01-12-2012, 02:33 PM
Joint Bayesian Model Selection and Estimation of Noisy Sinusoids via Reversible Jump MCMC
1Joint Bayesian Model Selection.pdf (Size: 285.96 KB / Downloads: 33)
Abstract
In this paper, the problem of joint Bayesian model
selection and parameter estimation for sinusoids in white Gaussian
noise is addressed. An original Bayesian model is proposed
that allows us to define a posterior distribution on the parameter
space. All Bayesian inference is then based on this distribution.
Unfortunately, a direct evaluation of this distribution and of
its features, including posterior model probabilities, requires
evaluation of some complicated high-dimensional integrals. We
develop an efficient stochastic algorithm based on reversible jump
Markov chain Monte Carlo methods to perform the Bayesian
computation. A convergence result for this algorithm is established.
In simulation, it appears that the performance of detection
based on posterior model probabilities outperforms conventional
detection schemes.
INTRODUCTION
MODEL selection is a fundamental data analysis task. It
has many applications in various fields of science and
engineering. Over the past two decades, many of these problems
have been addressed using information criteria such as
Akaike information criterion (AIC) [1] or Rissanen’s principle
of minimum description length (MDL) [23]. The widespread
use of these criteria is mainly due to their intrinsic simplicity.
AIC and MDL are applied by evaluating two terms: a data
term that requires the maximization of the likelihood and
a penalty term of the complexity of the model. Within a
Bayesian framework, model selection appears more difficult
as it involves the evaluation of Bayes factors, which typically
requires the computation of high-dimensional integrals with
no closed-form analytical expression. These computational
problems have limited the use of Bayesian model selection,
except for the cases for which asymptotic expansions of the
Bayes factors are valid [5].
BAYESIAN MODEL AND AIMS
We follow a Bayesian approach where the unknowns
and are regarded as being drawn from appropriate prior
distributions. These priors reflect our degree of belief of the
relevant values of the parameters [5]. We first propose a
model that sets up a probability distribution over the space of
possible structures of the signal. Subsequently, we specify the
detection/estimation aims. Finally, we exploit the analytical
properties of the model to obtain an expression, up to a
normalizing constant, of the posterior distribution .
SIMULATION RESULTS
In this section, we present an extensive Monte Carlo study
of the performance of the method and algorithm that we
have proposed to solve the problem of detection/estimation of
sinusoids embedded in noise. It is not possible to theoretically
evaluate such a performance as the quantities required (correct
detection and over and under estimation probabilities) are not
available in closed form. We thus propose a Monte Carlo
simulation study for two experiments.
CONCLUSIONS
In this paper, joint Bayesian model selection and parameter
estimation of sinusoids in white Gaussian noise have been
addressed. An original Bayesian model was proposed that
allows us to define a posterior distribution over the space
of possible structures of the signal. The evaluation of this
posterior distribution and of its features of interest requires numerical
methods. An efficient computational algorithm based
on reversible jump MCMC methods was derived to estimate
this posterior distribution. An extensive simulation study is
carried out, and results show that model selection based on
the posterior model probabilities performs better than
other classical criteria. This method is of great interest when
addressing scenarios for which a low SNR, small sample size,
or closely spaced frequencies are encountered. Of course, in
more favorable cases, computationally cheaper methods are a
good alternative.