added stats/Fisher-information-derivation.ipynb

zyxue · Apr 13, 2022 · da33881 · da33881
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commit da33881
Showing 1 changed file with 120 additions and 0 deletions.
diff --git a/stats/Fisher-information-derivation.ipynb b/stats/Fisher-information-derivation.ipynb
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+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Rederives https://en.wikipedia.org/wiki/Fisher_information#Definition."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "given $f(X; \\theta)$ is the maximum likelihood of $\\theta$ based on data $X$,"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "\\begin{align*}\n",
+    "\\frac{\\partial}{\\partial \\theta} \\log f(X; \\theta) \n",
+    "&= \\frac{\\frac{\\partial}{\\partial \\theta} f(X; \\theta)}{ f(X;\\theta)} \\\\\n",
+    "\\frac{\\partial^2}{\\partial \\theta^2} \\log f(X; \\theta) \n",
+    "&= \\frac{\\left( \\frac{\\partial^2}{\\partial \\theta^2} f(X; \\theta) \\right ) f(X;\\theta) - \\frac{\\partial}{\\partial \\theta} f(X; \\theta) \\frac{\\partial}{\\partial \\theta} f(X ;\\theta) }{ f(X;\\theta) ^2} \\\\\n",
+    "&= \\frac{\\frac{\\partial^2}{\\partial \\theta^2} f(X; \\theta)}{ f(X;\\theta)} - \\left( \\frac{\\frac{\\partial} {\\partial \\theta} f(X; \\theta)}{f(X ;\\theta)} \\right )^2 \\\\\n",
+    "&= \\frac{\\frac{\\partial^2}{\\partial \\theta^2} f(X; \\theta)}{ f(X;\\theta)} - \\left( \\frac{\\partial}{\\partial \\theta} \\log f(X; \\theta) \\right )^2\n",
+    "\\end{align*}"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "The expectation of the first part is equal to zero"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "\\begin{align*}\n",
+    "\\mathbb{E} \\left[ \\frac{\\frac{\\partial^2}{\\partial \\theta^2} f(X; \\theta)}{ f(X;\\theta)} \\bigg| \\theta \\right ]\n",
+    "&= \\int_{X} \\frac{\\frac{\\partial^2}{\\partial \\theta^2} f(X; \\theta)}{ f(X;\\theta)} f(X; \\theta) dX \\\\\n",
+    "&= \\int_{X} \\frac{\\partial^2}{\\partial \\theta^2} f(X; \\theta) dX \\\\\n",
+    "&= \\frac{\\partial^2}{\\partial \\theta^2} \\int_X f(X;\\theta) dX \\\\\n",
+    "&= \\frac{\\partial^2}{\\partial \\theta^2} 1 \\\\\n",
+    "&= 0\n",
+    "\\end{align*}"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "So"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "\\begin{align*}\n",
+    "\\mathbb{E}\\left[ \\frac{\\partial}{\\partial \\theta} \\log L(X; \\theta) \\bigg | \\theta \\right ]\n",
+    "&= - \\mathbb{E}\\left[ \\left( \\frac{\\partial}{\\partial \\theta} \\log f(X; \\theta) \\right )^2 \\bigg | \\theta \\right]\n",
+    "\\end{align*}"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "which is the negative of variance of the score $\\frac{\\partial}{\\partial \\theta} \\log f(X; \\theta)$ since the expectation of the score is also 0:"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "\\begin{align*}\n",
+    "\\mathbb{E}\\left[ \\frac{\\partial}{\\partial \\theta} \\log f(X; \\theta) | \\theta \\right] \n",
+    "&= \\mathbb{E}\\left[ \\frac{ \\frac{\\partial}{\\partial \\theta}  f(X; \\theta)}{f(X;\\theta)} \\bigg| \\theta \\right] \n",
+    "&= \\int_X \\frac{ \\frac{\\partial}{\\partial \\theta}  f(X; \\theta)}{f(X;\\theta)} f(X;\\theta) dX \\\\\n",
+    "&= \\int_X \\frac{\\partial}{\\partial \\theta}  f(X; \\theta) dX \\\\\n",
+    "&= \\frac{\\partial}{\\partial \\theta} \\int_X f(X; \\theta) dX \\\\\n",
+    "&= \\frac{\\partial}{\\partial \\theta} 1 \\\\\n",
+    "&= 0\n",
+    "\\end{align*}"
+   ]
+  },
+  {
+   "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": "3.8.6"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 4
+}