Multiple-Try RJMCMC (GMTRJ) Algorithm

• The paper introduces GMTRJ as an extension of RJMCMC that incorporates multiple candidate proposals with rigorous weighting for improved chain mixing.
• It details the use of dimension-matching transformations and Metropolis–Hastings–Green acceptance probabilities to maintain detailed balance.
• Empirical results demonstrate that variants like GMTRJ-quad and GMTRJ-man significantly reduce integrated autocorrelation time compared to standard methods.

The Generalized Multiple-Try Reversible Jump (GMTRJ) algorithm is an extension of the Reversible Jump Markov Chain Monte Carlo (RJMCMC) method designed for efficient Bayesian model selection and estimation in scenarios involving models with differing parameter dimensions. GMTRJ incorporates a multiple-try mechanism, enabling multiple candidate proposals at each model jump, with rigorous selection and weighting strategies that ensure detailed balance and improved chain mixing. This approach systematically generalizes both the traditional RJMCMC (Green 1995) and the Multiple-Try Metropolis (MTM) algorithm (Liu, Liang & Wong 2000), and is formalized by Pandolfi, Bartolucci, and Friel (Pandolfi et al., 2010).

1. Canonical Framework and Objectives

Let $\{M_1, \ldots, M_M\}$ denote the set of candidate models, with model $M_m$ characterized by parameter vector $\theta_m \in \Theta_m \subseteq \mathbb{R}^{d_m}$. For observed data $y$, the joint posterior over models and parameters is given by

$$\pi(m,\theta_m) \propto L(y\,|\,m,\theta_m)\,p(\theta_m\,|\,m)\,p(m),$$

where $L$, $p(\cdot\,|\,m)$, and $p(m)$ denote likelihood, conditional prior, and model prior, respectively. The principal aim is to construct an MCMC sampler that (i) traverses the model index space, (ii) updates model-specific parameters, and (iii) remains efficient under variable parameter dimensions $d_m$. GMTRJ achieves these goals by introducing a multiple-try mechanism within the reversible jump paradigm, exploiting flexible selection weights and dimension-matching transformations (Pandolfi et al., 2010).

2. Multiple-Try Proposal Mechanism

At each between-model move, rather than proposing a single parameter vector for the target model, GMTRJ generates $K$ parameter proposals via auxiliary variable augmentation and deterministic, invertible mappings. The procedure is as follows:

• Model Proposal: Given current state $(m, \theta)$, propose a new model $m' \ne m$ with probability $h(m \to m')$.
• Dimension-Matching and Candidates: For $j = 1, \ldots, K$, draw $u^{(j)} \sim T_{m \to m'}(\cdot\,|\,\theta)$ and form $$(\theta^{(j)}, u'^{(j)}) = g_{m \to m'}(\theta, u^{(j)})$$, ensuring $d(\theta) + d(u) = d(\theta') + d(u')$ for invertibility.
• Weighted Selection: One candidate is selected with probability proportional to strictly positive weights:

$$P(J = j) = \frac{w_{m \to m'}^*(\theta, \theta^{(j)})}{\sum_{j=1}^K w_{m \to m'}^*(\theta, \theta^{(j)})}$$

where $w_{m \to m'}^*(\theta, \theta')$ is a user-defined positive weight function. Selection probabilities may employ a local quadratic approximation of the log-posterior (e.g., GMTRJ-quad) for computational efficiency (Pandolfi et al., 2010).

3. Acceptance Probability and Detailed Balance

Upon selection of proposal $(m',\theta')=(m',\theta^{(J)})$ with matched auxiliary variables, the algorithm guarantees detailed balance with a Metropolis–Hastings–Green acceptance step. This involves:

• Drawing $K-1$ reverse auxiliary proposals $v_\ell \sim T_{m' \to m}(\cdot\,|\,\theta')$, then completing the reverse candidate set $$(\theta_r^\ell, u_r^\ell) = g_{m' \to m}(\theta', v_\ell)$$ for $\ell=1,\ldots,K-1$, and $(\theta_r^K, u_r^K) = (\theta, u^{(J)})$.
• Computing reverse selection probabilities using corresponding reverse weights.
• The acceptance probability is

$$\alpha = \min \Bigg\{ 1, \frac{\pi(m',\theta') h(m' \to m) T_{m' \to m}(u'|\theta') p_{m' \to m}^K} {\pi(m,\theta) h(m \to m') T_{m \to m'}(u | \theta) p_{m \to m'}^J} \left| \frac{\partial(\theta',u')}{\partial(\theta,u)} \right| \Bigg\}$$

where $p_{m' \to m}^K$ and $p_{m \to m'}^J$ are reverse and forward selection probabilities given by normalized weights, and the Jacobian accounts for the change-of-variables in the transformation [(Pandolfi et al., 2010), Eq (3.2)].

