Let ? denote an I ? J matrix of cell probabilities for the joint distribution of X and Y. Suppose that there exist I ? 1 column vectors ?_1k and J ? 1 column vectors ?_2k of probabilities, k = 1,...,q, and a set of probabilities {?_k} such that

Explain why this implies that there is a latent variable Z such that X and Y are conditionally independent, given Z.

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Π=Σ ,πημπρκ. Σ Ριπιμπο Π k=1