Minimally and vaguely informative priors to combat practical parameter non-identifiability of hidden Markov models exemplified by ion channel data

Hidden Markov Model (HMM) inference for time-series data from ion channels or other biomolecules is challenging. We argue that inference on partially observed chemical reaction networks (CRNs) suffers from practical parameter non-identifiability (non-PI) that often goes unnoticed in maximum likelihood (ML) inferences. Limitations in the signal bandwidth and a poor signal-to-noise ratio only add to the non-PI problem. We study the role of the prior distribution in the face of non-PI. In particular, we advocate using minimally informative (MI) priors and additional restrictions on the parameter space that can be derived from physical considerations. Using patch clamp (PC) ion-channel measurements as a prototypical time series, we demonstrate Bayesian strategies for alleviating non-PI problems with sharpened prior information.

In Bayesian statistics, the prior can substantially modulate the posterior. We demonstrate that non-PI can be severely harmful when using uniform priors on the rate matrix of HMMs, which are implicitly assumed in ML. We show that MI priors enable meaningful HMM inference with data whose quality can be one to two orders of magnitude worse than required to reach the same accuracy with uniform priors. However, we also demonstrate that non-PI pathologies can persist even with a prior MI. In this case, the MI prior alleviates but does not entirely resolve the problem of improper posteriors. For complex HMMs, stronger prior assumptions are needed to render the posterior proper.

We propose to confine the parameters to a sampling box whose limits are physically reasonable and derived from theory. This fusion of data and physical information allows for meaningful inferences even for the most complex HMM with data of the lowest quality that we tested. However, hard theoretical limits, such as diffusion-limited binding rates, are rarely available. As an alternative, we test a vague prior on the ratios of each pair of binding rates and additionally unbinding rates, thereby softly linking them. This implicitly assumes finite cooperativity and introduces a bias towards non-cooperativity. However, in contrast to the standard practice of choosing equal chemical rates, which supposes strict non-cooperativity, this additional prior still allows for cooperativity. Despite its vagueness, our prior renders the posterior either proper in a strict sense or sufficiently proper for all data sets we considered without imposing the assumption of non-cooperativity. Hence, our approach can infer how likely different degrees of cooperativity are. Combining theoretical upper limits and vague finite cooperativity assumptions dramatically improves inferences.