Efficient MCMC sampling in dynamic mixture models

Abstract

We show how to improve the efficiency of Markov Chain Monte Carlo (MCMC) simulations in dynamic mixture models by block-sampling the discrete latent variables. Two algorithms are proposed: the first is a multi-move extension of the single-move Gibbs sampler devised by Gerlach, Carter and Kohn (in J. Am. Stat. Assoc. 95, 819–828, 2000); the second is an adaptive Metropolis-Hastings scheme that performs well even when the number of discrete states is large. Three empirical examples illustrate the gain in efficiency achieved. We also show that visual inspection of sample partial autocorrelations of the discrete latent variables helps anticipating whether blocking can be effective.

Appendix: Converge of the adaptive MH algorithm

As above we denote S _t,t+h−1=(S _t ,…,S _t+h−1) the random vector defined on the finite space ℵ that contains all possible paths of length h. Our aim is to show that the distribution of $\{\mathbf{S}_{t,t+h-1}^{(n)}; n \ge 1 \}$ generated by the adaptive MH algorithm given in Sect. 2.4 with transition kernel:

converges to the correct target distribution, say π _w (S _t,t+h−1)=π(S _t,t+h−1), where w underlines the conditioning on $\mathbf{w} \equiv (\mathbf{s}_{1,t-1},\mathbf{s}_{t+h}^{T},\boldsymbol{\theta},\mathbf{y})$ that is omitted. Here we focus on a generic block since convergence in distribution of the full sequence {S ⁽ⁿ⁾;n≥1} to π(S) is ensured by the results in Tierney (1994).

The adaptive proposal $\tilde{q}_{n}(\mathbf{s}_{t,t+h-1})$ is such that:

$$ \tilde{q}_n(\mathbf{s}_{t,t+h-1}) = \delta q_0( \mathbf{s}_{t,t+h-1}) + (1-\delta)q_n(\mathbf{s}_{t,t+h-1}) $$

where 0<δ<1, q ₀(⋅) is the uniform distribution on ℵ, and the adaptive component q _n (s _t,t+h−1) is detailed in (2.6), (2.8), and (2.10) with the recursions (2.14)–(2.15) for the marginal and transition probabilities. The proof consists of verifying that this adaptive scheme satisfies the sufficient conditions given in Giordani and Kohn (2010):

for some constants K>0 and r>0.

Condition GK1 is immediate since the distributions π(s _t,t+h−1) and q _n (s _t,t+h−1) are defined on the discrete space ℵ, so they are bounded above at 1. Furthermore, the constant term q ₀(s _t,t+h−1) equals 1/N ^h, so the two ratios in GK1 are bounded above.

Condition GK2 needs more attention. The weight that the adaptive component attaches to a given block is a function of the marginal and transition probabilities and can be written as:

(A.1)

We first show that the marginal and transition probabilities q _n (s _t ) and q _n (s _t+1|s _t ) involved in Eq. (A.1) settle down asymptotically. The recursion for marginal probabilities (2.14) yields:

$$ q_n(s_t) - q_{n+1}(s_t) = \frac{1}{n} \bigl[q_n(s_t) - \mathsf{1} \bigl(s_t^{(n-1)}=s_t\bigr)\bigr] = O \bigl(n^{-1}\bigr) $$

whereas the one for transition probabilities (2.15) implies:

Giordani and Kohn (see Lemma 1 in Appendix, 2010) show that Condition GK1 implies uniform ergodicity, i.e. $T_{n}(\mathbf{s}_{t,t+h-1},\overline{\mathbf{s}}_{t,t+h-1}) \geq \epsilon \pi(\,\overline{\mathbf{s}}_{t,t+h-1})$ for some ϵ>0. This ensures that q _n (s _t ) is strictly positive asymptotically and thus q _n (s _t+1|s _t )−q _n+1(s _t+1|s _t )=O(n ⁻¹).

Let us write q _n (s _t+k |s _t+k−1)−q _n+1(s _t+k |s _t+k−1)=ϵ _k with ϵ _k =O(n ⁻¹), and let C _n denote the numerator of q _n (s _t,t+h−1) in Eq. (A.1). We have:

so C _n −C _n+1=ϵ _C =O(n ⁻¹). Let D _n denote the denominator of q _n (s _t,t+h−1) in (A.1). Since

$$D_n= \sum_{s_t}\cdots\sum_{s_{t+h-1}} C_n,\qquad D_n - D_{n+1}=\epsilon_D = O\bigl(n^{-1}\bigr) $$

as well. Hence:

which proves Condition GK2.