4. Weight Function Choices and Algorithmic Variants

The choice of $w^*(\theta, \theta')$ governs computational trade-offs:

• GMTRJ-I: $$w^*(\theta, \theta') = \pi(\theta', m') T_{m' \to m}(u'|\theta')$$ (recovers “MTM-I”).
• GMTRJ-inv: $$w^*(\theta, \theta') = \pi(\theta', m') / T_{m \to m'}(u|\theta)$$.
• GMTRJ-quad: Employs a quadratic expansion:

$$\log \pi(m',\theta') \approx \log \pi(m', \hat\theta_{m'}) + (\theta' - \hat\theta_{m'})^T s_{m'} - \tfrac{1}{2} (\theta' - \hat\theta_{m'})^T H_{m'} (\theta' - \hat\theta_{m'})$$

Only first and second derivatives at reference $\hat\theta_{m'}$ need to be computed, yielding computational gain for large $K$ (Pandolfi et al., 2010).

• GMTRJ-man: For latent class models or discrete parameters lacking Hessian structure, $$w^*(\theta, \theta')=L_\text{marginal}(y|m',\theta')$$, using only manifest likelihoods for efficiency.

This flexibility enables adaptation of GMTRJ to a range of model structures and computational bottlenecks.

5. Pseudocode Summary

The following summarizes the full GMTRJ sweep for state $(m_t, \theta_t)$ at iteration $t$:

1. With probability $p_w$, update $\theta_t$ by within-model Metropolis–Hastings; otherwise, proceed with a between-model GMTRJ move:
   • Draw $m'$ via $h(m_t\to m')$.
   • Generate $K$ candidates as specified and compute forward weights.
   • Sample candidate index $J$ according to weights and set proposal $(m',\theta')$.
   • Draw $K-1$ reverse proposals and compute reverse weights.
   • Compute acceptance probability $\alpha$ as above.
   • Accept $(m',\theta')$ with probability $\alpha$, else remain at $(m_t,\theta_t)$ (Pandolfi et al., 2010).

6. Empirical Performance and Illustrative Examples

Extensive empirical comparisons highlight the efficiency gains of GMTRJ, especially under the GMTRJ-quad and GMTRJ-man weighting schemes.

Example	Model Space	GMTRJ Variant	Relative Efficiency Gains
Logistic regression	Covariate selection ($M_1$–$M_5$)	GMTRJ-quad	3–4× lower integrated autocorrelation time (IAT) per CPU-second vs. standard RJ or DR
Finite latent class	Unknown $C$ classes	GMTRJ-man	$\sim$50% lower CPU-adjusted IAT with $K=5$ compared to RJ

In both logistic regression variable selection and latent class models, all GMTRJ variants recover correct posterior model probabilities with markedly improved CPU-normalized mixing relative to standard RJMCMC and delayed rejection algorithms. The cost per iteration remains similar for GMTRJ-quad as the quadratic approximation can be reused for all $K$ proposals; GMTRJ-man dramatically accelerates computations in models where latent allocations dominate likelihood evaluation (Pandolfi et al., 2010).

7. Structural Properties and Theoretical Implications

GMTRJ generalizes RJMCMC by embedding an arbitrary weighting scheme—satisfying positivity and local normalizability—within the multiple-try framework, while preserving detailed balance. This construction facilitates advanced posterior exploration via increased between-model acceptance and reduced autocorrelation. When weighting is based on quadratic local approximations, the computational overhead is minimal with respect to posterior evaluations, while efficiency, as measured by IAT and CPU-time, is substantially improved.

A plausible implication is that the GMTRJ methodology can be systematically adapted to novel model spaces and selection architectures by judicious choice of the selection weights, auxiliary variable mappings, and dimension-matching schemes, thereby extending the practical feasibility of Bayesian model search far beyond the scope of standard RJMCMC methods (Pandolfi et al., 2010).