pilipolio/GrokingTheBeta.ipynb

## 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": 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      }
     ],
     "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": 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OnDnKF1980WEfANy4cePGrRdbb9kdbKNzcI1eye+uf+7254mIyL3sdssYDAZY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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": 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ywC2HfZRKJVatWoXp06ejqakJCxYscPvJXiIi+pXbrvOPi4vD8ePHsWrVKuTk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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": 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56EJ1qQtXi10oK7Fm9GiIiYGpU5HRTKLbk0QgeoRKdSWny05TVFZU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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": {}
  }
 ]
}