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docs/SimpleBayesianNetwork.ipynb

@@ -0,0 +1,56 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Hoi"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "\n",
+    "$P(Q) = \\hat{Y}$\n",
+    "\n",
+    "```\n",
+    "\n",
+    "hoi \n",
+    "mooi man\n",
+    "```\n",
+    "\n",
+    "```mermaid\n",
+    "\n",
+    "graph TD\n",
+    "    Q((Q))\n",
+    "    Y((Y))\n",
+    "\n",
+    "    Q-->Y\n",
+    "    \n",
+    "```"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": ".venv",
+   "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": "3.10.7"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}

docs/index.md

@@ -1,6 +1,8 @@
+[TOC]
+
 # Original sum-product algorithm
 
-## Simple example Baysian network
+## Simple example Bayesian network
 
 First consider a simple Bayesian network, in which there is one hidden variable $Q$ and one observed variable $Y$.
 
@@ -78,7 +80,7 @@ f_1(q) = P(Q) \\
 f_2(q, y) = P(Y|Q)
 ```
 
-### Message definitions
+#### Message definitions
 
 ```math
 
@@ -88,8 +90,8 @@ d_3(y) & =
     P(\hat{Y}|Y)
     \qquad \qquad \qquad \qquad \qquad \qquad \qquad 
     & \begin{cases}
-        1 & \text{if } \hat{y} = y \\
-        0 & \text{if } \hat{y} \ne y \\
+        1 & \text{if } y = \hat{y} \\
+        0 & \text{if } y \ne \hat{y} \\
     \end{cases} \\
 a_3(y) & = 
     d_3(y)
@@ -115,4 +117,34 @@ b_1(q) & =
 
 \end{align}
 
+```
+
+#### Inference
+
+Since most messages depend on other message, a number of iterations is needed to calculate the final values for all messages (in this case 4 iterations: $d_3$→$a_3$→$b_2$→$a_2$→$b_3$).
+
+After that, the messages can be used to perform inference on the hidden variable $Q$:
+
+```math
+
+a_1(q)b_1(q) = P(\hat{Y}|Q)P(Q)
+
+```
+
+Normalizing yields the posterior distribution for $Q$:
+
+```math
+
+P(Q|\hat{Y}) = \frac{P(\hat{Y}|Q)P(Q)}{P(\hat{Y})} 
+     = \frac{P(\hat{Y}|Q)P(Q)}{\sum\limits_{q}{P(\hat{Y}|Q)P(Q)}} 
+    =  \frac{a_1(q)b_1(q)}{\sum\limits_{q}{a_1(q)b_1(q)}}
+
+```
+
+Furthermore, the messages can be used to calculate the conditional probability distributions between hidden and/or observed variables:
+
+```math
+
+
+
 ```