140 research outputs found
Discrete chain graph models
The statistical literature discusses different types of Markov properties for
chain graphs that lead to four possible classes of chain graph Markov models.
The different models are rather well understood when the observations are
continuous and multivariate normal, and it is also known that one model class,
referred to as models of LWF (Lauritzen--Wermuth--Frydenberg) or block
concentration type, yields discrete models for categorical data that are
smooth. This paper considers the structural properties of the discrete models
based on the three alternative Markov properties. It is shown by example that
two of the alternative Markov properties can lead to non-smooth models. The
remaining model class, which can be viewed as a discrete version of
multivariate regressions, is proven to comprise only smooth models. The proof
employs a simple change of coordinates that also reveals that the model's
likelihood function is unimodal if the chain components of the graph are
complete sets.Comment: Published in at http://dx.doi.org/10.3150/08-BEJ172 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Computing all roots of the likelihood equations of seemingly unrelated regressions
Seemingly unrelated regressions are statistical regression models based on
the Gaussian distribution. They are popular in econometrics but also arise in
graphical modeling of multivariate dependencies. In maximum likelihood
estimation, the parameters of the model are estimated by maximizing the
likelihood function, which maps the parameters to the likelihood of observing
the given data. By transforming this optimization problem into a polynomial
optimization problem, it was recently shown that the likelihood function of a
simple bivariate seemingly unrelated regressions model may have several
stationary points. Thus local maxima may complicate maximum likelihood
estimation. In this paper, we study several more complicated seemingly
unrelated regression models, and show how all stationary points of the
likelihood function can be computed using algebraic geometry.Comment: To appear in the Journal of Symbolic Computation, special issue on
Computational Algebraic Statistics. 11 page
Correction on Moments of minors of Wishart matrices
Correction on Moments of minors of Wishart matrices by M. Drton and A. Goia
(Ann. Statist. 36 (2008) 2261-2283), arXiv:math/0604488Comment: Published in at http://dx.doi.org/10.1214/12-AOS988 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Algebraic statistical models
Many statistical models are algebraic in that they are defined in terms of
polynomial constraints, or in terms of polynomial or rational parametrizations.
The parameter spaces of such models are typically semi-algebraic subsets of the
parameter space of a reference model with nice properties, such as for example
a regular exponential family. This observation leads to the definition of an
`algebraic exponential family'. This new definition provides a unified
framework for the study of statistical models with algebraic structure. In this
paper we review the ingredients to this definition and illustrate in examples
how computational algebraic geometry can be used to solve problems arising in
statistical inference in algebraic models
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
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