Artificial IntelligenceAIMA Exercises

Consider the query ${\textbf{P}}({Rain}{Sprinkler}{true},{WetGrass}{true})$ in Figure rain-clustering-figure(a) (page rain-clustering-figure) and how Gibbs sampling can answer it.
1. How many states does the Markov chain have?
2. Calculate the transition matrix ${\textbf{Q}}$ containing $q({\textbf{y}}$ $\rightarrow$ ${\textbf{y}}')$ for all ${\textbf{y}}$, ${\textbf{y}}'$.
3. What does ${\textbf{ Q}}^2$, the square of the transition matrix, represent?
4. What about ${\textbf{Q}}^n$ as $n\to \infty$?
5. Explain how to do probabilistic inference in Bayesian networks, assuming that ${\textbf{Q}}^n$ is available. Is this a practical way to do inference?

Consider the query ${\textbf{P}}({Rain}{Sprinkler}{true},{WetGrass}{true})$ in Figure rain-clustering-figure(a) (page rain-clustering-figure) and how Gibbs sampling can answer it.
1. How many states does the Markov chain have?
2. Calculate the transition matrix ${\textbf{Q}}$ containing $q({\textbf{y}}$ $\rightarrow$ ${\textbf{y}}')$ for all ${\textbf{y}}$, ${\textbf{y}}'$.
3. What does ${\textbf{ Q}}^2$, the square of the transition matrix, represent?
4. What about ${\textbf{Q}}^n$ as $n\to \infty$?
5. Explain how to do probabilistic inference in Bayesian networks, assuming that ${\textbf{Q}}^n$ is available. Is this a practical way to do inference?

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