43,081 research outputs found
Sequential Bayesian inference for implicit hidden Markov models and current limitations
Hidden Markov models can describe time series arising in various fields of
science, by treating the data as noisy measurements of an arbitrarily complex
Markov process. Sequential Monte Carlo (SMC) methods have become standard tools
to estimate the hidden Markov process given the observations and a fixed
parameter value. We review some of the recent developments allowing the
inclusion of parameter uncertainty as well as model uncertainty. The
shortcomings of the currently available methodology are emphasised from an
algorithmic complexity perspective. The statistical objects of interest for
time series analysis are illustrated on a toy "Lotka-Volterra" model used in
population ecology. Some open challenges are discussed regarding the
scalability of the reviewed methodology to longer time series,
higher-dimensional state spaces and more flexible models.Comment: Review article written for ESAIM: proceedings and surveys. 25 pages,
10 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
Langevin and Hamiltonian based Sequential MCMC for Efficient Bayesian Filtering in High-dimensional Spaces
Nonlinear non-Gaussian state-space models arise in numerous applications in
statistics and signal processing. In this context, one of the most successful
and popular approximation techniques is the Sequential Monte Carlo (SMC)
algorithm, also known as particle filtering. Nevertheless, this method tends to
be inefficient when applied to high dimensional problems. In this paper, we
focus on another class of sequential inference methods, namely the Sequential
Markov Chain Monte Carlo (SMCMC) techniques, which represent a promising
alternative to SMC methods. After providing a unifying framework for the class
of SMCMC approaches, we propose novel efficient strategies based on the
principle of Langevin diffusion and Hamiltonian dynamics in order to cope with
the increasing number of high-dimensional applications. Simulation results show
that the proposed algorithms achieve significantly better performance compared
to existing algorithms
Hamiltonian Monte Carlo Acceleration Using Surrogate Functions with Random Bases
For big data analysis, high computational cost for Bayesian methods often
limits their applications in practice. In recent years, there have been many
attempts to improve computational efficiency of Bayesian inference. Here we
propose an efficient and scalable computational technique for a
state-of-the-art Markov Chain Monte Carlo (MCMC) methods, namely, Hamiltonian
Monte Carlo (HMC). The key idea is to explore and exploit the structure and
regularity in parameter space for the underlying probabilistic model to
construct an effective approximation of its geometric properties. To this end,
we build a surrogate function to approximate the target distribution using
properly chosen random bases and an efficient optimization process. The
resulting method provides a flexible, scalable, and efficient sampling
algorithm, which converges to the correct target distribution. We show that by
choosing the basis functions and optimization process differently, our method
can be related to other approaches for the construction of surrogate functions
such as generalized additive models or Gaussian process models. Experiments
based on simulated and real data show that our approach leads to substantially
more efficient sampling algorithms compared to existing state-of-the art
methods
Variational Hamiltonian Monte Carlo via Score Matching
Traditionally, the field of computational Bayesian statistics has been
divided into two main subfields: variational methods and Markov chain Monte
Carlo (MCMC). In recent years, however, several methods have been proposed
based on combining variational Bayesian inference and MCMC simulation in order
to improve their overall accuracy and computational efficiency. This marriage
of fast evaluation and flexible approximation provides a promising means of
designing scalable Bayesian inference methods. In this paper, we explore the
possibility of incorporating variational approximation into a state-of-the-art
MCMC method, Hamiltonian Monte Carlo (HMC), to reduce the required gradient
computation in the simulation of Hamiltonian flow, which is the bottleneck for
many applications of HMC in big data problems. To this end, we use a {\it
free-form} approximation induced by a fast and flexible surrogate function
based on single-hidden layer feedforward neural networks. The surrogate
provides sufficiently accurate approximation while allowing for fast
exploration of parameter space, resulting in an efficient approximate inference
algorithm. We demonstrate the advantages of our method on both synthetic and
real data problems
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