvKUiPQ/+uhMPLIEZeJFhaqOVNWhQDUwJ7hQRLyROKj7YKIHZ1Treqo6MRLHC3Ie8LKqjlLVryJ8rBYTER9wLfBEM6s+APww8hGZ9sgSlIlHHwLHuXMWvSciTwArRSRZRB4RkZVuzeMUABGZLSIvi8gb7tw/P6/bkYjc4tbKVonIze57vUVkrYj8FVgOPASkuDWax911St3/RUT+4G6/UkQucd+f6tZInhORL0TkcWlkkD8RGSkin4gzx86LIpIjImcBNwPXich7jZ0AEfmNOHMUfeKOcICIdBCR50VkifvvJPf9ceLM9fWp+/9A9/0Ut4bzuYg8jTPuHiLiFZF5QZ/pe42EcCqw3H3Y1eceb6q7/Z0i8pug7+p0N6EZEyra84zYP/vXGv+AUvd/H85QPP+N8+R7GdDHLfs+8Ii7PAhnmJpkYDawE2fEjBRgFc7YeyfgjFCdhjOVwmpgFNAbZwSFCQ2P30g8FwJv44xE0Mk9Zhc3tiKcMcs8wMfApEY+1+fAFHf5l8A97vIdwK1NnAsFznGXfw/8P3f5ibpjAD2Bte5yJuBzl08HnneXb8EZPQFgOM40EnXn5e2g42U3EsMvgBuDXg/BGQ7pDOBTguZMcs/PCdH+GbJ/sffP/mox8SJFRFa4yx/i1GomAotVdZP7/iTgPgBV/UJEtuBMHQHOBXcfgIi84K6rwIvqDIpa9/7JOOOObVFn3pvmTAKeVNVaYJc4s8+OBYrd2Arcfa/ASXwL6jYUkSyci3/djLWPAs+Gccxq4F/u8jKcpABO8hkcVFHLFJEMIAt41L0XpLhD2OBMUncvgKp+LiKfu+9vBPqKyH3Aa8BbjcTQhaDx+dQZIugfwKvAierMr1RnN9DVjdWYepagTLyoUNWRwW+4F+Ky4LcOs33DMb+0mfXLDlMWEsZhyqqClmtpvd/HGlWt+zzB+/XgJIeKkACdRPOeqp4vIr2B+UHFh4yFpqoHRGQEzlQT3wH+C+d+U7AKnNppsGFAIU5NMliyu74xIewelDmWfABcDiAiA3Cauda5ZWeISK6IpOB0QFjorn+eiKSKSBrOhIUfNrHvGhFJaOT9D4BL3Ps2HXBqJWFNxaCqRcABETnZfetK4P3DbNKct4Ab6l6IyEh3MQvY7i7PDlo/+HwNxWnmQ0TyAY+qPg/8DGcq8IbWAscFHesCnCbUycC9IpIdtO4AnOZTY0JYgjLHkr8CXnFGnn4amK2qdbWYBcA/cCafe15Vl6rqcmAeTkJZhDNq96dN7PtB4PO6ThJBXsS5j/QZ8B/gh6r6dQtivhr4g9u8NhLnPtSRugkY43Z6WMPBno6/B+4UkYU498rqPACku8f+IQcTazdgvtssOQ/4cSPH+jdOMqpLaL8FvqnOlC1/Af7slnXCqf3G43Q25ijZaObmmCcis4ExqnpDc+ua8InIizgJ+cvDrPM9oFhVH2q7yEx7YTUoY0yk/Ains8ThFOJ0/jDmEFaDMsYYE5OsBmWMMSYmWYIyxhgTkyxBGWOMiUmWoIwxxsQkS1DGGGNi0v8H9p3vSECHTxAAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "posterior.plot(label='grid posterior', color='C1')\n",
    "posterior_conjugate.plot(label='conjugate posterior',\n",
    "                        color='C4', ls=':')\n",
    "\n",
    "decorate_euro(title='Posterior distribution of x')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "They are the same other than small differences due to floating-point approximations.\n",
    "\n",
    "The examples so far are problems we have already solved, so let's try something new."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-04-16T19:40:55.111685Z",
     "iopub.status.busy": "2021-04-16T19:40:55.111135Z",
     "iopub.status.idle": "2021-04-16T19:40:55.113945Z",
     "shell.execute_reply": "2021-04-16T19:40:55.113557Z"
    },
    "tags": [
     "hide-cell"
    ]
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.allclose(posterior, posterior_conjugate)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Lions and Tigers and Bears\n",
    "\n",
    "Suppose we visit a wild animal preserve where we know that the only animals are lions and tigers and bears, but we don't know how many of each there are.\n",
    "During the tour, we see 3 lions, 2 tigers, and one bear. Assuming that every animal had an equal chance to appear in our sample, what is the probability that the next animal we see is a bear?\n",
    "\n",
    "To answer this question, we'll use the data to estimate the prevalence of each species, that is, what fraction of the animals belong to each species.\n",
    "If we know the prevalences, we can use the multinomial distribution to compute the probability of the data.\n",
    "For example, suppose we know that the fraction of lions, tigers, and bears is 0.4, 0.3, and 0.3, respectively.\n",
    "\n",
    "In that case the probability of the data is:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-04-16T19:40:55.118407Z",
     "iopub.status.busy": "2021-04-16T19:40:55.117894Z",
     "iopub.status.idle": "2021-04-16T19:40:55.120481Z",
     "shell.execute_reply": "2021-04-16T19:40:55.120055Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.10368"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from scipy.stats import multinomial\n",
    "\n",
    "data = 3, 2, 1\n",
    "n = np.sum(data)\n",
    "ps = 0.4, 0.3, 0.3\n",
    "\n",
    "multinomial.pmf(data, n, ps)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, we could choose a prior for the prevalences and do a Bayesian update using the multinomial distribution to compute the probability of the data.\n",
