Ch8.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Chapter 8: Transformations\n",
    " \n",
    "This Jupyter notebook is the Python equivalent of the R code in section 8.8 R, pp. 373 - 375, [Introduction to Probability, 2nd Edition](https://www.crcpress.com/Introduction-to-Probability-Second-Edition/Blitzstein-Hwang/p/book/9781138369917), Blitzstein & Hwang.\n",
    "\n",
    "----"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Beta and Gamma distributions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy.stats import beta, gamma\n",
    "\n",
    "# to learn more about scipy.stats.beta, un-comment ouf the following line\n",
    "#print(beta.__doc__)\n",
    "\n",
    "# to learn more about scipy.stats.gamma, un-comment ouf the following line\n",
    "#rint(gamma.__doc__)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The Beta and Gamma distributions are implemented in [`scipy.stats.beta`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.beta.html) and [`scipy.stats.gamma`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.gamma.html), respectively.\n",
    "\n",
    "* To evaluate the $Beta(a, b)$ PDF or CDF at $x$, we use `beta.pdf(x, a, b)` and `beta.cdf(x, a, b)`. To generate $n$ realizations from the $Beta(a, b)$ distribution, we use `beta.rvs(a, b, size=n)`. \n",
    "* To evaluate the $Gamma(a, \\lambda)$ PDF or CDF at $x$, we use`gamma.pdf(x, a, scale=1/lambd)` or `gamma.cdf(x, a, scale=1/lambd)`. To generate $n$ realizations from the $Gamma(a, \\lambda)$ distribution, we use `gamma.rvs(a, scale=1/lambd, size=n)`. \n",
    "   * The $\\lambda$ parameter in $Gamma(a, \\lambda)$ corresponds in `gamma` to using `scale` = $\\frac{1}{\\lambda}$ and default value of `loc` = 0.\n",
    "\n",
    "For example, we can check that the $Gamma(3, 2)$ distribution has mean $\\frac{3}{2}$ and variance $\\frac{3}{4}$. To do this, we generate a large number of $Gamma(3, 2)$ random variables using `gamma.rvs`, then compute their mean and var using their corresponding methods in `numpy.array` (you could of course also use `numpy.mean` and `numpy.var`, passing in the array of r.v.):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "mean of Gamma(3, 2) = 1.5012880719327166\n",
      "variance of Gamma(3, 2) = 0.7512319256875035\n"
     ]
    }
   ],
   "source": [
    "# seed the random number generator\n",
    "np.random.seed(317811)\n",
    "\n",
    "alpha = 3.0\n",
    "lambd = 2.0\n",
    "\n",
    "y = gamma.rvs(alpha, scale=1/lambd, size=10**5)\n",
    "\n",
    "mean = y.mean()\n",
    "#mean = np.mean(y)\n",
    "print('mean of Gamma(3, 2) = {}'.format(mean))\n",
    "\n",
    "var = y.var()\n",
    "#var = np.var(y)\n",
    "print('variance of Gamma(3, 2) = {}'.format(var))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**NOTE**: [`lambda`](https://docs.python.org/3/tutorial/controlflow.html#lambda-expressions) in Python is a reserved keyword for declaring small, anonymous functions, and so we cannot used the name `lambda` for variable or functions.\n",
    "\n",
    "Try changing the `numpy.random.seed` input value in the code block above, and hit SHIFT+ENTER to re-execute the code block. Did you get values that are close to 1.5 and 0.75, respectively?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Convolution of Uniforms\n",
    "\n",
    "Using [`scipy.stats.uniform`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.uniform.html) and [`matplotlib.pyplot.hist`](https://matplotlib.org/api/_as_gen/matplotlib.pyplot.hist.html), we can quickly verify that for $X, Y \\stackrel{i.i.d.}{\\sim} Unif(0, 1)$, the distribution of $T = X + Y$ is triangular in shape:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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hk8qDVfXQxEqSLZnmgViSpM3XsEnlH5O8E3hyezb9J4DPjC4sSdJiNGxSOZbuufJXAb9K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\nvipzAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "np.random.seed(514229)\n",
    "\n",
    "from scipy.stats import uniform\n",
    "\n",
    "x = uniform.rvs(size=10**5)\n",
    "y = uniform.rvs(size=10**5)\n",
    "\n",
    "t = x + y\n",
    "\n",
    "plt.hist(t, bins=20)\n",
    "\n",
    "plt.title(r'Histogram of $T = X + Y$')\n",
    "plt.xlabel(r'$x + y$')\n",
    "plt.ylabel('Frequency')\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The histogram looks like an ascending and then descending staircase, a discrete approximation to a triangle."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Bayes' billiards\n",
    "\n",
