{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Estimador de máxima verossimilhança da distribuição geométrica\n",
    "Neste notebook do Jupyter vamos demonstrar a dedução matemática do estimador de máxima verossimilhança do parâmetro da distribuição geométrica. Para ver os resultados dos comandos de cada etapa coloque o mouse ao lado esquerdo da célula que tem os comandos e clique em 'Run this cell'\n",
    "## Preparação\n",
    "Antes de começar importamos a biblioteca de matemática simbólica do Python. Ela permite criar expressões, como funções e suas derivadas, e manipulá-las simbolicamente."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import *\n",
    "init_printing()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## A função de densidade probabilística\n",
    "Para uma dada observação $x_i$ a função de massa da distribuição geométrica é"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle p \\left(1 - p\\right)^{{x}_{i}}$"
      ],
      "text/plain": [
       "         x[i]\n",
       "p⋅(1 - p)    "
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x, p, i, n = symbols('x p i n')\n",
    "f = p*(1-p)**(Indexed(x,i))\n",
    "f"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Defina a função de log-verossimilhança negativa\n",
    "É a soma dos logarítimos dos valores da função acima sobre todos os valores dos dados ($x_i$). Primeiro calculamos o logarítmo da função de massa:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\log{\\left(p \\right)} + \\log{\\left(1 - p \\right)} {x}_{i}$"
      ],
      "text/plain": [
       "log(p) + log(1 - p)⋅x[i]"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "lf = ln(f)\n",
    "lf=expand_log(lf, force=true)\n",
    "lf"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "e em seguida definimos a sua soma sobre todos os valores de $x_i$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\sum_{i=1}^{n} \\left(\\log{\\left(p \\right)} + \\log{\\left(1 - p \\right)} {x}_{i}\\right)$"
      ],
      "text/plain": [
       "  n                             \n",
       " ___                            \n",
       " ╲                              \n",
       "  ╲                             \n",
       "  ╱   (log(p) + log(1 - p)⋅x[i])\n",
       " ╱                              \n",
       " ‾‾‾                            \n",
       "i = 1                           "
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "LL = Sum(lf, (i,1,n))\n",
    "LL"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Estimativa de máxima verossimilhança (mle)\n",
    "É o valor do parâmetro $p$ que maximiza a função de log-verossimilhança (ou minimiza a de log-verossimilhança negativa). Sabemos que as funções têm derivadas zero em seus pontos máximos e mínimos.\n",
    "Então o truque para achar a expressão do mle é encontrar a expressão para o valor de $p$\n",
    "que faz a derivada da função de log-verossimilhança ser igual a zero. São duas etapas:\n",
    "\n",
    "### Calcule a derivada da função de log-verossimlhança em relação ao parâmetro\n",
    "Aqui usamos o comando 'diff' para obter a derivada da log-verossimilhança em função do parâmetro $p$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{n}{p} + \\frac{\\sum_{i=1}^{n} {x}_{i}}{p - 1}$"
      ],
      "text/plain": [
       "      n       \n",
       "     ___      \n",
       "     ╲        \n",
       "      ╲       \n",
       "      ╱   x[i]\n",
       "     ╱        \n",
       "     ‾‾‾      \n",
       "n   i = 1     \n",
       "─ + ──────────\n",
       "p     p - 1   "
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dLL = diff(LL,p)\n",
    "dLL=simplify(expand(dLL))\n",
    "dLL"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Resolvemos a derivada para o parâmetro\n",
    "Agora igualamos a derivada da função de log-verossimilhança a zero e resolvemos esta equação para o parãmetro $p$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ \\frac{n}{n + \\sum_{i=1}^{n} {x}_{i}}\\right]$"
      ],
      "text/plain": [
       "⎡      n       ⎤\n",
       "⎢──────────────⎥\n",
       "⎢      n       ⎥\n",
       "⎢     ___      ⎥\n",
       "⎢     ╲        ⎥\n",
       "⎢      ╲       ⎥\n",
       "⎢n +   ╱   x[i]⎥\n",
       "⎢     ╱        ⎥\n",
       "⎢     ‾‾‾      ⎥\n",
       "⎣    i = 1     ⎦"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve(dLL,p)"
   ]
  }
 ],
 "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": "3.10.8"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}