statistics/Distributions-Beta.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "from numpy import random\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import seaborn as sns\n",
    "import warnings\n",
    "from scipy.stats import binom\n",
    "from scipy.stats import beta\n",
    "\n",
    "warnings.filterwarnings( 'ignore' )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Beta( 1, 1 )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "a = 1\n",
    "b = 1\n",
    "x = np.arange( 0, 1, 0.01 )\n",
    "y = beta.pdf( x, a, b )\n",
    "plt.plot( x, y )\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Beta( 2, 2 )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "a = 2\n",
    "b = 2\n",
    "x = np.arange( 0, 1, 0.01 )\n",
    "y = beta.pdf( x, a, b )\n",
    "plt.plot( x, y )\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Haldane's Prior - Beta( 0, 0 )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Another *uninformative prior* is the Beta(0, 0) prior (the [Haldane's prior probability](https://en.wikipedia.org/wiki/Beta_distribution#Haldane.27s_prior_probability_.28Beta.280.2C0.29.29)) where both $\\alpha=0$ and $\\beta=0$. This doesn't produce a \"proper\" prior distribution, per se, as it has infinite mass at 0 and 1, because the $0$'s will cause $1/0$ terms in the Beta function. The Beta(0, 0) can't be graphed easily (as shown below). \n",
    "\n",
    "But, if you want an uninformative prior, since you just need to do some basic arithmetic to calculate the updated $\\alpha$ and $\\beta$ terms for your posterior distribution as soon as you collect some data for each variation, it doesn't really matter much what the initial prior distribution looks like."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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vM00hWPaemf8bVwE/AZ7qHp+Y9J7eZK9/CvyM+d8sub07dxfwoe757zD/GyNzwA+Adw987u3d5x3iLP3NqHHuGfhr4L8Hfq5PARdMej8r+TMe+BpTFwL/FxOS1Dh/a0iSGmcIJKlxhkCSGmcIJKlxhkCSGmcIJKlxhkCSGve/5wv9yACcdLkAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "a = 0\n",
    "b = 0\n",
    "x = np.arange( 0, 1, 0.01 )\n",
    "y = beta.pdf( x, a, b )\n",
    "plt.plot( x, y )\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Jeffrey's Prior - Beta( 1/2, 1/2 )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Another Beta distribution that is often chosen as an uninformative prior is the Beta( 1/2, 1/2 ) ([the Jeffrey's prior](https://en.wikipedia.org/wiki/Beta_distribution#Jeffreys'_prior_probability_(Beta(1/2,1/2)_for_a_Bernoulli_or_for_a_binomial_distribution)https://en.wikipedia.org/wiki/Beta_distribution#Jeffreys'_prior_probability_(Beta(1/2,1/2)_for_a_Bernoulli_or_for_a_binomial_distribution))).\n",
    "\n",
    "This uninformative prior is useful when the likelihood of "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "a = 1/2\n",
    "b = 1/2\n",
    "x = np.arange( 0, 1, 0.01 )\n",
    "y = beta.pdf( x, a, b )\n",
    "plt.plot( x, y )\n",
    "plt.show()"
   ]
  }
 ],
 "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": "3.8.5"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}