Efficient Inference for Coupled Hidden Markov Models in Continuous Time and Discrete Space

TL;DR

The paper tackles scalable posterior inference for systems of interacting continuous-time Markov chains (IPS) with high dimensionality. It introduces Latent IPS with a learnable generator and couples it with twisted sequential Monte Carlo (tSMC) to tilt the path measure toward informative regions using a learnable twist potential. TwistNet and a wake–sleep learning scheme enable amortized learning of the twist and model parameters, improving efficiency in high-dimensional settings. The approach is demonstrated on latent SIRS dynamics on graphs up to 256 nodes and neural wildfire-spread models on 64×64 grids, where it outperforms baselines in posterior fidelity and parameter recovery, highlighting its potential for scalable Bayesian inference in complex discrete-state, continuous-time systems.

Abstract

Systems of interacting continuous-time Markov chains are a powerful model class, but inference is typically intractable in high dimensional settings. Auxiliary information, such as noisy observations, is typically only available at discrete times, and incorporating it via a Doob's $h-$transform gives rise to an intractable posterior process that requires approximation. We introduce Latent Interacting Particle Systems, a model class parameterizing the generator of each Markov chain in the system. Our inference method involves estimating look-ahead functions (twist potentials) that anticipate future information, for which we introduce an efficient parameterization. We incorporate this approximation in a twisted Sequential Monte Carlo sampling scheme. We demonstrate the effectiveness of our approach on a challenging posterior inference task for a latent SIRS model on a graph, and on a neural model for wildfire spread dynamics trained on real data.

Figures (13)

• Figure 1: Example of a latent IPS as a state space model. Here, the latent trajectory is sampled from a continuous-time SIR model, and potentials are emission distributions of partially masked observations.
• Figure 1: Parameter estimates and relative parameter error (RPE) for the SIRS model with 32 nodes, mean $\pm$ 2 standard deviations across 10 random seeds.
• Figure 2: Latent trajectory reconstruction, measured by cross-entropy loss on the test set of ground truth trajectories with respect to the posterior approximations. Error bars correspond to two standard errors computed across trajectories.
• Figure 3: Parameter estimates and relative parameter error (RPE) for the SIRS model with 64 nodes, mean $\pm$ 2 standard deviations across 10 random seeds.
• Figure 4: Observed active fire maps from WildfireSpreadTS gerard2023wildfirespreadts and snapshot of approximate posterior samples taken at observation times. A pixel can only turn active if another one in its Moore neighborhood is, and transitions from unburned to burned are impossible. Observations are noisy, as satellite imagery for wildfires can be obscured by smoke and clouds schroeder2014new.

Theorems & Definitions (5)

• Proposition 1
• Proposition 2: Doob’s $h$-transform for IPS
• Theorem 3
• Lemma 4
• proof