8,346 research outputs found
On Particle Learning
This document is the aggregation of six discussions of Lopes et al. (2010)
that we submitted to the proceedings of the Ninth Valencia Meeting, held in
Benidorm, Spain, on June 3-8, 2010, in conjunction with Hedibert Lopes' talk at
this meeting, and of a further discussion of the rejoinder by Lopes et al.
(2010). The main point in those discussions is the potential for degeneracy in
the particle learning methodology, related with the exponential forgetting of
the past simulations. We illustrate in particular the resulting difficulties in
the case of mixtures.Comment: 14 pages, 9 figures, discussions on the invited paper of Lopes,
Carvalho, Johannes, and Polson, for the Ninth Valencia International Meeting
on Bayesian Statistics, held in Benidorm, Spain, on June 3-8, 2010. To appear
in Bayesian Statistics 9, Oxford University Press (except for the final
discussion
A Bayesian information criterion for singular models
We consider approximate Bayesian model choice for model selection problems
that involve models whose Fisher-information matrices may fail to be invertible
along other competing submodels. Such singular models do not obey the
regularity conditions underlying the derivation of Schwarz's Bayesian
information criterion (BIC) and the penalty structure in BIC generally does not
reflect the frequentist large-sample behavior of their marginal likelihood.
While large-sample theory for the marginal likelihood of singular models has
been developed recently, the resulting approximations depend on the true
parameter value and lead to a paradox of circular reasoning. Guided by examples
such as determining the number of components of mixture models, the number of
factors in latent factor models or the rank in reduced-rank regression, we
propose a resolution to this paradox and give a practical extension of BIC for
singular model selection problems
Local mixture models of exponential families
Exponential families are the workhorses of parametric modelling theory. One
reason for their popularity is their associated inference theory, which is very
clean, both from a theoretical and a computational point of view. One way in
which this set of tools can be enriched in a natural and interpretable way is
through mixing. This paper develops and applies the idea of local mixture
modelling to exponential families. It shows that the highly interpretable and
flexible models which result have enough structure to retain the attractive
inferential properties of exponential families. In particular, results on
identification, parameter orthogonality and log-concavity of the likelihood are
proved.Comment: Published at http://dx.doi.org/10.3150/07-BEJ6170 in the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Mixtures of g-priors in Generalized Linear Models
Mixtures of Zellner's g-priors have been studied extensively in linear models
and have been shown to have numerous desirable properties for Bayesian variable
selection and model averaging. Several extensions of g-priors to Generalized
Linear Models (GLMs) have been proposed in the literature; however, the choice
of prior distribution of g and resulting properties for inference have received
considerably less attention. In this paper, we unify mixtures of g-priors in
GLMs by assigning the truncated Compound Confluent Hypergeometric (tCCH)
distribution to 1/(1 + g), which encompasses as special cases several mixtures
of g-priors in the literature, such as the hyper-g, Beta-prime, truncated
Gamma, incomplete inverse-Gamma, benchmark, robust, hyper-g/n, and intrinsic
priors. Through an integrated Laplace approximation, the posterior distribution
of 1/(1 + g) is in turn a tCCH distribution, and approximate marginal
likelihoods are thus available analytically, leading to "Compound
Hypergeometric Information Criteria" for model selection. We discuss the local
geometric properties of the g-prior in GLMs and show how the desiderata for
model selection proposed by Bayarri et al, such as asymptotic model selection
consistency, intrinsic consistency, and measurement invariance may be used to
justify the prior and specific choices of the hyper parameters. We illustrate
inference using these priors and contrast them to other approaches via
simulation and real data examples. The methodology is implemented in the R
package BAS and freely available on CRAN
Model selection in High-Dimensions: A Quadratic-risk based approach
In this article we propose a general class of risk measures which can be used
for data based evaluation of parametric models. The loss function is defined as
generalized quadratic distance between the true density and the proposed model.
These distances are characterized by a simple quadratic form structure that is
adaptable through the choice of a nonnegative definite kernel and a bandwidth
parameter. Using asymptotic results for the quadratic distances we build a
quick-to-compute approximation for the risk function. Its derivation is
analogous to the Akaike Information Criterion (AIC), but unlike AIC, the
quadratic risk is a global comparison tool. The method does not require
resampling, a great advantage when point estimators are expensive to compute.
The method is illustrated using the problem of selecting the number of
components in a mixture model, where it is shown that, by using an appropriate
kernel, the method is computationally straightforward in arbitrarily high data
dimensions. In this same context it is shown that the method has some clear
advantages over AIC and BIC.Comment: Updated with reviewer suggestion
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