Exercise 14.21 · AIMA 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?

Exercise 14.21

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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