By Gian-Carlo Rota, Kenneth Baclawski

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In the finite discrete case it is particularly simple to describe. The long-run distribution of the I state chain with transition matrix BA′ will coincide with (p1. , p2. , . . , pI. ), the X marginal distribution of P . 2 , . . J ), the Y marginal of P . If we denote the long-run distributions corresponding to BA′ and A′ B by π = (π1 , π2 , . . , πI ) and η = (η1 , η2 , . . , ηJ ), respectively, then, provided A and B are compatible, we will have aij ηj = bij πi , i = 1, 2, . . , I, J = 1, 2, .

J. 83) will not all be equal. 85) ηA′ B = η. 86) needs to be solved since the solutions are related by η = πB. 87) will almost always exist, whether or not A and B are compatible. 83) holds. Liu (1996) briefly discusses the difference between the arrays (aij ηj ) and (bij τi ) in the incompatible case in the context of Gibbs sampler simulations. 12 (Markovian measure of incompatibility; incompatible case). 49). 88) ⎛ 42 2. 019 for the Markovian measure of incompatibility. ✷ The actual value assumed by D does not appear to have a ready interpretation.

1 (Compatibility of conditional probability matrices). (Arnold and Gokhale (1994)). Suppose that A and B contain only positive elements, then the following statements are equivalent: (i) A and B are compatible. (ii) For every 2 × 2 subtable of A and the corresponding subtable of B, the cross-product ratios are equal. (iii) A and B have identical uniform marginal representations. 2 Review and Extensions of Compatibility Results 23 It is evident that if two conditional matrices are compatibles their UMR matrices and the UMR of the compatible joint probability distribution P coincide.

### An Introduction to Probability and Random Processes by Gian-Carlo Rota, Kenneth Baclawski

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