1,967 research outputs found
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
Equations of States in Statistical Learning for a Nonparametrizable and Regular Case
Many learning machines that have hierarchical structure or hidden variables
are now being used in information science, artificial intelligence, and
bioinformatics. However, several learning machines used in such fields are not
regular but singular statistical models, hence their generalization performance
is still left unknown. To overcome these problems, in the previous papers, we
proved new equations in statistical learning, by which we can estimate the
Bayes generalization loss from the Bayes training loss and the functional
variance, on the condition that the true distribution is a singularity
contained in a learning machine. In this paper, we prove that the same
equations hold even if a true distribution is not contained in a parametric
model. Also we prove that, the proposed equations in a regular case are
asymptotically equivalent to the Takeuchi information criterion. Therefore, the
proposed equations are always applicable without any condition on the unknown
true distribution
Incoherent boundary conditions and metastates
In this contribution we discuss the role which incoherent boundary conditions
can play in the study of phase transitions. This is a question of particular
relevance for the analysis of disordered systems, and in particular of spin
glasses. For the moment our mathematical results only apply to ferromagnetic
models which have an exact symmetry between low-temperature phases. We give a
survey of these results and discuss possibilities to extend them to some
situations where many pure states can coexist. An idea of the proofs as well as
the reformulation of our results in the language of Newman-Stein metastates are
also presented.Comment: Published at http://dx.doi.org/10.1214/074921706000000176 in the IMS
Lecture Notes--Monograph Series
(http://www.imstat.org/publications/lecnotes.htm) by the Institute of
Mathematical Statistics (http://www.imstat.org
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