Exercise 14.24 [soccer-rpm-exercise]
Three soccer teams $A$, $B$, and $C$, play each other once. Each match is between two teams, and can be won, drawn, or lost. Each team has a fixed, unknown degree of quality—an integer ranging from 0 to 3—and the outcome of a match depends probabilistically on the difference in quality between the two teams.
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Construct a relational probability model to describe this domain, and suggest numerical values for all the necessary probability distributions.
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Construct the equivalent Bayesian network for the three matches.
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Suppose that in the first two matches $A$ beats $B$ and draws with $C$. Using an exact inference algorithm of your choice, compute the posterior distribution for the outcome of the third match.
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Suppose there are $n$ teams in the league and we have the results for all but the last match. How does the complexity of predicting the last game vary with $n$?
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Investigate the application of MCMC to this problem. How quickly does it converge in practice and how well does it scale?