Computational issues in parameter estimation for hidden Markov models with Template Model Builder

TL;DR

This paper enhances parameter estimation for hidden Markov models by integrating Template Model Builder in R, enabling efficient uncertainty quantification and optimizer performance improvements, demonstrated through multiple examples and simulations.

Contribution

It introduces a practical method for confidence interval estimation of smoothing probabilities using TMB, and evaluates optimizer performance, highlighting efficiency gains and potential hybrid approaches.

Findings

TMB enables fast gradient and Hessian computations for HMMs.

Confidence intervals for smoothing probabilities can be obtained without bootstrap.

Using TMB accelerates optimizer convergence and improves accuracy.

Abstract

A popular way to estimate the parameters of a hidden Markov model (HMM) is direct numerical maximization (DNM) of the (log-)likelihood function. The advantages of employing the TMB (Kristensen et al., 2016) framework in R for this purpose were illustrated recently Bacri et al. (2022). In this paper, we present extensions of these results in two directions. First, we present a practical way to obtain uncertainty estimates in form of confidence intervals (CIs) for the so-called smoothing probabilities at moderate computational and programming effort via TMB. Our approach thus permits to avoid computer-intensive bootstrap methods. By means of several examples, we illustrate patterns present for the derived CIs. Secondly, we investigate the performance of popular optimizers available in R when estimating HMMs via DNM. Hereby, our focus lies on the potential benefits of employing TMB. Investigated criteria via a number of simulation studies are convergence speed, accuracy, and the impact of (poor) initial values. Our findings suggest that all optimizers considered benefit in terms of speed from using the gradient supplied by TMB. When supplying both gradient and Hessian from TMB, the number of iterations reduces, suggesting a more efficient convergence to the maximum of the log-likelihood. Last, we briefly point out potential advantages of a hybrid approach.