Beta_Model_Python.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "219f3866",
   "metadata": {},
   "source": [
    "# Method of Moments and Maximum Likelihood estimation\n",
    "\n",
    "### The example of a Beta model"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "68b5aec2",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0.03613599 0.61631475 0.00599853 ... 0.23542418 0.00548306 0.82538336]\n",
      "[3.02722947]\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "C:\\Users\\Public\\Documents\\Wondershare\\CreatorTemp\\ipykernel_17896\\627348098.py:77: RuntimeWarning: divide by zero encountered in log\n",
      "  loglikelihood = np.log(L)- (L * xi)\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 850x400 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "np.random.seed(2023)\n",
    "\n",
    "from scipy.stats import beta\n",
    "# one realization of a Beta(1/theta, 1) with theta = 3\n",
    "theta = 3\n",
    "beta.rvs(a = 1/theta, size = 1, b = 1)\n",
    "# array([0.03613599])\n",
    "\n",
    "import statistics\n",
    "# on average, we should get one.\n",
    "statistics.mean(beta.rvs(a = 1/theta, size = 100000, b = 1))\n",
    "# 0.24922457399140857\n",
    "\n",
    "# Method of Moments estimator\n",
    "def MoMBeta(x):\n",
    "  \n",
    "  n = len(x)\n",
    "  sample_moment = np.sum(x) / n\n",
    "  \n",
    "  theta_mom = (1 / sample_moment) -1\n",
    "  alpha_mom = 1 / theta_mom\n",
    "  \n",
    "  return theta_mom, alpha_mom\n",
    "\n",
    "# generate artificial data, sample of size 10,000\n",
    "np.random.seed(2023)\n",
    "x = beta.rvs(a = 1/3, size = 100000, b = 1)\n",
    "\n",
    "# apply MoMgamma()\n",
    "MoMBeta(x = x)\n",
    "# (2.9968621974937864, 0.3336823430974835)\n",
    "\n",
    "#---------------------------------\n",
    "# 2. Maximum Likelihood estimation\n",
    "#---------------------------------\n",
    "\n",
    "# generate a sample of size 5000 from a Beta distribution\n",
    "np.random.seed(2023)\n",
    "n = 5000\n",
    "theta = 3 # true value of the parameter, that we wish to estimate\n",
    "xi = beta.rvs(a = 1/theta, size = 5000, b = 1)\n",
    "print(xi)\n",
    "# [0.03613599 0.61631475 0.00599853 ... 0.23542418 0.00548306 0.82538336]\n",
    "\n",
    "# Closed-form MLE\n",
    "theta_hat_formula = 1 / ( - (n / np.sum(np.log(xi))) )  \n",
    "theta_hat_formula \n",
    "# [1] 3.0272294817747585\n",
    "\n",
    "#T numerical optimization\n",
    "from scipy import stats\n",
    "\n",
    "def llikelihood(Theta):  \n",
    "    # log-likelihood function\n",
    "    ll = -np.sum(stats.beta.logpdf(xi, a = 1/Theta, b = 1))\n",
    "    return ll\n",
    "\n",
    "from scipy.optimize import minimize\n",
    "\n",
    "# Numerical approximation of the MLE using minimize()\n",
    "mle = minimize(llikelihood, \n",
    "                   x0 = 4, \n",
    "                   method = 'BFGS')\n",
    "print(mle.x)\n",
    "# [3.02722947]\n",
    "\n",
    "# Plot the Log-Likelihood function\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "# set a range of possible parameter values\n",
    "possibletheta = np.linspace(start = 0, stop = 10, num = 5000)\n",
    "\n",
    "# compute the log-likelihood function for all possible parameter values\n",
    "LL = []\n",
    "for L in possibletheta:\n",
    "    loglikelihood = np.log(L)- (L * xi)\n",
    "    LL.append(loglikelihood.sum())\n",
    "\n",
    "# plotting using matplotlib\n",
    "px = 1/plt.rcParams['figure.dpi'] \n",
    "plt.figure(figsize=(850*px, 400*px))\n",
    "plt.plot(possibletheta, LL, color = 'black')\n",
    "plt.axvline(x = mle.x,  color = 'red')\n",
    "plt.ylim([-30000, 6000])\n",
    "plt.xlabel('lambda')\n",
    "plt.ylabel('Log-Likelihood')\n",
    "plt.title('Log-Likelihood (function of theta)', fontsize = 15, y = 1.1, loc = 'left')\n",
    "plt.suptitle('Maximum is reached at theta = 3.02722947', y = 0.95, color = 'darkred', x = 0.283)\n",
    "caption = 'artificial dataset of size 5,000'\n",
    "plt.text(6.5,-38000, caption, fontsize = 8)\n",
    "plt.grid(color = 'whitesmoke', linestyle = '-', linewidth = 1.2)"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "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": "3.10.9"
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 },
 "nbformat": 4,
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}