    "\n",
    "But there's an easier way, because the multinomial distribution has a conjugate prior: the Dirichlet distribution."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The Dirichlet Distribution\n",
    "\n",
    "The Dirichlet distribution is a multivariate distribution, like the multivariate normal distribution we used in <<_MultivariateNormalDistribution>> to describe the distribution of penguin measurements.  \n",
    "\n",
    "In that example, the quantities in the distribution are pairs of flipper length and culmen length, and the parameters of the distribution are a vector of means and a matrix of covariances.\n",
    "\n",
    "In a Dirichlet distribution, the quantities are vectors of probabilities, $\\mathbf{x}$, and the parameter is a vector, $\\mathbf{\\alpha}$.\n",
    "\n",
    "An example will make that clearer.  SciPy provides a `dirichlet` object that represents a Dirichlet distribution.\n",
    "Here's an instance with $\\mathbf{\\alpha} = 1, 2, 3$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-04-16T19:40:55.123839Z",
     "iopub.status.busy": "2021-04-16T19:40:55.123293Z",
     "iopub.status.idle": "2021-04-16T19:40:55.125400Z",
     "shell.execute_reply": "2021-04-16T19:40:55.125006Z"
    }
   },
   "outputs": [],
   "source": [
    "from scipy.stats import dirichlet\n",
    "\n",
    "alpha = 1, 2, 3\n",
    "dist = dirichlet(alpha)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Since we provided three parameters, the result is a distribution of three variables.\n",
    "If we draw a random value from this distribution, like this:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-04-16T19:40:55.128896Z",
     "iopub.status.busy": "2021-04-16T19:40:55.128370Z",
     "iopub.status.idle": "2021-04-16T19:40:55.130736Z",
     "shell.execute_reply": "2021-04-16T19:40:55.131081Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0.53566485, 0.35129   , 0.11304515]])"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dist.rvs()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-04-16T19:40:55.134432Z",
     "iopub.status.busy": "2021-04-16T19:40:55.133988Z",
     "iopub.status.idle": "2021-04-16T19:40:55.136611Z",
     "shell.execute_reply": "2021-04-16T19:40:55.136273Z"
    },
    "tags": [
     "hide-cell"
    ]
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dist.rvs().sum()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The result is an array of three values. \n",
    "They are bounded between 0 and 1, and they always add up to 1, so they can be interpreted as the probabilities of a set of outcomes that are mutually exclusive and collectively exhaustive.\n",
    "\n",
    "Let's see what the distributions of these values look like.  I'll draw 1000 random vectors from this distribution, like this:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-04-16T19:40:55.139365Z",
     "iopub.status.busy": "2021-04-16T19:40:55.138950Z",
     "iopub.status.idle": "2021-04-16T19:40:55.141151Z",
     "shell.execute_reply": "2021-04-16T19:40:55.141600Z"
    }
   },
   "outputs": [],
   "source": [
    "sample = dist.rvs(1000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-04-16T19:40:55.145904Z",
     "iopub.status.busy": "2021-04-16T19:40:55.145144Z",
     "iopub.status.idle": "2021-04-16T19:40:55.148345Z",
     "shell.execute_reply": "2021-04-16T19:40:55.147900Z"
    },
    "tags": [
     "hide-cell"
    ]
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1000, 3)"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sample.shape"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The result is an array with 1000 rows and three columns.  I'll compute the `Cdf` of the values in each column."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-04-16T19:40:55.153307Z",
     "iopub.status.busy": "2021-04-16T19:40:55.152394Z",
     "iopub.status.idle": "2021-04-16T19:40:55.159392Z",
     "shell.execute_reply": "2021-04-16T19:40:55.158863Z"
    }
   },
   "outputs": [],
   "source": [
    "from empiricaldist import Cdf\n",
    "\n",
    "cdfs = [Cdf.from_seq(col) \n",