    "In the Bayes' billiards story, we have $n$ white balls and 1 gray ball, throw them onto the unit interval completely at random, and count the number of white balls to the left of the gray ball. Letting $p$ be the position of the gray ball and $X$ be the number of white balls to the left of the gray ball, we have\n",
    "\n",
    "\\begin{align}\n",
    "  p &\\sim Unif(0, 1) \\\\\n",
    "  X|p &\\sim Bin(n, p)\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "By performing this experiment a large number of times, we can verify the results we derived in this chapter about the marginal PMF of $X$ and the posterior PDF of $p \\text{ given } X = x$. We'll let the number of simulations be called `nsims`, to avoid a name conflict with the number of white balls, `n`, which we set equal to 10:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "nsims = 10**5\n",
    "n = 10"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We simulate 10<sup>5</sup> values of $p$, then simulate 10<sup>5</sup> values from the conditional distribution of $X$ given $p$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "np.random.seed(832040)\n",
    "\n",
    "p = uniform.rvs(size=nsims)\n",
    "\n",
    "from scipy.stats import binom\n",
    "\n",
    "x = binom.rvs(n, p, size=nsims)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Notice that we feed the entire array `p` into `binom.rvs`. This means that the first element of `x` is generated using the first element of `p`, the second element of `x` is generated using the second element of `p`, and so forth. Thus, conditional on a particular element of `p`, the corresponding element of `x` is Binomial, but the elements of `p` are themselves Uniform, exactly as the model specifies.\n",
    "\n",
    "According to the Bayes' billiards argument, the marginal distribution of $X$ should be Discrete Uniform on the integers 0 through $n$. Is this in fact the case? We can make a histogram of `x` to check! Because the distribution of $X$ is discrete, we can specify `bins=np.arange(0, n+1, 0.5), align='left'` in the call to `matplotlib.pyplot.hist` so that each bar is centered at an integer value:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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7htonDG8jSRqzUU+Ofxb4bwzuiHv10GOrquq0qtq/qpYxOLl9eVX9J+DLwMtat9XApW35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\nAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(x, bins=np.arange(0, n+1, 0.5), align='left')\n",
    "\n",
    "plt.title(r'Histogram of $x$')\n",
    "plt.xlabel(r'$x$')\n",
    "plt.ylabel('Frequency')\n",
    "# also set the x-axis ticks to show integers 0, 1, 2, ..., 10\n",
    "plt.xticks(range(0, 10+1))\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Indeed, all the histogram bars are approximately equal in height, consistent with a Discrete Uniform distribution. \n",
    "\n",
    "Now for the posterior distribution of $p$ given $X = x$. Conditioning is very simple since we are using `numpy.array`. To consider only the simulated values of $p$ where the value of $X$ was 3, we use [`numpy.where(x==3)`](https://docs.scipy.org/doc/numpy/reference/generated/numpy.where.html) to find the indices where `x` = 3, and then use that within the square brackets of `p` to index only those corresponding values. In particular, we can create a histogram of these values using `matplotlib.pyplot.hist` to see what the posterior distribution of $p$ given $X = 3$ looks like.\n",
    "\n",
    "According to the Beta-Binomial conjugacy result, the true posterior distribution is $p|X = 3 \\sim Beta(4, 8)$. We can plot the histogram of `p[numpy.where(x==3)]` next to a histogram of simulated values from the $Beta(4, 8)$ distribution to confirm that they look similar:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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v2zdJ3pRka5ILkzx8d+9XkrRkjTRPSZLUa9ii6Ye9xUqSRwA/6ui/K28EPlpV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\nHSyaJEmSJKmDRZMkSZIkdfj/oIUcqN7x43kAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 1008x432 with 2 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "np.random.seed(1346269)\n",
    "\n",
    "fig = plt.figure(figsize=(14, 6))\n",
    "\n",
    "p_given_x_3 = p[np.where(x==3)]\n",
    "\n",
    "ax1 = fig.add_subplot(121)\n",
    "ax1.hist(p_given_x_3, bins=20, color='#e66101')\n",
    "ax1.set_title(r'Histogram of $\\tt{p[numpy.where(x==3)]}$')\n",
    "ax1.set_ylabel('Frequency')\n",
    "\n",
    "brv = beta.rvs(4.0, 8.0, size=10**4)\n",
    "\n",
    "ax2 = fig.add_subplot(122)\n",
    "ax2.hist(brv, bins=20, color='#5e3c99')\n",
