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1) gives the same infimum (see [4]). 1) we see that for fixed A, Ji(A; j^) is a nonnegative, convex function of JUL and 0^h(A; JLL)^<». Moreover, if X is a Polish space, then h(A; /n) is lower semicontinuous in ^ in the weak topology. The following lemma was proved in [6]. 1. For any (X, 2) and any two probability measures A, jn on (X, 2), ft (A; jn) is finite if and only if (1) ju«A, and (2) d|Lt/dA=/(x) is such that /(xHog/MeL^A). Under these circumstances, Remark. Suppose

D. 7), h(a;/3) is, of course, the entropy of |3 with respect to a, as introduced in Section 10, and p(t, x, •) is the transition function for the Markov process0,P x. Let n be a positive integer, let h = l/n and consider the grid {jh}, j = 0, ±1, ±2, Define a Markov chain on this grid with stationary marginal /ut and transition probability px(dy). d. , values of the Markov chain at adjacent grid points. Let Q(h) be the measure induced by this interpolation procedure from the Markov chain on the grid.

In particular, if H(Q)<<» and /x is the marginal of Q, then /n«p(l,x, •) uniformly for x in compact sets. Proof. 1, so we want to prove a«p(l, x, •) uniformly for x in compact sets. Let A c X be such that a(A) = 5>0. By hypothesis I, p(l, x, A)^e(x, 6). By hypothesis II (Section 9), p(l, x, •) as a mapping from X—> L^(a) is continuous, and therefore p(l, x', A)^e(x, S)/2 for all x' in a neighborhood of x. The result now follows by the usual compactness argument. 3. Let Q&Ms(£l) be such that H(Q)

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A case study in non-centering for data augmentation: Stochastic epidemics by Neal P.

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