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Random-scan Gibbs is reversible and invariant

Statement

Let $m \geq 1$ , let $P_{1}, \dots, P_{m}$ be exact block-update kernels for one target probability $π$ , and let $w_{j} \geq 0$ with $\sum_{j = 1}^{m} w_{j} = 1$ . Then $P_{r a n d} = \sum_{j = 1}^{m} w_{j} P_{j}$ is reversible with respect to $π$ and leaves $π$ invariant. No irreducibility or convergence conclusion follows from this statement.

Facts & Assumptions

Given: The target, exact block updates, and weights in the Statement.

[F1]

Each exact block update is reversible and invariant for the target. Each exact Gibbs block update is reversible and invariant.

[F2]

Random scan uses the kernel $P_{r a n d} (x, A) = \sum_{j} w_{j} P_{j} (x, A)$ . Random and systematic Gibbs scan schemes.

[F3]

Reversibility is the two-event detailed-balance identity. Reversibility for a probability kernel.

[F4]

Invariance is equality $π P_{r a n d} = π$ . Invariant and stationary distribution for a Markov kernel.

[F5]

The random-scan mixture has probability-measure sections and measurable evaluations. Random and systematic scan kernels are well-defined.

[F6]

The nonnegative integral is additive for finite sums. Additivity of the nonnegative Lebesgue integral.

[F7]

The nonnegative integral is homogeneous for nonnegative scalar multiples. Monotonicity and nonnegative homogeneity of the nonnegative integral.

Proof

Given: The hypotheses in the Statement and facts [F1]–[F7].

1.1F1F2F6F7

By [F2] and nonnegativity of the weights, for every measurable $A, C$ , $\int_{A} P_{r a n d} (x, C) π (d x) = \sum_{j = 1}^{m} w_{j} \int_{A} P_{j} (x, C) π (d x)$ . This follows by applying [F7] to each scalar multiple and [F6] repeatedly to the finite sum.

2.1F1F2F3step 1.1

Reversibility of each $P_{j}$ in [F1] changes the last expression to $\sum_{j = 1}^{m} w_{j} \int_{C} P_{j} (y, A) π (d y) = \int_{C} P_{r a n d} (y, A) π (d y)$ . Thus the detailed-balance identity [F3] holds for every $A, C$ .

3.1F3F4F5step 2.1∎

Set $C$ equal to the whole state space in step 2.1 and use $P_{r a n d} (y, S) = 1$ from [F5]. The resulting identity is $\int P_{r a n d} (y, A) π (d y) = π (A)$ , which is invariance by [F4]. The proof establishes stationarity only; it does not establish irreducibility, uniqueness, ergodicity, or a rate.

Depends on

Used by

An invariant Gibbs target can still give reducible dynamics Example

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Sources

Art B. Owen, Monte Carlo Theory, Methods and Examples, Chapter 12 (standard reference, not scraped)
Gareth O. Roberts and Jeffrey S. Rosenthal, General State Space Markov Chains and MCMC Algorithms (standard reference, not scraped)