87,316 research outputs found
Efficient simulation and integrated likelihood estimation in state space models
We consider the problem of implementing simple and efficient Markov chain Monte Carlo (MCMC) estimation algorithms for state space models. A conceptually transparent derivation of the posterior distribution of the states is discussed, which also leads to an efficient simulation algorithm that is modular, scalable and widely applicable. We also discuss a simple approach for evaluating the integrated likelihood, defined as the density of the data given the parameters but marginal of the state vector. We show that this high-dimensional integral can be easily evaluated with minimal computational and conceptual difficulty. Two empirical applications in macroeconomics demonstrate that the methods are versatile and computationally undemanding. In one application, involving a time-varying parameter model, we show that the methods allow for efficient handling of large state vectors. In our second application, involving a dynamic factor model, we introduce a new blocking strategy which results in improved MCMC mixing at little cost. The results demonstrate that the framework is simple, flexible and efficient
Auxiliary Likelihood-Based Approximate Bayesian Computation in State Space Models
A computationally simple approach to inference in state space models is
proposed, using approximate Bayesian computation (ABC). ABC avoids evaluation
of an intractable likelihood by matching summary statistics for the observed
data with statistics computed from data simulated from the true process, based
on parameter draws from the prior. Draws that produce a 'match' between
observed and simulated summaries are retained, and used to estimate the
inaccessible posterior. With no reduction to a low-dimensional set of
sufficient statistics being possible in the state space setting, we define the
summaries as the maximum of an auxiliary likelihood function, and thereby
exploit the asymptotic sufficiency of this estimator for the auxiliary
parameter vector. We derive conditions under which this approach - including a
computationally efficient version based on the auxiliary score - achieves
Bayesian consistency. To reduce the well-documented inaccuracy of ABC in
multi-parameter settings, we propose the separate treatment of each parameter
dimension using an integrated likelihood technique. Three stochastic volatility
models for which exact Bayesian inference is either computationally
challenging, or infeasible, are used for illustration. We demonstrate that our
approach compares favorably against an extensive set of approximate and exact
comparators. An empirical illustration completes the paper.Comment: This paper is forthcoming at the Journal of Computational and
Graphical Statistics. It also supersedes the earlier arXiv paper "Approximate
Bayesian Computation in State Space Models" (arXiv:1409.8363
Marginal Likelihood Estimation with the Cross-Entropy Method
We consider an adaptive importance sampling approach to estimating the marginal likelihood, a quantity that is fundamental in Bayesian model comparison and Bayesian model averaging. This approach is motivated by the difficulty of obtaining an accurate estimate through existing algorithms that use Markov chain Monte Carlo (MCMC) draws, where the draws are typically costly to obtain and highly correlated in high-dimensional settings. In contrast, we use the cross-entropy (CE) method, a versatile adaptive Monte Carlo algorithm originally developed for rare-event simulation. The main advantage of the importance sampling approach is that random samples can be obtained from some convenient density with little additional costs. As we are generating independent draws instead of correlated MCMC draws, the increase in simulation effort is much smaller should one wish to reduce the numerical standard error of the estimator. Moreover, the importance density derived via the CE method is in a well-defined sense optimal. We demonstrate the utility of the proposed approach by two empirical applications involving women's labor market participation and U.S. macroeconomic time series. In both applications the proposed CE method compares favorably to existing estimators
Data-driven modelling of biological multi-scale processes
Biological processes involve a variety of spatial and temporal scales. A
holistic understanding of many biological processes therefore requires
multi-scale models which capture the relevant properties on all these scales.
In this manuscript we review mathematical modelling approaches used to describe
the individual spatial scales and how they are integrated into holistic models.
We discuss the relation between spatial and temporal scales and the implication
of that on multi-scale modelling. Based upon this overview over
state-of-the-art modelling approaches, we formulate key challenges in
mathematical and computational modelling of biological multi-scale and
multi-physics processes. In particular, we considered the availability of
analysis tools for multi-scale models and model-based multi-scale data
integration. We provide a compact review of methods for model-based data
integration and model-based hypothesis testing. Furthermore, novel approaches
and recent trends are discussed, including computation time reduction using
reduced order and surrogate models, which contribute to the solution of
inference problems. We conclude the manuscript by providing a few ideas for the
development of tailored multi-scale inference methods.Comment: This manuscript will appear in the Journal of Coupled Systems and
Multiscale Dynamics (American Scientific Publishers
Ensemble Kalman methods for high-dimensional hierarchical dynamic space-time models
We propose a new class of filtering and smoothing methods for inference in
high-dimensional, nonlinear, non-Gaussian, spatio-temporal state-space models.
The main idea is to combine the ensemble Kalman filter and smoother, developed
in the geophysics literature, with state-space algorithms from the statistics
literature. Our algorithms address a variety of estimation scenarios, including
on-line and off-line state and parameter estimation. We take a Bayesian
perspective, for which the goal is to generate samples from the joint posterior
distribution of states and parameters. The key benefit of our approach is the
use of ensemble Kalman methods for dimension reduction, which allows inference
for high-dimensional state vectors. We compare our methods to existing ones,
including ensemble Kalman filters, particle filters, and particle MCMC. Using a
real data example of cloud motion and data simulated under a number of
nonlinear and non-Gaussian scenarios, we show that our approaches outperform
these existing methods
A non-Gaussian continuous state space model for asset degradation
The degradation model plays an essential role in asset life prediction and condition based maintenance. Various degradation models have been proposed. Within these models, the state space model has the ability to combine degradation data and failure event data. The state space model is also an effective approach to deal with the multiple observations and missing data issues. Using the state space degradation model, the deterioration process of assets is presented by a system state process which can be revealed by a sequence of observations. Current research largely assumes that the underlying system development process is discrete in time or states. Although some models have been developed to consider continuous time and space, these state space models are based on the Wiener process with the Gaussian assumption. This paper proposes a Gamma-based state space degradation model in order to remove the Gaussian assumption. Both condition monitoring observations and failure events are considered in the model so as to improve the accuracy of asset life prediction. A simulation study is carried out to illustrate the application procedure of the proposed model
Latent Gaussian modeling and INLA: A review with focus on space-time applications
Bayesian hierarchical models with latent Gaussian layers have proven very
flexible in capturing complex stochastic behavior and hierarchical structures
in high-dimensional spatial and spatio-temporal data. Whereas simulation-based
Bayesian inference through Markov Chain Monte Carlo may be hampered by slow
convergence and numerical instabilities, the inferential framework of
Integrated Nested Laplace Approximation (INLA) is capable to provide accurate
and relatively fast analytical approximations to posterior quantities of
interest. It heavily relies on the use of Gauss-Markov dependence structures to
avoid the numerical bottleneck of high-dimensional nonsparse matrix
computations. With a view towards space-time applications, we here review the
principal theoretical concepts, model classes and inference tools within the
INLA framework. Important elements to construct space-time models are certain
spatial Mat\'ern-like Gauss-Markov random fields, obtained as approximate
solutions to a stochastic partial differential equation. Efficient
implementation of statistical inference tools for a large variety of models is
available through the INLA package of the R software. To showcase the practical
use of R-INLA and to illustrate its principal commands and syntax, a
comprehensive simulation experiment is presented using simulated non Gaussian
space-time count data with a first-order autoregressive dependence structure in
time
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