Segment 39. MCMC and Gibbs Sampling

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To Calculate

1. Suppose the domain of a model are the five integers $1,2,3,4,5\$ , and that your proposal distribution is: "When Failed to parse (unknown error): x_1 = 2,3,4 , choose with equal probability Failed to parse (unknown error): x_2 = x_1 \pm 1 . For Failed to parse (unknown error): x_1=1 always choose Failed to parse (unknown error): x_2 =2 . For Failed to parse (unknown error): x_1=5 always choose Failed to parse (unknown error): x_2 =4 . What is the ratio of Failed to parse (unknown error): q 's that goes into the acceptance probability Failed to parse (unknown error): \alpha(x_1,x_2) for all the possible values of Failed to parse (unknown error): x_1 and Failed to parse (unknown error): x_2 ?

2. Suppose the domain of a model is Failed to parse (unknown error): -\infty < x < \infty and your proposal distribution is (perversely),

$cases}\tfrac{7}{2}\exp[-7(x_2-x_1)],\quad & x_2 \ge x_1 \\ \tfrac{5}{2}\exp[-5(x_1-x_2)],\quad & x_2 < x_1 \end{cases$

Sketch this distribution as a function of Failed to parse (unknown error): x_2-x_1 . Then, write down an expression for the ratio of Failed to parse (unknown error): q 's that goes into the acceptance probability Failed to parse (unknown error): \alpha(x_1,x_2) .

1. Suppose an urn contains 7 large orange balls, 3 medium purple balls, and 5 small green balls. When balls are drawn randomly, the larger ones are more likely to be drawn, in the proportions large:medium:small = 6:4:3. You want to draw exactly 6 balls, one at a time without replacement. How would you use Gibbs sampling to learn: (a) How often do you get 4 orange plus 2 of the same (non-orange) color? (b) What is the expectation (mean) of the product of the number of purple and number of green balls drawn?

2. How would you do the same problem computationally but without Gibbs sampling?

3. How would you do the same problem non-stochastically (e.g., obtain answers to 12 significant figures)? (Hint: This is known as the Wallenius non-central hypergeometric distribution.)