MCMC implementation for Bayesian hidden semi-Markov models with illustrative applications

Abstract

Hidden Markov models (HMMs) are flexible, wellestablished models useful in a diverse range of applications. However, one potential limitation of such models lies in their inability to explicitly structure the holding times of each hidden state. Hidden semi-Markov models (HSMMs) are more useful in the latter respect as they incorporate additional temporal structure by explicit modelling of the holding times. However, HSMMs have generally received less attention in the literature, mainly due to their intensive computational requirements. Here a Bayesian implementation of HSMMs is presented. Recursive algorithms are proposed in conjunction with Metropolis-Hastings in such a way as to avoid sampling from the distribution of the hidden state sequence in the MCMC sampler. This provides a computationally tractable estimation framework for HSMMs avoiding the limitations associated with the conventional EM algorithm regarding model flexibility. Performance of the proposed implementation is demonstrated through simulation experiments as well as an illustrative application relating to recurrent failures in a network of underground water pipes where random effects are also included into the HSMM to allow for pipe heterogeneity.Keywords HSMM · random effects · MCMC · recursive algorithms · Bayesian model · water pipes.

A Hidden Semi-Markov Model for Characterizing Regime Shifts in Ocean Density Variability

“…If both these distributions are geometric, then this is an HMM where the scalar parameters 0< ϕ S , ϕ T <1 define the 2×2 transition matrix. Note that τ ≠0 so self‐transitions are not allowed, as this would conflict with the very definition of holding times between well‐defined regimes (Economou et al ., ). This implies that neither M S nor M T is a special case of M ST , which is a point that we return to later.…”

“…For a given value of θ , Economou et al . () provided an efficient algorithm for computing the sum in equation and used this to fit HSMMs in a Bayesian setting by using MCMC sampling—and here we adopt this algorithm to fit model M ST as a Bayesian model. This is the so‐called forward algorithm (in HMM jargon), which sequentially computes the probability distribution of the latent states at each time step t given the data up to t , from which the likelihood is computed as a by‐product.…”

“…For estimating the latent state sequence, a backward algorithm can be utilized after model fitting (Economou et al ., ), to compute the probability distribution p { C ( t )| ρ , θ } of the latent states at each time step t given all the data (not just the data up to t ). Monte Carlo sampling can then be used to integrate out θ to obtain the predictive distribution p { C ( t )| ρ } of the latent states at each time step.…”

“…We follow the example in Economou et al . () and assume that the holding time distribution for each state is a zero‐truncated Poisson distribution so thatwith mean holding time . The implications of this modelling choice are checked later, using out‐of‐sample prediction.…”

Modeling Content Structures of Domain-Specific Texts with RUP-HDP-HSMM and Its Applications

“…Based on the results of the forward algorithm, we can sample the whole sequence of hidden states z i in d i at a time. Inspired by recursive algorithm given in [16], we give the modified version using RUP to generate duration times of hidden states for a text. Let…”