Trying to understand the Beta distribution, its link with the binomial and some benefits associated with the bayesian framework.

GrokingTheBeta.ipynb

{
 "metadata": {
  "name": "GrokingTheBeta"
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Groking the Beta distribution"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Most of that notebook was gathered from this thorough CS introduction to the [beta-binomial conjugate](http://www.cs.cmu.edu/~10701/lecture/technote2_betabinomial.pdf), the wikipedia page about [conjugate prior](http://en.wikipedia.org/wiki/Conjugate_prior) examplified with the beta-binomial prior and posterior distributions, and also those concise [slides](http://courses.engr.illinois.edu/cs598jhm/sp2010/Slides/Lecture02HO.pdf)."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Estimating the $p$ parameter from a Binomial\n",
      "--------------------------------------------\n",
      "\n",
      "Starting from the number of success $k$ from a binomial distribution $k|n,p$ where the number of samples $n$ is fixed and the success probability $p$ is the hidden parameter to be learned :\n",
      "\n",
      "<center> $k|p,n \\sim \\text{Bin}(p,n)$ </center>\n",
      "\n",
      "<center>$ P(k|p,n) = {{n}\\choose{k}} p^k (1-p)^{n-k}$</center>\n",
      "\n",
      "We observe $s = 100$ samples $(k_i)_{i = 1, \\ldots, s}$ from that distribution.  "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "n = 50\n",
      "p = .7\n",
      "size = 100\n",
      "samples = np.random.binomial(n, p, size)\n",
      "print 'samples.shape = {} drawn from the Bin({},{})'.format(samples.shape, p, n)\n",
      "plt.hist(samples, bins=20);"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "samples.shape = (100L,) drawn from the Bin(0.7,50)\n"
       ]
      },
      {
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAXIAAAD9CAYAAAChtfywAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\ncI4iBwDn/h93IscoWhReRgAAAABJRU5ErkJggg==\n"
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Question : how to estimate $p$?\n",
      "-------------------------------\n",
      "\n",
      "### the frequentist approach : $\\hat{p} = \\frac{1}{n s} \\sum \\limits_{i=0}^n k_i $"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "freq_estimation_p = np.mean(samples) / n\n",
      "freq_estimation_p"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 11,
       "text": [
        "0.70340000000000003"
       ]
      }
     ],
     "prompt_number": 11
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### The bayesian approach :\n",
      "\n",
      "$$ \\overbrace{P \\hspace{.1cm} (p|x)}^{\\text{posterior}} \\hspace{.3cm} = \\frac{\\overbrace{P(p)}^{\\text{prior}} \\hspace{.3cm}  \\overbrace{P \\hspace{.1cm} (x|p)}^{\\text{original distribution}}}{\\underbrace{P \\hspace{.1cm} (x)}_{\\text{marginal}}}  $$\n",
      "\n",
      "Choosing a prior :\n",
      "\n",
      " - modelling our belief (yeah, right ...)\n",
      " - realistic/flexible $\\rightarrow$ itself with parameters\n",
      " - proving algebraic conveniences $\\rightarrow$ [conjugated](http://en.wikipedia.org/wiki/Conjugate_prior), i.e. from the same distribution family\n",
      "\n",
      "And here enters the $\\text{Beta}$ distribution (see [here](http://www.cs.cmu.edu/~10701/lecture/technote2_betabinomial.pdf) for the constant part and a summary about the $\\Gamma$ function):\n",
      "\n",
      "<center> $ p | \\beta_1, \\beta_2 \\sim \\text{Beta}(\\beta_1, \\beta_2)$</center>\n",
      "\n",
      "<center> $ P(p) \\propto p ^ {\\beta_1 - 1} (1 - p) ^ {\\beta_2 - 1}$</center>\n",
      "\n",
      "The posterior distribution is then :\n",
      "\n",
      "$$ P(p|X) \\propto P(p) P (X|p) $$\n",
      "\n",
      "$$ P(p|X) \\propto p^{\\beta_1 - 1} (1 - p) ^ {\\beta_2 - 1} \\times p^k (1-p)^{n-k} $$\n",
      "\n",
