\begin{problem}{Sampling}
Consider the following Bayes Net and corresponding probability tables.

\vspace{-0.5cm}

\begin{center}
 \includegraphics[width=0.7\linewidth]{figures/sampling_bn.png} 
\end{center}

Consider the case where we are sampling to approximate the query $P(R \mid +f,+m)$.

\begin{question}[12]
Fill in the following table with the probabilities of \emph{drawing} each respective sample given that we are
using each of the following sampling techniques.

\begin{center}
\begin{tabular}{| p{3.5cm} | p{5.5cm} | p{5.5cm}|}
    \hline
	\textbf{Method} & $<+r, +e, -w, +m, +f>$ &  $ <+r, -e, +w, -m, +f> $ \\ [3mm]
	\hline
	Prior sampling &\solution{}{\SixAPriorOne} & 
	\solution{}{\SixAPriorTwo}\\[17pt]
	\hline
	Likelihood weighting & \solution{}{\SixALikelihoodOne}& \solution{}{\SixALikelihoodTwo}\\[17pt]
	\hline
\end{tabular}
\end{center}

\end{question}

\begin{question}[4] 
\begin{center}
    \textit{''Sampling is important.''} -- Gibby Gibson, \textit{iCarly}, S2 Ep. 81
\end{center}

We are going to use Gibbs sampling to estimate the probability of getting the sample $<+r, +e, -w, +m, +f>$. We will start from the sample $<-r, -e, -w, +m, +f>$ and resample $E$ first then $R$. What is the probability of drawing sample $<+r, +e, -w, +m, +f>$?

\solutionspace{4.5cm}{17cm}{}{\Sixb}\\

\end{question}

\end{problem}