    "        for col in sample.transpose()]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The result is a list of `Cdf` objects that represent the marginal distributions of the three variables.  Here's what they look like."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-04-16T19:40:55.177226Z",
     "iopub.status.busy": "2021-04-16T19:40:55.176730Z",
     "iopub.status.idle": "2021-04-16T19:40:55.349780Z",
     "shell.execute_reply": "2021-04-16T19:40:55.349286Z"
    },
    "tags": [
     "hide-input"
    ]
   },
   "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": [
    "for i, cdf in enumerate(cdfs):\n",
    "    label = f'Column {i}'\n",
    "    cdf.plot(label=label)\n",
    "    \n",
    "decorate()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Column 0, which corresponds to the lowest parameter, contains the lowest probabilities.\n",
    "Column 2, which corresponds to the highest parameter, contains the highest probabilities.\n",
    "\n",
    "As it turns out, these marginal distributions are beta distributions.\n",
    "The following function takes a sequence of parameters, `alpha`, and computes the marginal distribution of variable `i`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-04-16T19:40:55.353284Z",
     "iopub.status.busy": "2021-04-16T19:40:55.352778Z",
     "iopub.status.idle": "2021-04-16T19:40:55.355939Z",
     "shell.execute_reply": "2021-04-16T19:40:55.355549Z"
    }
   },
   "outputs": [],
   "source": [
    "def marginal_beta(alpha, i):\n",
    "    \"\"\"Compute the ith marginal of a Dirichlet distribution.\"\"\"\n",
    "    total = np.sum(alpha)\n",
    "    return make_beta(alpha[i], total-alpha[i])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can use it to compute the marginal distribution for the three variables."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-04-16T19:40:55.360731Z",
     "iopub.status.busy": "2021-04-16T19:40:55.360306Z",
     "iopub.status.idle": "2021-04-16T19:40:55.361946Z",
     "shell.execute_reply": "2021-04-16T19:40:55.362286Z"
    }
   },
   "outputs": [],
   "source": [
    "marginals = [marginal_beta(alpha, i)\n",
    "             for i in range(len(alpha))]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The following plot shows the CDF of these distributions as gray lines and compares them to the CDFs of the samples."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-04-16T19:40:55.413354Z",
     "iopub.status.busy": "2021-04-16T19:40:55.398818Z",
     "iopub.status.idle": "2021-04-16T19:40:55.550204Z",
     "shell.execute_reply": "2021-04-16T19:40:55.550685Z"
    },
    "tags": [
     "hide-input"
    ]
   },
   "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": [
    "xs = np.linspace(0, 1, 101)\n",
    "\n",
    "for i in range(len(alpha)):\n",
    "    label = f'Column {i}'\n",
    "    \n",
    "    pmf = pmf_from_dist(marginals[i], xs)\n",
    "    pmf.make_cdf().plot(color='C5')\n",
    "    \n",
    "    cdf = cdfs[i]\n",
    "    cdf.plot(label=label, ls=':')\n",
    "\n",
    "decorate()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This confirms that the marginals of the Dirichlet distribution are beta distributions.\n",
    "And that's useful because the Dirichlet distribution is the conjugate prior for the multinomial likelihood function.\n",
    "\n",
    "If the prior distribution is Dirichlet with parameter vector `alpha` and the data is a vector of observations, `data`, the posterior distribution is Dirichlet with parameter vector `alpha + data`.\n",
    "\n",
    "As an exercise at the end of this chapter, you can use this method to solve the Lions and Tigers and Bears problem."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Summary\n",
    "\n",
    "After reading this chapter, if you feel like you've been tricked, I understand.  It turns out that many of the problems in this book can be solved with just a few arithmetic operations.  So why did we go to all the trouble of using grid algorithms?\n",
    "\n",
    "Sadly, there are only a few problems we can solve with conjugate priors; in fact, this chapter includes most of the ones that are useful in practice.\n",
