\begin{frame}
	\myheading{Module 17.2: How do we represent a joint distribution}
\end{frame}

\begin{frame}
	\begin{columns}
		\column{0.5\textwidth}
		\begin{overlayarea}{\textwidth}{\textheight}
			\vspace{0.2in}
			\begin{center}
				\begin{tikzpicture}[thick,scale=0.9, every node/.style={scale=0.9}]
					\node[] at (0,6.3) {$Y$ (yes/no)};
				    \node[hidden_neuron] at (0,5.6) {Oil};
				    \node[hidden_neuron] at (-3,3.7) {Salinity};
				    \node[] at (-3,4.4) {$X_{1}$ (high/low)};
				    \node[hidden_neuron] at (0,3.7) {Pressure};
				    \node[] at (0,4.4) {$X_{2}$ (high/low)};
				    \node[hidden_neuron] at (3,3.7) {Depth};
				    \node[] at (3,4.4) {$X_{3}$ (deep/shallow)} ;
				    \node[hidden_neuron] at (2,2.2) {Biodiversity};
				    \node[] at (-2,2.9) {$X_{4}$ (high/low)} ;
				    \node[hidden_neuron] at (-2,2.2) {Temperature};
				    \node[] at (2,2.9) {$X_{5}$ (high/low)} ;
				    \node[hidden_neuron] at (0,0.5) {Density};
				    \node[] at (0,1.2) {$X_{6}$ (high/low)} ;
				\end{tikzpicture}
				\[P(Y, X_1,X_2,X_3,X_4,X_5,X_6)\]
			\end{center}
		\end{overlayarea}
		\column{0.5\textwidth}
		\begin{overlayarea}{\textwidth}{\textheight}
			\vspace{0.2in}
			\begin{itemize}\justifying
				\onslide<1->{\item Let us return to the case of $n$ random variables}
				\vspace{0.1in}
				\onslide<2->{\item For simplicity assume each of these variables can take binary values}
				\vspace{0.1in}
				\onslide<3->{\item To specify the joint distribution, we need to specify $2^{n} - 1$ values. Why not $(2^{n})$?}
				\vspace{0.1in}
				\onslide<4->{\item If we specify these $2^{n} - 1$ values, we have an explicit representation for the joint distribution}
			\end{itemize}
		\end{overlayarea}
	\end{columns}
\end{frame}

\begin{frame}
	\begin{columns}
		\column{0.5\textwidth}
		\begin{overlayarea}{\textwidth}{\textheight}
			\vspace{0.2in}
			\begin{table}[h!]
				\centering
				\begin{tabular}{|c c c c c c c|} 
				\hline
				$X_1$ & $X_2$ & $X_3$ & $X_4$ &... & $X_n$ & $P$ \\ 
				\hline\hline
				0 & 0 & 0 & 0 &... & 0& 0.01 \\ 
				1 & 0 & 0 & 0 &... & 0& 0.03 \\ 
				0 & 1 & 0 & 0 &... & 0& 0.05 \\ 
				1 & 1 & 0 & 0 &... & 0& 0.1 \\ 
				  &   &   &   &... &  &   \\
				  &   &   &   &... &  &  \\
				 &   &   &   &... &  &  \\
				1 & 1 & 1 & 1 &... & 1 & 0.002\\ 
				\hline
				\end{tabular}
			\end{table}
			(Once the first $2^n-1$ values are specified 
			the last value is deterministic as the values 
			need to sum to 1)
		\end{overlayarea}
		\column{0.5\textwidth}
		\begin{overlayarea}{\textwidth}{\textheight}
			\vspace{0.2in}
			\onslide<1->{\textbf{Challenges with explicit representation}}
			\begin{itemize}\justifying
				\onslide<2->{\item \textbf{Computational:} Expensive to manipulate and too large to to store}
				\vspace{0.1in}
				\onslide<3->{\item \textbf{Cognitive:} Impossible to acquire so many numbers from a human}
				\vspace{0.1in}
				\onslide<4->{\item \textbf{Statistical:} Need huge amounts of data to learn the parameters}
			\end{itemize}
		\end{overlayarea}
	\end{columns}
\end{frame}

