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Adaptive MCMC marginal convergence under diminishing adaptation with uniform ergodicity or containment

Statement

Let $(X, X)$ and $(Γ, G)$ , the jointly measurable probability-kernel family $(P_{γ})_{γ \in Γ}$ , common invariant probability $π$ , and adaptive process $(X_{n}, Γ_{n})$ satisfy Diminishing adaptation. Fix an initial law $μ_{0}$ for $(X_{0}, Γ_{0})$ ; a fixed initial pair is the Dirac-law special case. If diminishing adaptation holds under this law and either of the following conditions holds, then $∥ L_{μ_{0}} (X_{n}) - π ∥_{T V} ⟶ 0.$

Simultaneous uniform ergodicity: for every $ϵ > 0$ there is an integer $N (ϵ) \geq 1$ such that $\sup_{x \in X, γ \in Γ} ∥ P_{γ}^{N (ϵ)} (x, \cdot) - π ∥_{T V} \leq ϵ .$
Containment: for every $ϵ, δ > 0$ there is an integer $N$ such that $P_{μ_{0}} ⁣ (M_{ϵ} (X_{n}, Γ_{n}) \leq N) \geq 1 - δ for every n \geq 0,$ where $M_{ϵ}$ is the fixed-kernel convergence time from Containment for adaptive MCMC.

The conclusion is convergence of the one-time marginal distributions. It does not itself assert a strong law, a central limit theorem, or convergence of sample averages.

Facts & Assumptions

Given: A common target probability $π$ , probability kernels $(P_{γ})$ satisfying $π P_{γ} = π$ , an initial law $μ_{0}$ , and an adaptive process whose conditional transition at time $n$ is $P_{Γ_{n}} (X_{n}, \cdot)$ .

[F1]

Diminishing adaptation: $D_{n} = \sup_{x} ∥ P_{Γ_{n + 1}} (x, \cdot) - P_{Γ_{n}} (x, \cdot) ∥_{T V}$ is measurable, lies in $[0, 1]$ , and tends to zero in probability.

[F2]

Containment for adaptive MCMC: $M_{ϵ} (x, γ)$ is the first positive iterate at which the fixed kernel $P_{γ}$ is within $ϵ$ of $π$ , and containment gives a uniform-in- $n$ probability bound for it.

[F3]

Invariant and stationary distribution for a Markov kernel: $π P_{γ} = π$ for every fixed $γ$ .

[F4]

Total variation distance for probability laws: total variation is the supremum of setwise probability differences.

[F5]

Total variation duality, kernel contraction, and coupling bound: Markov kernels contract total variation; hence $∥ P_{γ}^{m + 1} (x, \cdot) - π ∥_{T V} \leq ∥ P_{γ}^{m} (x, \cdot) - π ∥_{T V}$ .

[F6]

Finite-block perturbation bound for adaptive MCMC: over a fixed block of $N$ steps, the adaptive marginal differs from the mixture obtained by freezing $Γ_{n}$ by at most $\sum_{i = 0}^{N - 1} \sum_{j = 0}^{i - 1} E D_{n + j}$ .

Proof

Given: The hypotheses and facts [F1]–[F6]. All probabilities and expectations below use the specified initial law $μ_{0}$ .

1.1F1algebra

Since $0 \leq D_{n} \leq 1$ and $D_{n} \to 0$ in probability, for each fixed $j \geq 0$ we have $E D_{n + j} \to 0$ : for any $t > 0$ , $E D_{n + j} \leq t + P (D_{n + j} > t)$ ; first let $n \to \infty$ and then $t ↓ 0$ .

1.2F3F4F6given

Assume simultaneous uniform ergodicity and fix a desired tolerance $η > 0$ . Choose $N = N (η / 2)$ as in alternative 1. For the frozen mixture $Q_{n, N}$ from [F6], every state and parameter has $∥ P_{Γ_{n}}^{N} (X_{n}, \cdot) - π ∥_{T V} \leq η / 2$ , so convexity of the setwise supremum gives $∥ Q_{n, N} - π ∥_{T V} \leq η / 2$ .

1.3F2F4F5given

Now assume containment. Fix $η > 0$ , take $ϵ = δ = η / 3$ , and choose $N$ from alternative 2. On the event $M_{ϵ} (X_{n}, Γ_{n}) \leq N$ , the definition gives an iterate $m \leq N$ within $ϵ$ of $π$ ; [F5] propagates this bound to the $N$ th iterate. On the complementary event total variation is at most one. Therefore the frozen mixture satisfies $∥ Q_{n, N} - π ∥_{T V} \leq ϵ + δ = 2 η / 3$ for every $n$ .

2.1F4F6step 1.1step 1.2algebra

For this fixed $N$ , [F6] and step 1.1 imply $∥ L (X_{n + N}) - Q_{n, N} ∥_{T V} \to 0$ , since its upper bound is a finite sum of expectations tending to zero. The triangle inequality and step 1.2 yield ${lim sup}_{n \to \infty} ∥ L (X_{n + N}) - π ∥_{T V} \leq η / 2$ . As $n + N$ ranges over all sufficiently large times and $η > 0$ is arbitrary, the asserted convergence follows in alternative 1.

3.1F4F6step 1.1step 1.3algebra∎

The same fixed-block bound [F6] tends to zero by step 1.1. The triangle inequality gives ${lim sup}_{n \to \infty} ∥ L (X_{n + N}) - π ∥_{T V} \leq 2 η / 3 < η$ . Again $N$ is fixed and $η$ is arbitrary, proving the asserted convergence in alternative 2.

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Sources

Gareth O. Roberts and Jeffrey S. Rosenthal, Coupling and Ergodicity of Adaptive MCMC (standard reference, not scraped)
Gersende Fort, Adaptive MCMC: theory and methods, Warwick lecture notes (standard reference, not scraped)