    "\n",
    "For the vast majority of problems, there is no conjugate prior and no shortcut to compute the posterior distribution.\n",
    "That's why we need grid algorithms and the methods in the next two chapters, Approximate Bayesian Computation (ABC) and Markov chain Monte Carlo methods (MCMC)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercises\n",
    "\n",
    "**Exercise:** In the second version of the World Cup problem, the data we use for the update is not the number of goals in a game, but the time until the first goal.\n",
    "So the probability of the data is given by the exponential distribution rather than the Poisson distribution. \n",
    "\n",
    "But it turns out that the gamma distribution is *also* the conjugate prior of the exponential distribution, so there is a simple way to compute this update, too.\n",
    "The PDF of the exponential distribution is a function of $t$ with $\\lambda$ as a parameter.\n",
    "\n",
    "$$\\lambda e^{-\\lambda t}$$\n",
    "\n",
    "Multiply the PDF of the gamma prior by this likelihood, confirm that the result is an unnormalized gamma distribution, and see if you can derive its parameters.\n",
    "\n",
    "Write a few lines of code to update `prior_gamma` with the data from this version of the problem, which was a first goal after 11 minutes and a second goal after an additional 12 minutes."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "tags": [
     "hide-cell"
    ]
   },
   "source": [
    "Remember to express these quantities in units of games, which are approximately 90 minutes."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-04-16T19:40:55.555009Z",
     "iopub.status.busy": "2021-04-16T19:40:55.554432Z",
     "iopub.status.idle": "2021-04-16T19:40:55.556035Z",
     "shell.execute_reply": "2021-04-16T19:40:55.556555Z"
    }
   },
   "outputs": [],
   "source": [
    "# Solution\n",
    "\n",
    "\"\"\"\n",
    "The unnormalized posterior is\n",
    "\n",
    "\\lambda^{\\alpha-1+1} e^{-(\\beta + t) \\lambda}\n",
    "\n",
    "which is an unnormalized gamma distribution with parameters\n",
    "`alpha+1` and `beta+t`, which means that we observed 1 goal\n",
    "in elapsed time `t`.\n",
    "\n",
    "So we can use the same update function and call it like this:\n",
    "\"\"\"\n",
    "\n",
    "data = 1, 11/90\n",
    "posterior1 = update_gamma(prior_gamma, data)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-04-16T19:40:55.560564Z",
     "iopub.status.busy": "2021-04-16T19:40:55.560072Z",
     "iopub.status.idle": "2021-04-16T19:40:55.562807Z",
     "shell.execute_reply": "2021-04-16T19:40:55.563397Z"
    }
   },
   "outputs": [],
   "source": [
    "# Solution\n",
    "\n",
    "# Here's the second update\n",
    "\n",
    "data = 1, 12/90\n",
    "posterior2 = update_gamma(posterior1, data)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-04-16T19:40:55.567516Z",
     "iopub.status.busy": "2021-04-16T19:40:55.566992Z",
     "iopub.status.idle": "2021-04-16T19:40:55.569654Z",
     "shell.execute_reply": "2021-04-16T19:40:55.569261Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1.4, 2.1386138613861387, 2.7079646017699113)"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Solution\n",
    "\n",
    "prior_gamma.mean(), posterior1.mean(), posterior2.mean()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-04-16T19:40:55.600659Z",
     "iopub.status.busy": "2021-04-16T19:40:55.592219Z",
     "iopub.status.idle": "2021-04-16T19:40:55.771012Z",
     "shell.execute_reply": "2021-04-16T19:40:55.770585Z"
    }
   },
   "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": [
    "# Solution\n",
    "\n",
    "# And here's what the posteriors look like\n",
    "\n",
    "pmf_from_dist(prior_gamma, lams).plot(color='C5', label='prior')\n",
    "pmf_from_dist(posterior1, lams).plot(label='after 1 goal')\n",
    "pmf_from_dist(posterior2, lams).plot(label='after 2 goals')\n",
    "\n",