      "$$ P(p|X) \\propto p^{\\beta_1 + k - 1} (1 - p) ^ {\\beta_2 + n - k - 1} $$\n",
      "\n",
      "which also follows a $\\text{Beta}$ distribution (hence the beta-binomial model):\n",
      "\n",
      "$$ p|X \\sim \\text{Beta} (\\beta_1 + k, \\beta_2 + n( - k) $$\n",
      "\n",
      "and back to estimate the probability $p$, to predict for example the next occurence of $x$, we take the posterior mean, i.e. the mean of that new $\\text{Beta}$ distribution (see the [wikipedia page](http://en.wikipedia.org/wiki/Beta_distribution)) :\n",
      "\n",
      "$$ \\hat{p} = \\mathbb{E}[p|X] = \\frac{\\beta_1 + k} {\\beta_1 + \\beta_2 + n}$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Choosing priors parameters $\\beta_1$ and $\\beta_2$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "For example, let's assume we know from a priori information that the probability $p$ is neither $0$ or $1$. "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from scipy.stats import beta\n",
      "p_s = np.linspace(0,1,100)\n",
      "beta_pdf = beta(1.25, 1.25).pdf(x=p_s)\n",
      "plt.plot(p_s, beta_pdf);"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAXcAAAD9CAYAAABHnDf0AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\n"
      }
     ],
     "prompt_number": 37
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "What do we benefit from the bayesian framework?  \n",
      "-----------------------------------------------\n",
      "\n",
      "Having a priori about our parameters, even not fully correct (biaising our estimation) can drasicaly reduce the noise of our estimation (reducing the variance). In some case, regularization can be seen as a bayesian a priori. Let's illustrate that :"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "p = 0.7\n",
      "repeats = 100\n",
      "varying_n = np.logspace(1, 3, num = 30)\n",
      "\n",
      "data_with_varying_n = np.array([np.random.binomial(n, p, size = repeats) for n in varying_n])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 89
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Frequentist estimation\n",
      "freq_estimations = data_with_varying_n.T / varying_n\n",
      "# Bayesian estimation with our prior\n",
      "bayesian_estimations = (data_with_varying_n.T + 1.25) / (varying_n + 2.5)\n",
      "\n",
      "p = np.vstack((freq_estimations.ravel(), bayesian_estimations.ravel()))\n",
      "plt.hist(np.vstack((freq_estimations.ravel(), bayesian_estimations.ravel())).T, bins=20);\n",
      "plt.legend(['frequentist', 'with prior']);"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAX8AAAD9CAYAAABUS3cAAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nRDLE8CcikiGGPxGRDP0PCbdfVXNCXhUAAAAASUVORK5CYII=\n"
      }
     ],
     "prompt_number": 90
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.plot(np.mean((freq_estimations - .7)**2, axis=0), label='frequentist')\n",
      "plt.plot(np.mean((bayesian_estimations - .7)**2, axis=0), label='with prior')\n",
      "plt.legend();"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAYIAAAD9CAYAAACx+XApAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\ncZYAAAAASUVORK5CYII=\n"
      }
     ],
     "prompt_number": 88
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Notes :\n",
      "-------\n",
      "\n",
      "- Hyper parameters seen as \"pseudo-observations\" or \"pseudo-counts\" see those [slides](http://courses.engr.illinois.edu/cs598jhm/sp2010/Slides/Lecture02HO.pdf)).\n",
      "\n",
      "- Random-pause based profiling of code in that [stackexchange question](http://stats.stackexchange.com/questions/4659/relationship-between-binomial-and-beta-distributions)\n",
      "\n",
      "- The uniform-beta conjugate and application to quantile interval error estimation in this [blog post](http://probabilityandstats.wordpress.com/2010/02/21/the-order-statistics-and-the-uniform-distribution/)"
     ]
    }
   ],
   "metadata": {}
  }
 ]
}