    "ax2.set_title(r'Histogram of $Beta(4, \\, 8)$, $10^4$ simulated values')\n",
    "ax2.set_ylabel('Frequency')\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For the side-by-side graph above, we obtain an instance of [`matplotlib.pyplot.figure`](https://matplotlib.org/api/_as_gen/matplotlib.pyplot.figure.html) from `pyplot`, specifying the figure's width and height. Next, we use [`figure.add_subplot(121)`](https://matplotlib.org/api/_as_gen/matplotlib.pyplot.subplot.html) to create graph axes in a figure with 1 row and 2 columns, in the 1st column position from the left, for the histogram of `p[numpy.where(x==3)]`. Likewise, `figure.add_subplot(122)` creates the graph axes for the histogram for $Beta(4, 8)$ in the 2nd column position of the 1-row, 2-column figure."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Simulating order statistics\n",
    "\n",
    "Simulating order statistics in Python/NumPy/SciPy is easy: we simply simulate i.i.d. r.v.s and sort them in order. For example,"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-2.05800632 -1.14300612 -0.09108752 -0.08188422  0.45810851  0.55556877\n",
      "  0.59398037  0.9312065   0.98367818  1.12202827]\n"
     ]
    }
   ],
   "source": [
    "np.random.seed(2178309)\n",
    "\n",
    "from scipy.stats import norm\n",
    "\n",
    "rv = np.sort(norm.rvs(size=10))\n",
    "print(rv)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "produces one realization of $X_{(1)}, \\ldots, X_{(10)}$, where $X_{(1)}, \\ldots, X_{(10)}$ are i.i.d. $N(0, 1)$. If we want to plot a histogram of realizations of, say, $X_{(9)}$, we'll need to use iteration. Let's try using [Python's list comprehensions](https://docs.python.org/3/tutorial/datastructures.html#list-comprehensions):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "order_stats has shape: (10000, 10)\n"
     ]
    }
   ],
   "source": [
    "np.random.seed(3524578)\n",
    "\n",
    "rv_arr = [np.sort(norm.rvs(size=10)) for _ in range(10**4)]\n",
    "order_stats = np.array(rv_arr)\n",
    "\n",
    "print('order_stats has shape: {}'.format(order_stats.shape))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The 10<sup>4</sup> iterations are done with\n",
    "\n",
    "    rv_arr = [np.sort(norm.rvs(size=10)) for _ in range(10**4)]\n",
    "\n",
    "This Python list comprehension does the following:\n",
    "* loops 10<sup>4</sup> times with `for _ in range(10**4)`, where we ignore the iteration counter by using `_`\n",
    "   * as we loop, we do not need to know which particular iteration we are on\n",
    "* for each iteration, calculate `np.sort(norm.rvs(size=10))`\n",
    "* gather up the results of all the iterations into a Python list (array), assigning this resulting list to `rv_arr`\n",
    "* this one-liner list comprehension is equivalent to all this code:\n",
    "<pre>\n",
    "    rv_arr = []\n",
    "    for _ in range(10**4):\n",
    "        rv = np.sort(norm.rvs(size=10))\n",
    "        rv_arr.append(rv)\n",
    "</pre>\n",
    "\n",
    "This creates a matrix, `order_stats`, with 10<sup>4</sup> rows and 10 columns. The _i_<sup>th</sup> column of the matrix contains 10<sup>4</sup> realizations of $X_{(i+1)}$ (don't forget that Python is zero-indexed, so the 8<sup>th</sup> column is $X_{(9)}$!). Now we can create a histogram of $X_{(9)}$, simply by selecting column 8 of the matrix:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x9 = order_stats[:, 8]\n",
    "\n",
    "plt.hist(x9, bins=20)\n",
    "\n",
    "plt.xticks(np.arange(-1, 4, 1))\n",
    "plt.title(r'Histogram of $X_{(9)}$')\n",
    "plt.ylabel('Frequency')\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can also compute summaries like `numpy.mean(x9)` and `numpy.var(x9)`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "mean of x9 = 1.0092010380841254\n",
      "variance of x9 = 0.2108867888683199\n"
     ]
    }
   ],
   "source": [
    "mean_x9 = np.mean(x9)\n",
    "print('mean of x9 = {}'.format(mean_x9))\n",
    "\n",
    "var_x9 = np.var(x9)\n",
    "print('variance of x9 = {}'.format(var_x9))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "----\n",
    "\n",
    "Joseph K. Blitzstein and Jessica Hwang, Harvard University and Stanford University, &copy; 2019 by Taylor and Francis Group, LLC"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.3"
  },
  "notebook_info": {
   "author": "Joseph K. Blitzstein, Jessica Hwang",
   "chapter": "8. Transformations",
   "creator": "buruzaemon",
   "section": "8.8",
   "title": "Introduction to Probability, 1st Edition"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}