5,934 research outputs found
A Tractable State-Space Model for Symmetric Positive-Definite Matrices
Bayesian analysis of state-space models includes computing the posterior
distribution of the system's parameters as well as filtering, smoothing, and
predicting the system's latent states. When the latent states wander around
there are several well-known modeling components and
computational tools that may be profitably combined to achieve these tasks.
However, there are scenarios, like tracking an object in a video or tracking a
covariance matrix of financial assets returns, when the latent states are
restricted to a curve within and these models and tools do not
immediately apply. Within this constrained setting, most work has focused on
filtering and less attention has been paid to the other aspects of Bayesian
state-space inference, which tend to be more challenging. To that end, we
present a state-space model whose latent states take values on the manifold of
symmetric positive-definite matrices and for which one may easily compute the
posterior distribution of the latent states and the system's parameters, in
addition to filtered distributions and one-step ahead predictions. Deploying
the model within the context of finance, we show how one can use realized
covariance matrices as data to predict latent time-varying covariance matrices.
This approach out-performs factor stochastic volatility.Comment: 22 pages: 16 pages main manuscript, 4 pages appendix, 2 pages
reference
Flexible covariance estimation in graphical Gaussian models
In this paper, we propose a class of Bayes estimators for the covariance
matrix of graphical Gaussian models Markov with respect to a decomposable graph
. Working with the family defined by Letac and Massam [Ann.
Statist. 35 (2007) 1278--1323] we derive closed-form expressions for Bayes
estimators under the entropy and squared-error losses. The family
includes the classical inverse of the hyper inverse Wishart but has many more
shape parameters, thus allowing for flexibility in differentially shrinking
various parts of the covariance matrix. Moreover, using this family avoids
recourse to MCMC, often infeasible in high-dimensional problems. We illustrate
the performance of our estimators through a collection of numerical examples
where we explore frequentist risk properties and the efficacy of graphs in the
estimation of high-dimensional covariance structures.Comment: Published in at http://dx.doi.org/10.1214/08-AOS619 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Particle Learning and Smoothing
Particle learning (PL) provides state filtering, sequential parameter
learning and smoothing in a general class of state space models. Our approach
extends existing particle methods by incorporating the estimation of static
parameters via a fully-adapted filter that utilizes conditional sufficient
statistics for parameters and/or states as particles. State smoothing in the
presence of parameter uncertainty is also solved as a by-product of PL. In a
number of examples, we show that PL outperforms existing particle filtering
alternatives and proves to be a competitor to MCMC.Comment: Published in at http://dx.doi.org/10.1214/10-STS325 the Statistical
Science (http://www.imstat.org/sts/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Particle Learning for General Mixtures
This paper develops particle learning (PL) methods for the estimation of general mixture models. The approach is distinguished from alternative particle filtering methods in two major ways. First, each iteration begins by resampling particles according to posterior predictive probability, leading to a more efficient set for propagation. Second, each particle tracks only the "essential state vector" thus leading to reduced dimensional inference. In addition, we describe how the approach will apply to more general mixture models of current interest in the literature; it is hoped that this will inspire a greater number of researchers to adopt sequential Monte Carlo methods for fitting their sophisticated mixture based models. Finally, we show that PL leads to straight forward tools for marginal likelihood calculation and posterior cluster allocation.Business Administratio
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