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Gradient Free Parameter Estimation for Hidden Markov Models with Intractable Likelihoods

Author

Listed:
  • Elena Ehrlich

    (Imperial College London)

  • Ajay Jasra

    (National University of Singapore)

  • Nikolas Kantas

    (National University of Singapore
    University College London)

Abstract

In this article we focus on Maximum Likelihood estimation (MLE) for the static model parameters of hidden Markov models (HMMs). We will consider the case where one cannot or does not want to compute the conditional likelihood density of the observation given the hidden state because of increased computational complexity or analytical intractability. Instead we will assume that one may obtain samples from this conditional likelihood and hence use approximate Bayesian computation (ABC) approximations of the original HMM. Although these ABC approximations will induce a bias, this can be controlled to arbitrary precision via a positive parameter ϵ, so that the bias decreases with decreasing ϵ. We first establish that when using an ABC approximation of the HMM for a fixed batch of data, then the bias of the resulting log- marginal likelihood and its gradient is no worse than $\mathcal{O}(n\epsilon)$ , where n is the total number of data-points. Therefore, when using gradient methods to perform MLE for the ABC approximation of the HMM, one may expect parameter estimates of reasonable accuracy. To compute an estimate of the unknown and fixed model parameters, we propose a gradient approach based on simultaneous perturbation stochastic approximation (SPSA) and Sequential Monte Carlo (SMC) for the ABC approximation of the HMM. The performance of this method is illustrated using two numerical examples.

Suggested Citation

  • Elena Ehrlich & Ajay Jasra & Nikolas Kantas, 2015. "Gradient Free Parameter Estimation for Hidden Markov Models with Intractable Likelihoods," Methodology and Computing in Applied Probability, Springer, vol. 17(2), pages 315-349, June.
  • Handle: RePEc:spr:metcap:v:17:y:2015:i:2:d:10.1007_s11009-013-9357-4
    DOI: 10.1007/s11009-013-9357-4
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    References listed on IDEAS

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    5. Pitt, Michael K., 2002. "Smooth particle filters for likelihood evaluation and maximisation," Economic Research Papers 269464, University of Warwick - Department of Economics.
    6. repec:dau:papers:123456789/5724 is not listed on IDEAS
    7. George Poyiadjis & Arnaud Doucet & Sumeetpal S. Singh, 2011. "Particle approximations of the score and observed information matrix in state space models with application to parameter estimation," Biometrika, Biometrika Trust, vol. 98(1), pages 65-80.
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    Cited by:

    1. Patrick L. McDermott & Christopher K. Wikle & Joshua Millspaugh, 2017. "Hierarchical Nonlinear Spatio-temporal Agent-Based Models for Collective Animal Movement," Journal of Agricultural, Biological and Environmental Statistics, Springer;The International Biometric Society;American Statistical Association, vol. 22(3), pages 294-312, September.
    2. Johan Dahlin & Mattias Villani & Thomas B. Schon, 2015. "Bayesian optimisation for fast approximate inference in state-space models with intractable likelihoods," Papers 1506.06975, arXiv.org, revised Jun 2017.

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