Parallel sequential Monte Carlo samplers and estimation of the number of states in a Hidden Markov Model

Abstract

The majority of modelling and inference regarding Hidden Markov Models (HMMs) assumes that the number of underlying states is known a priori. However, this is often not the case and thus determining the appropriate number of underlying states for a HMM is of considerable interest. This paper proposes the use of a parallel Sequential Monte Carlo samplers framework to approximate the posterior distribution of the number of states. This requires no additional computational effort if approximating parameter posteriors conditioned on the number of states is also necessary. The proposed methodology is evaluated on a comprehensive set of simulated data and shown to compare favorably with other state of the art methods. An application to business cycle analysis is also presented.

Beyond Whittle: Nonparametric Correction of a Parametric Likelihood with a Focus on Bayesian Time Series Analysis

“…Because the domain of the parameter space for stationary and causal AR(p)-processes has a rather complicated shape (namely the roots of the characteristic polynomial need to lie outside the unit circle), we use a reparametrization suggested by Nam et al (2014) based on the partial autocorrelation ρ = (ρ 1 , . .…”

The Uncertainty of Storm Season Changes: Quantifying the Uncertainty of Autocovariance Changepoints

“…Typically, in more general time series, this is not the case, and recent methods to estimate H include Robert, Rydén, and Titterington (2000), Mackay (2002), Chopin (2007), and Zhou, Johansen, and Aston (2012). In particular, we advocate the SMC-based method proposed by Nam, Aston, and Johansen (2014), which expands upon the SMC samplers framework outlined in this manuscript to estimate the posterior distribution for the number of states. This requires no additional cost if various parameter posteriors are being approximated, each assuming a different number of states being present.…”

“…It is worth mentioning that even though the order of the HMM needs to be chosen a priori the methodology is reasonably robust to the choice of the order. Indeed, as shown in Nam, Aston, and Johansen (2014), model selection to choose this order can be implemented given the algorithm used in the underlying analysis proposed here. In addition, we assume that the underlying unobserved MC, X k , is first-order Markov, although extensions to an qth-order Markov chain are permitted via the use of embedding arguments.…”