    "decorate_rate(title='World Cup Problem, Germany v Brazil')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise:** For problems like the Euro problem where the likelihood function is binomial, we can do a Bayesian update with just a few arithmetic operations, but only if the prior is a beta distribution.\n",
    "\n",
    "If we want a uniform prior, we can use a beta distribution with `alpha=1` and `beta=1`.\n",
    "But what can we do if the prior distribution we want is not a beta distribution?\n",
    "For example, in <<_TrianglePrior>> we also solved the Euro problem with a triangle prior, which is not a beta distribution.\n",
    "\n",
    "In these cases, we can often find a beta distribution that is a good-enough approximation for the prior we want.\n",
    "See if you can find a beta distribution that fits the triangle prior, then update it using `update_beta`.\n",
    "\n",
    "Use `pmf_from_dist` to make a `Pmf` that approximates the posterior distribution and compare it to the posterior we just computed using a grid algorithm.  How big is the largest difference between them?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "tags": [
     "hide-cell"
    ]
   },
   "source": [
    "Here's the triangle prior again."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-04-16T19:40:55.777176Z",
     "iopub.status.busy": "2021-04-16T19:40:55.776518Z",
     "iopub.status.idle": "2021-04-16T19:40:55.779638Z",
     "shell.execute_reply": "2021-04-16T19:40:55.779125Z"
    },
    "tags": [
     "hide-cell"
    ]
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2500"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from empiricaldist import Pmf\n",
    "\n",
    "ramp_up = np.arange(50)\n",
    "ramp_down = np.arange(50, -1, -1)\n",
    "\n",
    "a = np.append(ramp_up, ramp_down)\n",
    "xs = uniform.qs\n",
    "\n",
    "triangle = Pmf(a, xs, name='triangle')\n",
    "triangle.normalize()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "tags": [
     "hide-cell"
    ]
   },
   "source": [
    "And here's the update."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-04-16T19:40:55.784901Z",
     "iopub.status.busy": "2021-04-16T19:40:55.784432Z",
     "iopub.status.idle": "2021-04-16T19:40:55.787098Z",
     "shell.execute_reply": "2021-04-16T19:40:55.786719Z"
    },
    "tags": [
     "hide-cell"
    ]
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.007008842590059086"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "k, n = 140, 250\n",
    "likelihood = binom.pmf(k, n, xs)\n",
    "\n",
    "posterior = triangle * likelihood\n",
    "posterior.normalize()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "tags": [
     "hide-cell"
    ]
   },
   "source": [
    "To get you started, here's the beta distribution that we used as a uniform prior."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-04-16T19:40:55.792272Z",
     "iopub.status.busy": "2021-04-16T19:40:55.791535Z",
     "iopub.status.idle": "2021-04-16T19:40:55.794600Z",
     "shell.execute_reply": "2021-04-16T19:40:55.794106Z"
    },
    "tags": [
     "hide-cell"
    ]
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.5"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "alpha = 1\n",
    "beta = 1\n",
    "\n",
    "prior_beta = make_beta(alpha, beta)\n",
    "prior_beta.mean()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "tags": [
     "hide-cell"
    ]
   },
   "source": [
    "And here's what it looks like compared to the triangle prior."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-04-16T19:40:55.815338Z",
     "iopub.status.busy": "2021-04-16T19:40:55.810991Z",
     "iopub.status.idle": "2021-04-16T19:40:55.957815Z",
     "shell.execute_reply": "2021-04-16T19:40:55.958298Z"
    },
    "tags": [
     "hide-cell"
    ]
   },
   "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": [
    "prior_pmf = pmf_from_dist(prior_beta, xs)\n",
    "\n",
    "triangle.plot(label='triangle')\n",
    "prior_pmf.plot(label='beta')\n",
    "\n",
    "decorate_euro('Prior distributions')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "tags": [
     "hide-cell"
    ]
   },
   "source": [
