Consider the data-association problem described in box 12.D: We have two sets of objects U = {u 1…

Consider the data-association problem described in box 12.D: We have two sets of objects U = {u1, . . . , uk} and another V = {v1, . . . , vm}, and we wish to map U’s to V’s. We have a set of observed features Bi for each object ui, and a set of hidden attributes Aj for each vj. We have a prior P(Aj), and a set of factors φi(Aj, Bi, Ci) such that φi(aj, bi, Ci) = 1 for all aj, bi if Ci ≠ j. The model contains no other potentials. We wish to compute the posterior over Aj using collapsed Gibbs sampling, where we sample the Ci’s but maintain a closed-form posterior over the Aj ’s. Provide a sampling scheme for this task, showing clearly both the sampling distribution for the Ci variables and the computation of the closed form over the Ai variables given the assignment to the Ci’s.