Exact inference for a class of non-linear hidden Markov models

Exact inference for hidden Markov models requires the evaluation of all distributions of interest - filtering, prediction, smoothing and likelihood - with a finite computational effort. This article provides sufficient conditions for exact inference for a class of hidden Markov models on general state spaces given a set of discretely collected indirect observations linked non linearly to the signal, and a set of practical algorithms for inference. The conditions we obtain are concerned with the existence of a certain type of dual process, which is an auxiliary process embedded in the time reversal of the signal, that in turn allows to represent the distributions and functions of interest as finite mixtures of elementary densities or products thereof. We describe explicitly how to update recursively the parameters involved, yielding qualitatively similar results to those obtained with Baum--Welch filters on finite state spaces. We then provide practical algorithms for implementing the recursions, as well as approximations thereof via an informed pruning of the mixtures, and we show superior performance to particle filters both in accuracy and computational efficiency. The code for optimal filtering, smoothing and parameter inference is made available in the Julia package DualOptimalFiltering.

[1]  Omiros Papaspiliopoulos,et al.  Optimal filtering and the dual process , 2013, 1305.4571.

[2]  Omiros Papaspiliopoulos,et al.  Conjugacy properties of time-evolving Dirichlet and gamma random measures , 2016, 1607.02896.

[3]  Marco Ferrante,et al.  Finite dimensional filters for nonlinear stochastic difference equations with multiplicative noises , 1998 .

[4]  David B. Dunson,et al.  Bayesian Higher Order Hidden Markov Models , 2018, 1805.12201.

[5]  Marco Ferrante On the existence of finite-dimensional filters in discrete time , 1992 .

[6]  Simo Särkkä,et al.  Bayesian Filtering and Smoothing , 2013, Institute of Mathematical Statistics textbooks.

[7]  John M. Olin Calculating posterior distributions and modal estimates in Markov mixture models , 1996 .

[8]  L. Devroye Non-Uniform Random Variate Generation , 1986 .

[9]  Bani K. Mallick,et al.  Prediction of protein interdomain linker regions by a hidden Markov model , 2005, Bioinform..

[10]  Mathieu Kessler,et al.  Random scale perturbation of an AR(1) process and its properties as a nonlinear explicit filter , 2004 .

[11]  Valentine Genon-Catalot,et al.  A non-linear explicit filter , 2003 .

[12]  S. Ethier,et al.  Markov Processes: Characterization and Convergence , 2005 .

[13]  Fredrik Johansson,et al.  Nemo/Hecke: Computer Algebra and Number Theory Packages for the Julia Programming Language , 2017, ISSAC.

[14]  Yi Li,et al.  Bayesian Hidden Markov Modeling of Array CGH Data , 2008, Journal of the American Statistical Association.

[15]  C. Yau,et al.  Bayesian non‐parametric hidden Markov models with applications in genomics , 2011 .

[16]  Haikady N. Nagaraja,et al.  Inference in Hidden Markov Models , 2006, Technometrics.

[17]  Andrew J. Read,et al.  Hidden Markov models reveal complexity in the diving behaviour of short-finned pilot whales , 2017, Scientific Reports.

[18]  S. Tavaré,et al.  Line-of-descent and genealogical processes, and their applications in population genetics models. , 1984, Theoretical population biology.

[19]  S. Jansen,et al.  On the notion(s) of duality for Markov processes , 2012, 1210.7193.

[20]  Roland Langrock,et al.  Nonparametric inference in hidden Markov models using P‐splines , 2013, Biometrics.

[21]  Jason R. W. Merrick,et al.  A Hellinger distance approach to MCMC diagnostics , 2014 .

[22]  Robert C. Griffiths,et al.  The Transition Function of a Fleming-Viot Process , 1993 .

[23]  David Haussler,et al.  Using Dirichlet Mixture Priors to Derive Hidden Markov Models for Protein Families , 1993, ISMB.

[24]  Fabio Spizzichino,et al.  Sufficient conditions for finite dimensionality of filters in discrete time: a Laplace transform-based approach , 2001 .

[25]  Matteo Ruggiero,et al.  Predictive inference with Fleming–Viot-driven dependent Dirichlet processes , 2020, Bayesian Analysis.

[26]  Paul A. Jenkins,et al.  EXACT SIMULATION OF THE WRIGHT – FISHER DIFFUSION , 2017 .

[27]  Wolfgang J. Runggaldier,et al.  On necessary conditions for the existence of finite-dimensional filters in discrete time , 1990 .

[28]  Christopher Yau,et al.  Statistical Inference in Hidden Markov Models Using k-Segment Constraints , 2013, Journal of the American Statistical Association.

[29]  Valentine Genon-Catalot,et al.  Computable infinite-dimensional filters with applications to discretized diffusion processes , 2005 .

[30]  Song-xi Chen,et al.  Probability Density Function Estimation Using Gamma Kernels , 2000 .

[31]  Sawitzki GÜnther,et al.  Finite dimensional filter systems in discrete time , 1981 .

[32]  R. Griffiths,et al.  Lines of descent in the diffusion approximation of neutral Wright-Fisher models. , 1980, Theoretical population biology.

[33]  Nicolas Chopin,et al.  An Introduction to Sequential Monte Carlo , 2020 .

[34]  Michael I. Jordan,et al.  A Sticky HDP-HMM With Application to Speaker Diarization , 2009, 0905.2592.

[35]  G. Roberts,et al.  Retrospective exact simulation of diffusion sample paths with applications , 2006 .

[36]  James D. Hamilton Analysis of time series subject to changes in regime , 1990 .