67 research outputs found
An example and some questions fo Bayesian nonparametric statistics, from a predictive point of view
In this note we raise some questions for Bayesian nonparametric statistics starting from an example. The problems is described by Coram an Diaconis (2001) and regards studying, by probabilistic techniques, the correspondence between the eigenvalues of random unitary matrices and the complex zeros of Riemann's zeta function. This is an intriguing problem, and we will only underline some of the questions which might arise if we look at it from a (non parametric) Bayesian point of view.
State Space Models in R
We give an overview of some of the software tools available in R, either as built- in functions or contributed packages, for the analysis of state space models. Several illustrative examples are included, covering constant and time-varying models for both univariate and multivariate time series. Maximum likelihood and Bayesian methods to obtain parameter estimates are considered.
Bayes and empirical Bayes: do they merge?
Bayesian inference is attractive for its coherence and good frequentist
properties. However, it is a common experience that eliciting a honest prior
may be difficult and, in practice, people often take an {\em empirical Bayes}
approach, plugging empirical estimates of the prior hyperparameters into the
posterior distribution. Even if not rigorously justified, the underlying idea
is that, when the sample size is large, empirical Bayes leads to "similar"
inferential answers. Yet, precise mathematical results seem to be missing. In
this work, we give a more rigorous justification in terms of merging of Bayes
and empirical Bayes posterior distributions. We consider two notions of
merging: Bayesian weak merging and frequentist merging in total variation.
Since weak merging is related to consistency, we provide sufficient conditions
for consistency of empirical Bayes posteriors. Also, we show that, under
regularity conditions, the empirical Bayes procedure asymptotically selects the
value of the hyperparameter for which the prior mostly favors the "truth".
Examples include empirical Bayes density estimation with Dirichlet process
mixtures.Comment: 27 page
Non parametric mixture priors based on an exponential random scheme
We propose a general procedure for constructing nonparametric priors for Bayesian inference. Under very general assumptions,the proposed prior selects absolutely continuous distribution functions, hence it can be useful with continuous data. We use the notion of Feller-type approximation, with a random scheme based on the natural exponential family, in order to construct a large class of distribution functions. We show how one can assign a probability to such a class and discuss the main properties of the proposed prior, named Feller prior. Feller priors are related to mixture models with unknown number of components or, more generally,to mixtures with unknown weight distribution. Two illustrations relative to the estimation of a density and of a mixing distribution are carried out with respect to well known data-set in order to evaluate the performance ofour procedure. Computations are performed using a modified version of an MCMC algorithm which is briefly described.Bernstein Polynomials, density estimation, Feller operators, Hierarchical models, Mixture Models, Non-parametric Bayesian Inference
A Closed-Form Filter for Binary Time Series
Non-Gaussian state-space models arise in several applications. Within this
framework, the binary time series setting is a source of constant interest due
to its relevance in many studies. However, unlike Gaussian state-space models,
where filtering, predictive and smoothing distributions are available in
closed-form, binary state-space models require approximations or sequential
Monte Carlo strategies for inference and prediction. This is due to the
apparent absence of conjugacy between the Gaussian states and the likelihood
induced by the observation equation for the binary data. In this article we
prove that the filtering, predictive and smoothing distributions in dynamic
probit models with Gaussian state variables are, in fact, available and belong
to a class of unified skew-normals (SUN) whose parameters can be updated
recursively in time via analytical expressions. Also the functionals of these
distributions depend on known functions, but their calculation requires
intractable numerical integration. Leveraging the SUN properties, we address
this point via new Monte Carlo methods based on independent and identically
distributed samples from the smoothing distribution, which can naturally be
adapted to the filtering and predictive case, thereby improving
state-of-the-art approximate or sequential Monte Carlo inference in
small-to-moderate dimensional studies. A scalable and optimal particle filter
which exploits the SUN properties is also developed to deal with online
inference in high dimensions. Performance gains over competitors are outlined
in a real-data financial application
Predictive Constructions Based on Measure-Valued Pólya Urn Processes
Measure-valued Pólya urn processes (MVPP) are Markov chains with an additive structure
that serve as an extension of the generalized k-color Pólya urn model towards a continuum of pos-
sible colors. We prove that, for any MVPP on a Polish space , the normalized sequence
agrees with the marginal predictive distributions of some random process .
Moreover, , where is a random transition kernel on ; thus, if
represents the contents of an urn, then X n denotes the color of the ball drawn with distribution
and - the subsequent reinforcement. In the case , for some
non-negative random weights ... , the process is better understood as a randomly reinforced extension of Blackwell and MacQueen’s Pólya sequence. We study the asymptotic properties
of the predictive distributions and the empirical frequencies of under different assumptions
on the weights. We also investigate a generalization of the above models via a randomization of the
law of the reinforcement
A nonparametric latent space model for dynamic relational networks
Recent years have seen a growing interest in the study of social networks and relational data and, in particular, of their evolution over time. In the context of static networks, a commonly used statistical model defines a latent social space and assumes the relationship between two actors to be determined by the distance between them in such latent space. In this manner, it is possible to introduce additionalinformation about each actor and to quantify the residual dependence through a row-column exchangeability assumption on the adjacency matrix associated to the error terms of the model. The present paper analyzes the behavior of the stochastic model given by changes in a ''global sociability'' parameter which describes the dispersion of the residuals of the positions of the actors in the latent space. This justifies the definition of a Bayesian model for dynamic networks which extends the latent space representation through an infinite hidden Markov model on such positions
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