Complete Bayes unit report pre-advising meeting

report/report.tex

@@ -267,7 +267,7 @@ number of documented occurrences is frequently used in philosophical discussion
 are valid classifications even when the subset size results in overall fufilled terms to be infrequently categorized as the proposed subset.  Most people understand 
 these expressions but, when shown a table and how to calculate those ratios, the content enters the realm of collegiate instruction.
 
-\subsubsection{Bayes Theorem}
+\subsubsection{Bayes Theorem}\label{Bayes Theorem}
 
 The equation for Bayes Theorem is as follows:
 
@@ -334,6 +334,68 @@ applications of Bayes Theory calculate posterior probabilities continuously as n
 Bayesian Belief Networks are probablistic graphical models that preserve conditional dependence between random variables.  In spite of its name, 
 Bayesian Belief Networks do not necessarily apply Bayesian models, though they are a way to utilize Bayes Theorem for domains with greater complexity beyond a 
 single posterior probability.  In this type of network, edges are directed and the structure is utilized in a single direction.  This is in contrast to undirected
-Hidden Markov Models that do not assume the order of aquisition of random variables.
+Hidden Markov Models (to be covered in the next unit) that do not assume the order of aquisition of random variables.  While it may not be practical to calculate 
+the full conditional probability of a variable, Bayesian Belief Networks allow us to identify conditionally dependent variables that are weighted on the basis of 
+an earlier random variable.
+
+Following the example in the Bayes Theorem section of this report (\ref{Bayes Theorem}), let's suppose that a patient with a positive test takes a hypothetical 
+second test whose results are partially dependent on the first as they measure overlapping biological markers.  In this case, the results of the first test 
+is relevant to the second test:
+\vskip 5pt
+\begin{center}
+    \begin{tikzpicture}
+        \draw[black, thick] (-2, 4.5) rectangle (2, 5.5);
+        \node at (0, 5) (bio) {Biological Markers}; 
+
+        \draw[black, thick] (-1.5, 3) circle (0.75);
+        \node at (-1.5, 3) (T1) {Test 1};
+    
+        \draw[black, thick] (1.5, 3) circle (0.75);
+        \node at (1.5, 3) (T2) {Test 2}; 
+    
+        \draw[black, thick] (-2, 0) rectangle (2, 1);
+        \node at (0, 0.5) (DepRes) {Dependent Results}; 
+    
+        % Draw arrows from the bottom of the circles to the top of the rectangle
+        \draw[->] (T1.south) -- (DepRes.north);
+        \draw[->] (T2.south) -- (DepRes.north); 
+        \draw[->] (bio.south) -- (T1.north);
+        \draw[->] (bio.south) -- (T2.north);
+    \end{tikzpicture}
+\end{center}
+
+\vskip 5pt
+\begin{center}
+    \vskip 5pt
+    \begin{tabular}{| c | c | c |}
+        \hline
+        Test 1 Result & Test 2 Result & P(A) \\
+        \hline\hline
+        \multicolumn{3}{| c |}{Prior beliefs of test 1} \\
+        \hline
+        Unknown & Unknown & 10\% \\
+        Positive & Unknown & 67.857\% \\
+        Negative & Unknown & 0.581\% \\
+        \hline
+        \multicolumn{3}{| c |}{Prior beliefs of test 2} \\
+        \hline
+        Unknown & Positive & 55\% \\
+        Unknown & Negative & 1\% \\
+        \hline
+        \multicolumn{3}{| c |}{Dependent results from both tests} \\
+        \hline
+        Positive & Positive & 75\% \\
+        Positive & Negative & 1.5\% \\
+        Negative & Positive & 0.6\% \\
+        Negative & Negative & 0.087\% \\
+        \hline    
+    \end{tabular}
+\end{center}
+Note that this probability of positive results in both tests (which both have greater than 50\% of positives being true positives) is as certain as two positives 
+from two completely independent tests with 50\% of positives being true.  If the partial dependence was not included in the calculation, as would have occured in a 
+Naive Bayes model, the model's listed accuracy would be inflated.
+
+\newpage
+\section{Unit 4: Markov Chains}
 
 \end{document}