On inference concerning binary latent trait

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Let Z1 and Z2 be observable random variables. Assume they depend on latent trait U and are conditionally independent, given U. 1) How, and to what extent, the joint distribution of (U, Z1, Z2) can be recovered from that of (Z1,Z2)? 2) Suppose that, knowing Z1 and/or Z2, we are to make decision concerning U. What decision rule is the best? Both the problems are properly formalized and solved in the simple case of binary U.