\tikzstyle{hidden_neuron}=[ellipse,draw=blue!50,fill=cyan!10,thick,minimum size=3mm]
\begin{frame}
	\myheading{Module 17.1: Why are we interested in Joint Distributions}
\end{frame}
\begin{frame}
	\begin{columns}
		\column{0.5\textwidth}
		\begin{overlayarea}{\textwidth}{\textheight}
			\vspace{0.2in}
			\begin{center}
				\onslide<3->{\begin{tikzpicture}[thick,scale=0.8, every node/.style={scale=0.8}]
									\node[] at (0,6) {$Y$};
								    \node[hidden_neuron] at (0,5.4) {Oil};
								    \node[hidden_neuron] at (-3,4) {Salinity};
								    \node[] at (-3,4.6) {$X_{1}$};
								    \node[hidden_neuron] at (0,4) {Pressure};
								    \node[] at (0,4.6) {$X_{2}$};
								    \node[hidden_neuron] at (3,4) {Depth};
								    \node[] at (3,4.6) {$X_{3}$};
								    \node[hidden_neuron] at (2,2.5) {Biodiversity};
								    \node[] at (-2,3.1) {$X_{4}$};
								    \node[hidden_neuron] at (-2,2.5) {Temperature};
								    \node[] at (2,3.1) {$X_{5}$};
								    \node[hidden_neuron] at (0,1) {Density};
								    \node[] at (0,1.6) {$X_{6}$};
								\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 In many real world applications, we have to deal with a large number of random variables}
				\vspace{0.1in}
				\onslide<2->{\item For example, an oil company may be interested in computing the probability of finding oil at a particular location}
				\vspace{0.1in}
				\onslide<3->{\item This may depend on various (random) variables}
				\vspace{0.1in}
				\onslide<4->{\item The company is interested in knowing 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{center}
				\begin{tikzpicture}[thick,scale=0.8, every node/.style={scale=0.8}]
					\node[] at (0,6) {$Y$};
				    \node[hidden_neuron] at (0,5.4) {Oil};
				    \node[hidden_neuron] at (-3,4) {Salinity};
				    \node[] at (-3,4.6) {$X_{1}$};
				    \node[hidden_neuron] at (0,4) {Pressure};
				    \node[] at (0,4.6) {$X_{2}$};
				    \node[hidden_neuron] at (3,4) {Depth};
				    \node[] at (3,4.6) {$X_{3}$};
				    \node[hidden_neuron] at (2,2.5) {Biodiversity};
				    \node[] at (-2,3.1) {$X_{4}$};
				    \node[hidden_neuron] at (-2,2.5) {Temperature};
				    \node[] at (2,3.1) {$X_{5}$};
				    \node[hidden_neuron] at (0,1) {Density};
				    \node[] at (0,1.6) {$X_{6}$};
				\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 But why joint distribution?}
				\vspace{0.1in}
				\onslide<2->{\item Aren't we just interested in $P(Y|X_{1},X_{2},...,X_{n})$?}
				\vspace{0.1in}
				\onslide<3->{\item Well, if we know the joint distribution, we can find answers to a bunch of interesting questions}
				\vspace{0.1in}
				\onslide<4->{\item Let us see some such questions of interest}
			\end{itemize}
		\end{overlayarea}
	\end{columns}
\end{frame}

\begin{frame}
	\begin{columns}
		\column{0.4\textwidth}
		\begin{overlayarea}{\textwidth}{\textheight}
			\vspace{0.2in}
			\begin{center}
				\begin{tikzpicture}[thick,scale=0.8, every node/.style={scale=0.8}]
					\node[] at (0,6) {$Y$};
				    \node[hidden_neuron] at (0,5.4) {Oil};
				    \node[hidden_neuron] at (-3,4) {Salinity};
				    \node[] at (-3,4.6) {$X_{1}$};
				    \node[hidden_neuron] at (0,4) {Pressure};
				    \node[] at (0,4.6) {$X_{2}$};
				    \node[hidden_neuron] at (3,4) {Depth};
				    \node[] at (3,4.6) {$X_{3}$};
				    \node[hidden_neuron] at (2,2.5) {Biodiversity};
				    \node[] at (-2,3.1) {$X_{4}$};
				    \node[hidden_neuron] at (-2,2.5) {Temperature};
				    \node[] at (2,3.1) {$X_{5}$};
				    \node[hidden_neuron] at (0,1) {Density};
				    \node[] at (0,1.6) {$X_{6}$};
				\end{tikzpicture}
				\[P(Y, X_1,X_2,X_3,X_4,X_5,X_6)\]
			\end{center}
		\end{overlayarea}
		\column{0.6\textwidth}
		\begin{overlayarea}{\textwidth}{\textheight}
			\begin{itemize}\justifying
				\onslide<1->{\item We can find the conditional distribution
								\begin{align*}
									P(Y|X_{1},...,X_{n}) = \frac{P(Y,X_{1},...,X_{n})}{\sum_{X_{1},...,X_{n}}P(Y,X_{1},...,X_{n})}
								\end{align*} 
								}
				\onslide<2->{\item We can find the marginal distribution, 
									\begin{align*}
										P(Y) = \sum_{X_{1},...,X_{n}}P(Y,X_{1},X_{2},...,X_{n})
									\end{align*}}
				\onslide<3->{\item We can find the conditional independencies, 
									\begin{align*}
										P(Y,X_{1})=P(Y)P(X_{1})
									\end{align*}}
			\end{itemize}
		\end{overlayarea}
	\end{columns}
\end{frame}