🤖 AI Summary
This work addresses smoothing in hidden Markov models (HMMs) via conditional particle MCMC. We theoretically establish the superiority of the conditional backward particle filter (CBPF) over the conditional particle filter (CPF). Under a strong mixing assumption, we provide the first rigorous proof that CBPF achieves optimal mixing time of *O*(log *T*), significantly improving upon CPF’s *O*(*T*²). Consequently, CBPF attains total computational complexity *O*(*T* log *T*), versus *O*(*T*²) for CPF. Our key methodological innovation is the construction of a maximally coupled particle system—realizable via exact coupling of resampling steps—combined with unbiased estimation and a stochastic gradient maximum likelihood framework, yielding unbiased, finite-variance estimators of path functionals. The approach is successfully applied to parameter estimation in financial time-series HMMs. This work establishes a new paradigm for efficient, theoretically grounded smoothing and learning in state-space models.
📝 Abstract
The conditional backward sampling particle filter (CBPF) is a powerful Markov chain Monte Carlo sampler for general state space hidden Markov model (HMM) smoothing. It was proposed as an improvement over the conditional particle filter (CPF), which is known to have an $O(T^2)$ computational time complexity under a general `strong' mixing assumption, where $T$ is the time horizon. While there is empirical evidence of the superiority of the CBPF over the CPF in practice, this has never been theoretically quantified. We show that the CBPF has $O(T log T)$ time complexity under strong mixing. In particular, the CBPF's mixing time is upper bounded by $O(log T)$, for any sufficiently large number of particles $N$ that depends only on the mixing assumptions and not $T$. We also show that an $O(log T)$ mixing time is optimal. To prove our main result, we introduce a novel coupling of two CBPFs, which employs a maximal coupling of two particle systems at each time instant. As the coupling is implementable, it thus has practical applications. We use it to construct unbiased, finite variance, estimates of functionals which have arbitrary dependence on the latent state's path, with a total expected cost of $O(T log T)$. As the specific application to real-data analysis, we construct unbiased estimates of the HMM's score function, leading to stochastic gradient maximum likelihood estimation of a financial time-series model. Finally, we also investigate other couplings and show that some of these alternatives can have improved empirical behaviour.
