mle-of-uniform-distribution.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "from scipy.optimize import differential_evolution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.5590952009638621 1.4469720006899172\n",
      "0.5029192798674605 1.4866578874813645\n",
      "0.5002904764114741 1.4994683757767997\n",
      "0.5000709162887501 1.4999759888186115\n"
     ]
    }
   ],
   "source": [
    "a = 0.5\n",
    "b = 1.5\n",
    "n = 10000\n",
    "\n",
    "for n in [10, 100, 1000, 10000]:\n",
    "    x = np.random.uniform(a, b, size=n)\n",
    "    print(x.min(), x.max())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(x, density=True)\n",
    "plt.xlim([0, 2])\n",
    "plt.xlabel(\"$x$\")\n",
    "plt.ylabel(\"$p(x)$\");"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "def log_likelihood(theta_hat):\n",
    "    a_hat, b_hat = theta_hat\n",
    "    return -n * np.log(b_hat - a_hat)\n",
    "\n",
    "def negative_log_likelihood(theta_hat):\n",
    "    return -log_likelihood(theta_hat)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "     fun: -0.9537197940327512\n",
       " message: 'Optimization terminated successfully.'\n",
       "    nfev: 4233\n",
       "     nit: 140\n",
       " success: True\n",
       "       x: array([0.50007057, 1.4999752 ])"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bounds = [[-1e10, x.min()], [x.max(), 1e10]]\n",
    "differential_evolution(negative_log_likelihood, bounds=bounds)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "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": "3.6.5"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}