update

Author: tom mcandrew

outbreak_science/Computational_posterior.ipynb

@@ -1,5 +1,78 @@
 {
  "cells": [
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "376d3c08-3388-4a73-98d1-0dd20b1d16de",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "de95d2eb-303e-4105-8d15-43fd46c80242",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  },
+  {
+   "cell_type": "markdown",
+   "id": "40f1ba9e-f4e2-4892-8ea8-448a481f1b10",
+   "metadata": {},
+   "source": [
+    "\n",
+    "\\begin{align}\n",
+    "    \\beta &\\sim \\text{Beta}(2,5) ; \\;  \n",
+    "    \\gamma \\sim \\text{Beta}(2,8) \\\\ \n",
+    "    S_{0}&=990 ; I_{0} = 10; R_{0} = 0\\\\\n",
+    "    S,I,R,\\Delta I &= ODE( [S_{0},I_{0}, R_{0}; \\beta, \\gamma]  ) \\\\ \n",
+    "    (\\Delta i)_{t} &\\sim \\text{Pois}( \\Delta I_{t} )\n",
+    "\\end{align}\n",
+    "\n",
+    "\\begin{align}\n",
+    "    p( \\beta, \\gamma | \\mathcal{D} ) = \\frac{p(\\mathcal{D} |\\beta, \\gamma ) p(\\beta, \\gamma)}{ \\int_{\\beta=0}^{\\infty} \\int_{\\gamma=0}^{\\infty} p(\\mathcal{D} |\\beta, \\gamma ) p(\\beta, \\gamma) \\; d \\gamma d \\beta } \\\\ \n",
+    "\\end{align}\n",
+    "\n",
+    "\\begin{align}\n",
+    "    p(\\mathcal{D} |\\beta, \\gamma ) = \\text{Pois}( \\Delta I_{1} )[(\\Delta i)_{1}] \\times \\text{Pois}( \\Delta I_{2} )[(\\Delta i)_{2}] \\times \\cdots \\times  \\text{Pois}( \\Delta I_{T} )[(\\Delta i)_{T}] = \\prod_{t=1}^{T} \\text{Pois}( \\Delta I_{t} )[(\\Delta i)_{t}] \\\\ \n",
+    "\\end{align}\n",
+    "\n",
+    "\\begin{align}\n",
+    "    p(\\beta, \\gamma) &= p(\\beta) \\times p(\\gamma) \\\\ \n",
+    "    &= \\frac{1}{\\text{Beta}(2,5)} \\beta^{2-1} (1-\\beta)^{5-1} \\times  \\frac{1}{\\text{Beta}(2,8)} \\gamma^{2-1} (1-\\gamma)^{8-1}\n",
+    "\\end{align}\n",
+    "\n",
+    "\n",
+    "\\begin{align}\n",
+    "    p(\\mathcal{D} |\\beta, \\gamma )  &= \\prod_{t=1}^{T} \\text{Pois}( \\Delta I_{t} )[(\\Delta i)_{t}] \\times \\left[ \\frac{1}{\\text{Beta}(2,5)} \\beta^{2-1} (1-\\beta)^{5-1} \\times  \\frac{1}{\\text{Beta}(2,8)} \\gamma^{2-1} (1-\\gamma)^{8-1} \\right]\n",
+    "\\end{align}"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "0cfa96b5-aab0-410b-a068-8b60f5c3b2df",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "155442de-a1c4-446b-9ae1-9f38163972ed",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "52c8170a-5117-48e8-a7cc-5f0ea303ad9c",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  },
   {
    "cell_type": "code",
    "execution_count": 11,