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 $(\mu_n)_{n ≥ 0}$ on a Polish space $\mathbb{X}$, the normalized sequence $( \mu_n / \mu_n (\mathbb{X}) )_{n \ge 0}$ agrees with the marginal predictive distributions of some random process $(X_n)_{n \ge 1}$. Moreover, $\mu_n = \mu_{n − 1} + R_{X_n}, \ n \ge 1$, where $x \mapsto R_x$ is a random transition kernel on $\mathbb{X}$; thus, if $\mu_{n − 1}$ represents the contents of an urn, then X n denotes the color of the ball drawn with distribution $\mu_{n − 1} / \mu_{n − 1}(\mathbb{X})$ and $R_{X_{n}}$ - the subsequent reinforcement. In the case $R_{X_{n}} = W_n\delta_{X_n}$, for some non-negative random weights $W_1, \ W_2, \$ ... , the process $( X_n )_{n \ge 1}$ 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 $( X_n )_{n \ge 1}$ 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