Getting Started with Particle Metropolis-Hastings for Inference in Nonlinear Dynamical Models

This tutorial provides a gentle introduction to the particle Metropolis-Hastings (PMH) algorithm for parameter inference in nonlinear state-space models together with a software implementation in the statistical programming language R. We employ a step-by-step approach to develop an implementation of the PMH algorithm (and the particle filter within) together with the reader. This final implementation is also available as the package pmhtutorial in the CRAN repository. Throughout the tutorial, we provide some intuition as to how the algorithm operates and discuss some solutions to problems that might occur in practice. To illustrate the use of PMH, we consider parameter inference in a linear Gaussian state-space model with synthetic data and a nonlinear stochastic volatility model with real-world data.Comment: 41 pages, 7 figures. In press for Journal of Statistical Software. Source code for R, Python and MATLAB available at: https://github.com/compops/pmh-tutoria

Particle Metropolis-Hastings using gradient and Hessian information

Particle Metropolis-Hastings (PMH) allows for Bayesian parameter inference in nonlinear state space models by combining Markov chain Monte Carlo (MCMC) and particle filtering. The latter is used to estimate the intractable likelihood. In its original formulation, PMH makes use of a marginal MCMC proposal for the parameters, typically a Gaussian random walk. However, this can lead to a poor exploration of the parameter space and an inefficient use of the generated particles. We propose a number of alternative versions of PMH that incorporate gradient and Hessian information about the posterior into the proposal. This information is more or less obtained as a byproduct of the likelihood estimation. Indeed, we show how to estimate the required information using a fixed-lag particle smoother, with a computational cost growing linearly in the number of particles. We conclude that the proposed methods can: (i) decrease the length of the burn-in phase, (ii) increase the mixing of the Markov chain at the stationary phase, and (iii) make the proposal distribution scale invariant which simplifies tuning.Comment: 27 pages, 5 figures, 2 tables. The final publication is available at Springer via: http://dx.doi.org/10.1007/s11222-014-9510-

Quasi-Newton particle Metropolis-Hastings

Particle Metropolis-Hastings enables Bayesian parameter inference in general nonlinear state space models (SSMs). However, in many implementations a random walk proposal is used and this can result in poor mixing if not tuned correctly using tedious pilot runs. Therefore, we consider a new proposal inspired by quasi-Newton algorithms that may achieve similar (or better) mixing with less tuning. An advantage compared to other Hessian based proposals, is that it only requires estimates of the gradient of the log-posterior. A possible application is parameter inference in the challenging class of SSMs with intractable likelihoods. We exemplify this application and the benefits of the new proposal by modelling log-returns of future contracts on coffee by a stochastic volatility model with $\alpha$-stable observations.Comment: 23 pages, 5 figures. Accepted for the 17th IFAC Symposium on System Identification (SYSID), Beijing, China, October 201

Newton-based maximum likelihood estimation in nonlinear state space models

Maximum likelihood (ML) estimation using Newton's method in nonlinear state space models (SSMs) is a challenging problem due to the analytical intractability of the log-likelihood and its gradient and Hessian. We estimate the gradient and Hessian using Fisher's identity in combination with a smoothing algorithm. We explore two approximations of the log-likelihood and of the solution of the smoothing problem. The first is a linearization approximation which is computationally cheap, but the accuracy typically varies between models. The second is a sampling approximation which is asymptotically valid for any SSM but is more computationally costly. We demonstrate our approach for ML parameter estimation on simulated data from two different SSMs with encouraging results.Comment: 17 pages, 2 figures. Accepted for the 17th IFAC Symposium on System Identification (SYSID), Beijing, China, October 201

Impact of smoking and preoperative electrophysiology on outcome after open carpal tunnel release

Background: The aim was to evaluate the influence of smoking and preoperative electrophysiology on the outcome of open carpal tunnel release. Methods: This retrospective observational study evaluated the outcome in 493 patients (531 hands) primary operated for carpal tunnel syndrome. Data were collected from medical records, health evaluations, and QuickDASH questionnaires before surgery and 1 year after. Results: Smokers had a higher QuickDASH score preoperatively as well as postoperatively, but the change in total score did not differ. The odds of having a postoperative QuickDASH score >10 were 2.5 times higher in smoking patients than in non-smoking patients. In 124/493 patients (25%), no clinically significant improvement was seen. Normal and extreme preoperative electrophysiology values were associated with higher postoperative scores. No correlation was found between preoperative QuickDASH scores and preoperative electrophysiology values. Conclusions: Smokers with carpal tunnel syndrome experience more symptoms preoperatively. Smokers have remaining symptoms after surgery. There is no correlation between preoperative QuickDASH scores and preoperative electrophysiology values. Patients with normal or near to normal preoperative electrophysiology results have limited improvement after surgery