Online Likelihood-free Inference for Markov State-Space Models Using Sequential Monte Carlo

Abstract

Sequential Monte Carlo (SMC) methods (a.k.a particle filters) refer to a class of algo- rithms used for filtering problems in non-linear state-space models (SSMs). SMC methods approximate posterior distributions over latent states by propagating and resampling par- ticles, where each particle is associated with a weight representing its relative importance in approximating the posterior. Through iteratively updating particles and/or weights SMC methods gradually refine the particle-based approximation to reflect true posterior distribution. A key challenge in SMC methods arises when the likelihood, responsible for guiding particle weighting, is intractable. In such cases, Approximate Bayesian Computa- tion (ABC) methods can approximate the likelihood, bypassing the need for closed-form expressions. The particle SMC method of interest in this thesis is Chopin’s SMC2 frame- work Chopin et al. [2013] which uses a nested SMC approach. An “outer” operates on the parameter space θ, and an “inner” particle filter estimates the likelihood for a fixed param- eter θ, facilitating joint inference over the parameters and states. The framework proposed by Chopin required closed-form likelihoods and was intended for offline learning. This thesis proposes an ABC-SMC2 algorithm for online-inference in SSMs with intractable likelihoods. Our method uses Approximate Bayesian Computation (ABC) in the inner particle filter to approximate likelihoods via an ABC kernel, thus enabling inference with- out closed-form observation likelihoods. To address the challenges of online learning, we introduce an adaptive ε-scheduler for dynamically selecting the ABC kernel’s tolerance lev- els and a likelihood recalibration mechanism that retroactively refines posterior estimates using previously observed data. We validate our approach on three case studies using com- partment models governed by an ODE system: a toy linear ODE system, the non-linear Lotka–Volterra equations, and a high-dimensional SEIR model with real-world covariates. In these experiments, ABC-SMC2 outperforms fixed and adaptive ε-schedulers in terms of credible interval coverage, posterior accuracy, RMSE.