mle.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Maximum likelihood Estimation (MLE)\n",
    "based on http://python-for-signal-processing.blogspot.com/2012/10/maximum-likelihood-estimation-maximum.html\n",
    "## Simulate coin flipping\n",
    "- [Bernoulli distribution](https://en.wikipedia.org/wiki/Bernoulli_distribution) \n",
    "is the probability distribution of a random variable which takes the value 1 with probability $p$ and the value 0 with probability $q = 1 - p$\n",
    "- [scipy.stats.bernoulli](https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.bernoulli.html)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from scipy.stats import bernoulli \n",
    "\n",
    "np.random.seed(123456789)\n",
    "\n",
    "p_true = 1/2 # this is the value we will try to estimate from the observed data\n",
    "fp = bernoulli(p_true)\n",
    "\n",
    "def sample(n=10):\n",
    "    \"\"\"\n",
    "    simulate coin flipping\n",
    "    \"\"\"\n",
    "    return fp.rvs(n) # flip it n times\n",
    "\n",
    "xs = sample(100) # generate some samples"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "## Find maximum of Bernoulli distribution\n",
    "Single experiment\n",
    "$$\\phi(x) = p ^ {x} * (1 - p) ^ { 1 - x }$$\n",
    "Series of experiments\n",
    "$$\\mathcal{L}(p|x) = \\prod_{i=1}^{n} p^{x_{i}}*(p-1)^{1-x_{i}}$$\n",
    "### Hints\n",
    "- [sympy.diff()](http://docs.sympy.org/dev/modules/core.html#sympy.core.function.diff)\n",
    "- [sympy.expand()](http://docs.sympy.org/dev/modules/core.html#sympy.core.function.expand)\n",
    "- [sympy.expand_log()](http://docs.sympy.org/dev/modules/core.html#sympy.core.function.expand_log)\n",
    "- [sympy.solve()](http://docs.sympy.org/dev/modules/core.html#sympy.core.function.solve)\n",
    "- [sympy.symbols()](http://docs.sympy.org/dev/modules/core.html#symbols)\n",
    "- [sympy gotchas](http://docs.sympy.org/dev/tutorial/gotchas.html)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import sympy\n",
    "from sympy.abc import x\n",
    "\n",
    "p = sympy.symbols('p', positive=True)\n",
    "phi = p ** x * (1 - p) ** (1 - x)\n",
    "L = np.prod([phi.subs(x, i) for i in xs]) # objective function to maximize\n",
    "log_L = sympy.expand_log(sympy.log(L))\n",
    "sol = sympy.solve(sympy.diff(log_L, p), p)[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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lH2w6xItLq45vKwL5WckMzUknqbqGo5m7Gdk7gwHZacRaU5wJkiUZ0yUcKK3ik51HWL7z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\nkyWJstcAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f33e68a2390>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "\n",
    "x_space = np.linspace(1/100, 1, 100, endpoint=False)\n",
    "\n",
    "plt.plot(x_space,\n",
    "         list(map(sympy.lambdify(p, log_L, 'numpy'), x_space)),\n",
    "         sol,\n",
    "         log_L.subs(p, sol),\n",
    "         'o',\n",
    "         p_true,\n",
    "         log_L.subs(p, p_true),\n",
    "         's',\n",
    "        )\n",
    "plt.xlabel('$p$', fontsize=18)\n",
    "plt.ylabel('Likelihood', fontsize=18)\n",
    "plt.title('Estimate not equal to true value', fontsize=18)\n",
    "plt.grid(True)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Empirically examine the behavior of the maximum likelihood estimator \n",
    "- [evalf()](http://docs.sympy.org/dev/modules/core.html#module-sympy.core.evalf)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f33e868a4e0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def estimator_gen(niter=10, ns=100):\n",
    "    \"\"\"\n",
    "    generate data to estimate distribution of maximum likelihood estimator'\n",
    "    \"\"\"\n",
    "    x = sympy.symbols('x', real=True)\n",
    "    phi = p**x*(1-p)**(1-x)\n",
    "    for i in range(niter):\n",
    "        xs = sample(ns) # generate some samples from the experiment\n",
    "        L = np.prod([phi.subs(x,i) for i in xs]) # objective function to maximize\n",
    "        log_L = sympy.expand_log(sympy.log(L)) \n",
    "        sol = sympy.solve(sympy.diff(log_L, p), p)[0]\n",
    "        yield float(sol.evalf())\n",
    "    \n",
    "entries = list(estimator_gen(100)) # this may take awhile, depending on how much data you want to generate\n",
    "plt.hist(entries) # histogram of maximum likelihood estimator\n",
    "plt.title('$\\mu={:3.3f},\\sigma={:3.3f}$'.format(np.mean(entries), np.std(entries)), fontsize=18)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Dynamic of MLE by length sample sequence"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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N7YnANeEbuHve4e/N7I/A8+7+jJmlNLSviIi0rAaD392rzOwWYB6QDMxw9xVm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\nYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f33e7459828>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def estimator_dynamics(ns_space, num_tries = 20):\n",
    "    for ns in ns_space:\n",
    "        estimations = list(estimator_gen(num_tries, ns))\n",
    "        yield np.mean(estimations), np.std(estimations)\n",
    "    \n",
    "ns_space = list(range(10, 100, 5))\n",
    "entries = list(estimator_dynamics(ns_space))\n",
    "entries_mean = list(map(lambda e: e[0], entries))\n",
    "entries_std = list(map(lambda e: e[1], entries))\n",
    "\n",
    "plt.errorbar(ns_space, entries_mean, entries_std, fmt='-o')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "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": "3.6.0"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}