Combining chains of Bayesian models with Markov melding

1. Problem Statement

Bayesian inference practitioners often face the challenge of integrating multiple heterogeneous data sources into a single joint model. While specifying a unified joint model is theoretically ideal, it is often computationally intractable or methodologically difficult due to the complexity of the data structures.

• The Limitation of Existing Methods: Standard approaches like Markov melding (Goudie et al., 2019) allow for combining submodels that share a single common quantity ($\phi$) across all models. However, many real-world applications involve a chain structure where submodel $m$ shares a quantity with submodel $m-1$ and a different quantity with submodel $m+1$.
• The Gap: There was no generic framework to formally join chains of submodels while:
  1. Handling distinct domains and supports for the common quantities.
  2. Capturing prior dependencies between adjacent common quantities.
  3. Avoiding the computational burden of fitting a massive joint model in a single step.
  4. Moving beyond simplistic approximation strategies (e.g., using point estimates or normal approximations of subposteriors) which often lead to biased or overconfident inference.

2. Methodology: Chained Markov Melding

The authors propose Chained Markov Melding, an extension of the original Markov melding framework designed for $M$ submodels arranged in a chain.

A. Model Specification

Consider $M$ submodels $p_m(\theta_m, Y_m)$ where submodels $m$ and $m+1$ share a common quantity $\phi_{m \cap m+1}$.

• Common Quantities: Defined as $\phi = (\phi_{1 \cap 2}, \phi_{2 \cap 3}, \dots, \phi_{M-1 \cap M})$.
• Marginal Replacement: To combine the models, the authors extend the marginal replacement procedure. They replace the marginal priors of the common quantities in each submodel with a pooled prior $p_{pool}(\phi)$.
• The Melded Joint Model: The final joint distribution is constructed via Markov combination:
  $$p_{meld}(\phi, \psi, Y) = p_{pool}(\phi) \prod_{m=1}^M \frac{p_m(\phi_m, \psi_m, Y_m)}{p_m(\phi_m)}$$
  where $\psi_m$ are submodel-specific parameters.

B. Pooled Prior Construction

A critical contribution is defining how to pool priors when the common quantities differ between submodels (e.g., $\phi_{1 \cap 2}$ vs. $\phi_{2 \cap 3}$). The authors propose extensions to standard pooling methods:

1. Chained Logarithmic Pooling:
   $$p_{pool, \log}(\phi) \propto \prod_{m=1}^M p_m(\phi_m)^{\lambda_m}$$
   This allows for dependence between components of $\phi$ because the terms $p_m(\phi_m)$ often contain overlapping variables (e.g., $p_2$ contains both $\phi_{1 \cap 2}$ and $\phi_{2 \cap 3}$).
2. Chained Linear Pooling: A two-step process where linear pooling is applied to adjacent pairs, then multiplied. The authors note this assumes independence between non-adjacent components unless extended via copulas.
3. Chained Dictatorial Pooling: A method to select a single "authoritative" submodel prior for specific components of $\phi$ while pooling the rest, preserving dependencies where they exist in the authoritative model.

C. Posterior Estimation (Multi-Stage Sampler)

Fitting the full joint model directly is often infeasible. The authors propose a parallel multi-stage MCMC sampler (specifically for $M=3$ in their examples, but extensible):

• Stage 1 (Parallel): Independent MCMC runs are performed on the "end" submodels (e.g., $p_1$ and $p_3$) to generate samples from their subposteriors. This stage can be parallelized and reuses existing software implementations (e.g., Stan, JAGS).
• Stage 2 (Integration): A Metropolis-within-Gibbs sampler targets the full melded posterior.
  • Proposals for the parameters of the end submodels are drawn from the Stage 1 samples.
  • The middle submodel ($p_2$) is updated using standard proposals.
  • Acceptance Probability: Crucially, the acceptance ratio in Stage 2 depends only on the middle submodel and the pooled prior, avoiding the need to re-evaluate the likelihoods of the end submodels at every step. This significantly reduces computational cost.

D. Approximation Strategy

The paper also details a method using Normal Approximations for subposteriors. If the subposteriors of the common quantities can be approximated as Gaussian, the melded posterior can be derived analytically (up to the middle submodel), allowing for standard MCMC sampling without the multi-stage complexity.

3. Key Contributions

1. Theoretical Framework: Formalized the "Chained Markov Melding" approach, extending Markov melding to chain-structured dependencies where common quantities are not shared by all models.
2. Prior Pooling Extensions: Developed specific logarithmic and linear pooling functions capable of handling overlapping but distinct marginal distributions, including methods to preserve prior correlations between adjacent chain links.
3. Computational Efficiency: Introduced a parallel multi-stage sampler that:
   • Avoids evaluating all submodels simultaneously.
   • Allows for the reuse of existing submodel code.
   • Enables parallel computation in the first stage.
4. Uncertainty Quantification: Demonstrated that fully accounting for uncertainty in common quantities (rather than using point estimates) is critical to avoiding biased and overconfident inference.

4. Results and Case Studies

The methodology was validated using two distinct examples:

Case Study 1: Integrated Population Model (IPM) for Little Owls

• Setup: Combining three submodels: Capture-Recapture ($p_1$), Population Counts ($p_2$), and Fecundity ($p_3$).
• Goal: Estimate survival and reproduction rates.
• Findings:
  • The chained melded model produced results concordant with the original, fully specified joint IPM (which served as the ground truth).
  • Different pooling methods (Product-of-Experts vs. Logarithmic) yielded similar posteriors, suggesting the data was informative enough to dominate the prior choice.
  • The multi-stage sampler successfully replicated the full joint posterior, validating the computational approach.

Case Study 2: Survival Analysis with Uncertain Event Times

• Setup: Analyzing time-to-respiratory-failure (RF) in ICU patients.
  • $p_1$: Models P/F ratio data to derive the event time (uncertain quantity).
  • $p_2$: Survival model using the derived event time as the response.
  • $p_3$: Longitudinal model of fluid administration.
• Challenge: The event time is a non-invertible function of parameters in $p_1$, making standard joint modeling difficult.
• Findings:
  • Uncertainty Matters: Comparing the chained melding approach to a "plug-in" approach (fixing event times to point estimates) revealed significant differences.
  • The plug-in approach produced overconfident and biased estimates, particularly for the parameter linking fluid administration to survival risk ($\alpha$).
  • The chained melding approach correctly propagated the multimodal uncertainty from the event time derivation into the survival model, resulting in wider credible intervals and shifted posterior modes.
  • The multi-stage sampler successfully handled the computational hurdle of solving the event time optimization problem only in Stage 1, treating it as a post-processing step in Stage 2.

5. Significance

This paper provides a robust, generalizable solution for modular Bayesian inference. Its significance lies in:

• Scalability: It allows complex systems to be built from smaller, manageable submodels without sacrificing the rigor of full Bayesian inference.
• Flexibility: It accommodates diverse data types (e.g., count data, survival data, longitudinal data) and complex dependencies (chains) that were previously difficult to unify.
• Practicality: By enabling the reuse of existing submodel code and parallelizing computation, it lowers the barrier to entry for complex integrated modeling.
• Statistical Integrity: It highlights the dangers of "two-stage" approaches that ignore uncertainty in intermediate quantities, offering a principled alternative that preserves the full posterior distribution.

In summary, Chained Markov Melding offers a conceptually appealing and computationally feasible framework for integrating heterogeneous data sources in a chain structure, ensuring that uncertainty is properly quantified throughout the inference process.