A Spectral Analytic Comparison of Trace-class Data Augmentation Algorithms and their Sandwich Variants

Let fX(x) be an intractable probability density. If f(x, y) is a joint density whose x-marginal is fX(x), then f(x, y) can be used to build a data augmentation (DA) algorithm that simulates a Markov chain whose invariant density is fX(x). The move from the current state of the chain, Xn = x, to the new state, Xn+1, involves two simulation steps: Draw Y ∼ fY |X(·|x), call the result y, and then draw Xn+1 ∼ fX|Y (·|y). The sandwich algorithm is a variant that involves an extra step “sandwiched” between the two conditional draws. Let R(y, dy′) be any Markov transition function that is reversible with respect to the y-marginal, fY (y). The extra step entails drawing Y ′ ∼ R(y, ·), and then using this draw, call it y′, in place of y in the second step. In this paper, the DA and sandwich algorithms are compared in the case where the joint density, f(x, y), satisfies ∫ X ∫ Y fX|Y (x|y)fY |X(y|x) dy dx < ∞. This condition implies that the (positive) Markov operator associated with the DA Markov chain is traceclass. It is shown that, without any further assumptions, the sandwich algorithm always converges at least AMS 2000 subject classifications. Primary 60J27; secondary 62F15 Abbreviated title. Spectral Analysis of Data Augmentation