    "Now you take it from there."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-04-16T19:40:55.963468Z",
     "iopub.status.busy": "2021-04-16T19:40:55.962877Z",
     "iopub.status.idle": "2021-04-16T19:40:55.965132Z",
     "shell.execute_reply": "2021-04-16T19:40:55.965482Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.5595238095238095"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Solution\n",
    "\n",
    "data = 140, 250\n",
    "posterior_beta = update_beta(prior_beta, data)\n",
    "posterior_beta.mean()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-04-16T19:40:55.969495Z",
     "iopub.status.busy": "2021-04-16T19:40:55.969043Z",
     "iopub.status.idle": "2021-04-16T19:40:55.970756Z",
     "shell.execute_reply": "2021-04-16T19:40:55.971108Z"
    }
   },
   "outputs": [],
   "source": [
    "# Solution\n",
    "\n",
    "posterior_conjugate = pmf_from_dist(posterior_beta, xs)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-04-16T19:40:55.988979Z",
     "iopub.status.busy": "2021-04-16T19:40:55.983948Z",
     "iopub.status.idle": "2021-04-16T19:40:56.155440Z",
     "shell.execute_reply": "2021-04-16T19:40:56.155054Z"
    }
   },
   "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": [
    "# Solution\n",
    "\n",
    "posterior.plot(label='grid posterior', ls=':')\n",
    "posterior_conjugate.plot(label='conjugate posterior')\n",
    "\n",
    "decorate(xlabel='Proportion of heads (x)',\n",
    "         ylabel='Probability',\n",
    "         title='Posterior distribution of x')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-04-16T19:40:56.158948Z",
     "iopub.status.busy": "2021-04-16T19:40:56.158455Z",
     "iopub.status.idle": "2021-04-16T19:40:56.160890Z",
     "shell.execute_reply": "2021-04-16T19:40:56.161306Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Solution\n",
    "\n",
    "# The largest absolute difference is pretty small\n",
    "\n",
    "np.allclose(posterior, posterior_conjugate)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise:**  [3Blue1Brown](https://en.wikipedia.org/wiki/3Blue1Brown) is a YouTube channel about math; if you are not already aware of it, I recommend it highly.\n",
    "In [this video](https://www.youtube.com/watch?v=8idr1WZ1A7Q) the narrator presents this problem:\n",
    "\n",
    "> You are buying a product online and you see three sellers offering the same product at the same price.  One of them has a 100% positive rating, but with only 10 reviews.  Another has a 96% positive rating with 50 total reviews.  And yet another has a 93% positive rating, but with 200 total reviews.\n",
    ">\n",
    ">Which one should you buy from?\n",
    "\n",
    "Let's think about how to model this scenario.  Suppose each seller has some unknown probability, `x`, of providing satisfactory service and getting a positive rating, and we want to choose the seller with the highest value of `x`.\n",
    "\n",
    "This is not the only model for this scenario, and it is not necessarily the best.  An alternative would be something like item response theory, where sellers have varying ability to provide satisfactory service and customers have varying difficulty of being satisfied.\n",
    "\n",
    "But the first model has the virtue of simplicity, so let's see where it gets us.\n",
    "\n",
    "1. As a prior, I suggest a beta distribution with `alpha=8` and `beta=2`.  What does this prior look like and what does it imply about sellers?\n",
    "\n",
    "2. Use the data to update the prior for the three sellers and plot the posterior distributions.  Which seller has the highest posterior mean?\n",
    "\n",
    "3. How confident should we be about our choice?  That is, what is the probability that the seller with the highest posterior mean actually has the highest value of `x`?\n",
    "\n",
    "4. Consider a beta prior with `alpha=0.7` and `beta=0.5`.  What does this prior look like and what does it imply about sellers?\n",
    "\n",
    "5. Run the analysis again with this prior and see what effect it has on the results."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "tags": [
     "hide-cell"
    ]
   },
