76 research outputs found
Particle filter-based Gaussian process optimisation for parameter inference
We propose a novel method for maximum likelihood-based parameter inference in
nonlinear and/or non-Gaussian state space models. The method is an iterative
procedure with three steps. At each iteration a particle filter is used to
estimate the value of the log-likelihood function at the current parameter
iterate. Using these log-likelihood estimates, a surrogate objective function
is created by utilizing a Gaussian process model. Finally, we use a heuristic
procedure to obtain a revised parameter iterate, providing an automatic
trade-off between exploration and exploitation of the surrogate model. The
method is profiled on two state space models with good performance both
considering accuracy and computational cost.Comment: Accepted for publication in proceedings of the 19th World Congress of
the International Federation of Automatic Control (IFAC), Cape Town, South
Africa, August 2014. 6 pages, 4 figure
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 -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
- …