11 research outputs found
Bayes factors and the geometry of discrete hierarchical loglinear models
A standard tool for model selection in a Bayesian framework is the Bayes
factor which compares the marginal likelihood of the data under two given
different models. In this paper, we consider the class of hierarchical
loglinear models for discrete data given under the form of a contingency table
with multinomial sampling. We assume that the Diaconis-Ylvisaker conjugate
prior is the prior distribution on the loglinear parameters and the uniform is
the prior distribution on the space of models. Under these conditions, the
Bayes factor between two models is a function of their prior and posterior
normalizing constants. These constants are functions of the hyperparameters
which can be interpreted respectively as marginal counts and the
total count of a fictive contingency table.
We study the behaviour of the Bayes factor when tends to zero. In
this study two mathematical objects play a most important role. They are,
first, the interior of the convex hull of the support of the
multinomial distribution for a given hierarchical loglinear model together with
its faces and second, the characteristic function of this convex
set .
We show that, when tends to 0, if the data lies on a face of
of dimension , the Bayes factor behaves like
. This implies in particular that when the data is in
and in , i.e. when equals the dimension of model , the sparser
model is favored, thus confirming the idea of Bayesian regularization.Comment: 37 page
Dirichlet random walks
This article provides tools for the study of the Dirichlet random walk in
. By this we mean the random variable
where is Dirichlet distributed and where
are iid, uniformly distributed on the unit sphere of
and independent of In particular we compute explicitely in
a number of cases the distribution of Some of our results appear already
in the literature, in particular in the papers by G\'erard Le Ca\"{e}r (2010,
2011). In these cases, our proofs are much simpler from the original ones,
since we use a kind of Stieltjes transform of instead of the Laplace
transform: as a consequence the hypergeometric functions replace the Bessel
functions. A crucial ingredient is a particular case of the classical and non
trivial identity, true for : We extend these results to a study of
the limits of the Dirichlet random walks when the number of added terms goes to
infinity, interpreting the results in terms of an integral by a Dirichlet
process. We introduce the ideas of Dirichlet semigroups and of Dirichlet
infinite divisibility and characterize these infinite divisible distributions
in the sense of Dirichlet when they are concentrated on the unit ball of
{4mm}\noindent \textsc{Keywords:} Dirichlet processes, Stieltjes transforms,
random flight, distributions in a ball, hyperuniformity, infinite divisibility
in the sense of Dirichlet.
{4mm}\noindent \textsc{AMS classification}: 60D99, 60F99
The limiting behavior of some infinitely divisible exponential dispersion models
Consider an exponential dispersion model (EDM) generated by a probability on which is infinitely divisible with an unbounded L\'{e}vy
measure . The Jorgensen set (i.e., the dispersion parameter space) is
then , in which case the EDM is characterized by two
parameters: the natural parameter of the associated natural
exponential family and the Jorgensen (or dispersion) parameter . Denote by
the corresponding distribution and let is a r.v.
with distribution . Then if around zero we prove that the limiting law of as
is of a Pareto type (not depending on ) with the
form for and for . Such a result
enables an approximation of the distribution of for relatively small
values of the dispersion parameter of the corresponding EDM. Illustrative
examples are provided.Comment: 8 page
Random walks in the hyperbolic plane and the Minkowski question mark function
Si consideri il gruppo delle matrici invertibili 2x2 con coefficienti interi che agisce sul semipiano superiore del piano complesso mediante le corrispondenti trasformazioni di Mobius. Consideriamo una passeggiata casuale sul piano costruita scegliendo a caso tra nove matrici. Dimostriamo che la camminata casuale converge in legge ad una variabile casuale la cui la distribuzione è collegata in un senso semplice alla funzione di Minkowski ? Varie proprietà di questa legge limite possono essere ottenute da questa analisi.Consider the group of 2x2 invertible matrices with integer coefficients acting on the complex upper half-plane by the corresponding Mobius transformations. We consider a random walk on the plane constructed by choosing at random among nine matrices. We show that the random walk converges in law to a random variable whose distribution is related in a simple way to Minkowski's ? function. Various properties of this limiting law can be obtained from this analysis