   "source": [
    "Note: When you evaluate the beta distribution, you should restrict the range of `xs` so it does not include 0 and 1.  When the parameters of the beta distribution are less than 1, the probability density goes to infinity at 0 and 1.  From a mathematical point of view, that's not a problem; it is still a proper probability distribution.  But from a computational point of view, it means we have to avoid evaluating the PDF at 0 and 1. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-04-16T19:40:56.196209Z",
     "iopub.status.busy": "2021-04-16T19:40:56.185328Z",
     "iopub.status.idle": "2021-04-16T19:40:56.335967Z",
     "shell.execute_reply": "2021-04-16T19:40:56.336352Z"
    }
   },
   "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": [
    "# Solution\n",
    "\n",
    "# The first prior implies that most sellers are \n",
    "# satisfactory most of the time, but none are perfect.\n",
    "\n",
    "prior = make_beta(8, 2)\n",
    "\n",
    "xs = np.linspace(0.005, 0.995, 199)\n",
    "prior_pmf = pmf_from_dist(prior, xs)\n",
    "prior_pmf.plot(color='C5', label='prior')\n",
    "\n",
    "decorate(xlabel='Probability of positive rating',\n",
    "         ylabel='PDF')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-04-16T19:40:56.339519Z",
     "iopub.status.busy": "2021-04-16T19:40:56.339053Z",
     "iopub.status.idle": "2021-04-16T19:40:56.340656Z",
     "shell.execute_reply": "2021-04-16T19:40:56.341088Z"
    }
   },
   "outputs": [],
   "source": [
    "# Solution\n",
    "\n",
    "data1 = 10, 10\n",
    "data2 = 48, 50\n",
    "data3 = 186, 200"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-04-16T19:40:56.346127Z",
     "iopub.status.busy": "2021-04-16T19:40:56.345648Z",
     "iopub.status.idle": "2021-04-16T19:40:56.347288Z",
     "shell.execute_reply": "2021-04-16T19:40:56.347662Z"
    }
   },
   "outputs": [],
   "source": [
    "# Solution\n",
    "\n",
    "seller1 = update_beta(prior, data1)\n",
    "seller2 = update_beta(prior, data2)\n",
    "seller3 = update_beta(prior, data3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-04-16T19:40:56.353115Z",
     "iopub.status.busy": "2021-04-16T19:40:56.352217Z",
     "iopub.status.idle": "2021-04-16T19:40:56.355306Z",
     "shell.execute_reply": "2021-04-16T19:40:56.354930Z"
    }
   },
   "outputs": [],
   "source": [
    "# Solution\n",
    "\n",
    "seller1_pmf = pmf_from_dist(seller1, xs)\n",
    "seller2_pmf = pmf_from_dist(seller2, xs)\n",
    "seller3_pmf = pmf_from_dist(seller3, xs)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-04-16T19:40:56.374710Z",
     "iopub.status.busy": "2021-04-16T19:40:56.374205Z",
     "iopub.status.idle": "2021-04-16T19:40:56.555114Z",
     "shell.execute_reply": "2021-04-16T19:40:56.555467Z"
    }
   },
   "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": [
    "# Solution\n",
    "\n",
    "seller1_pmf.plot(label='seller 1')\n",
    "seller2_pmf.plot(label='seller 2')\n",
    "seller3_pmf.plot(label='seller 3')\n",
    "\n",
    "decorate(xlabel='Probability of positive rating',\n",
    "         ylabel='PDF',\n",
    "         xlim=(0.65, 1.0))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-04-16T19:40:56.559596Z",
     "iopub.status.busy": "2021-04-16T19:40:56.559152Z",
     "iopub.status.idle": "2021-04-16T19:40:56.561937Z",
     "shell.execute_reply": "2021-04-16T19:40:56.561579Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.9, 0.9333333333333333, 0.9238095238095239)"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Solution\n",
    "\n",
    "seller1.mean(), seller2.mean(), seller3.mean()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-04-16T19:40:56.565579Z",
     "iopub.status.busy": "2021-04-16T19:40:56.565155Z",
     "iopub.status.idle": "2021-04-16T19:40:56.571636Z",
     "shell.execute_reply": "2021-04-16T19:40:56.571163Z"
    }
   },
   "outputs": [],
   "source": [
    "# Solution\n",
    "\n",
    "iters = 10000\n",
    "a = np.empty((3, iters))\n",
    "\n",
    "a[0] = seller1.rvs(iters)\n",
    "a[1] = seller2.rvs(iters)\n",
    "a[2] = seller3.rvs(iters)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-04-16T19:40:56.575282Z",
     "iopub.status.busy": "2021-04-16T19:40:56.574731Z",
     "iopub.status.idle": "2021-04-16T19:40:56.583216Z",
     "shell.execute_reply": "2021-04-16T19:40:56.583564Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>probs</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>0.2948</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>0.4750</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>0.2302</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "0    0.2948\n",
       "1    0.4750\n",
       "2    0.2302\n",
       "dtype: float64"
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Solution\n",
    "\n",
    "from empiricaldist import Pmf\n",
    "\n",
    "best = np.argmax(a, axis=0)\n",
    "Pmf.from_seq(best)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise:** Use a Dirichlet prior with parameter vector `alpha = [1, 1, 1]` to solve the Lions and Tigers and Bears problem:\n",
    "\n",
    ">Suppose we visit a wild animal preserve where we know that the only animals are lions and tigers and bears, but we don't know how many of each there are.\n",
    ">\n",
    ">During the tour, we see three lions, two tigers, and one bear. Assuming that every animal had an equal chance to appear in our sample, estimate the prevalence of each species.\n",
    ">\n",
    ">What is the probability that the next animal we see is a bear?\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-04-16T19:40:56.586898Z",
     "iopub.status.busy": "2021-04-16T19:40:56.586274Z",
     "iopub.status.idle": "2021-04-16T19:40:56.588333Z",
     "shell.execute_reply": "2021-04-16T19:40:56.587929Z"
    }
   },
   "outputs": [],
   "source": [
    "# Solution\n",
    "\n",
    "prior_alpha = np.array([1, 1, 1])\n",
    "data = 3, 2, 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-04-16T19:40:56.591889Z",
     "iopub.status.busy": "2021-04-16T19:40:56.591119Z",
     "iopub.status.idle": "2021-04-16T19:40:56.593615Z",
     "shell.execute_reply": "2021-04-16T19:40:56.593186Z"
    }
   },
   "outputs": [],
   "source": [
    "# Solution\n",
    "\n",
    "posterior_alpha = prior_alpha + data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-04-16T19:40:56.598915Z",
     "iopub.status.busy": "2021-04-16T19:40:56.598290Z",
     "iopub.status.idle": "2021-04-16T19:40:56.601057Z",
     "shell.execute_reply": "2021-04-16T19:40:56.601527Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.2222222222222222"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Solution\n",
    "\n",
    "marginal_bear = marginal_beta(posterior_alpha, 2)\n",
    "marginal_bear.mean()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-04-16T19:40:56.605289Z",
     "iopub.status.busy": "2021-04-16T19:40:56.604526Z",
     "iopub.status.idle": "2021-04-16T19:40:56.607088Z",
     "shell.execute_reply": "2021-04-16T19:40:56.607591Z"
    }
   },
   "outputs": [],
   "source": [
    "# Solution\n",
    "\n",
    "dist = dirichlet(posterior_alpha)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-04-16T19:40:56.614252Z",
     "iopub.status.busy": "2021-04-16T19:40:56.613141Z",
     "iopub.status.idle": "2021-04-16T19:40:56.620618Z",
     "shell.execute_reply": "2021-04-16T19:40:56.620095Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>prob</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>lion</th>\n",
       "      <td>0.444444</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>tiger</th>\n",
       "      <td>0.333333</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>bear</th>\n",
       "      <td>0.222222</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "           prob\n",
       "lion   0.444444\n",
       "tiger  0.333333\n",
       "bear   0.222222"
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Solution\n",
    "\n",
    "import pandas as pd\n",
    "\n",
    "index = ['lion', 'tiger', 'bear']\n",
    "pd.DataFrame(dist.mean(), index, columns=['prob'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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