arXiv subjects

Fabrizio Leisen

Publications and source records attributed to Fabrizio Leisen.

46 records · Page 3Linked to original sources

arXiv2015

Compound random measures and their use in Bayesian nonparametrics

A new class of dependent random measures which we call {\it compound random measures} are proposed and the use of normalized versions of these random measures as priors in Bayesian nonparametric mixture models is considered. Their tractability allows the properties of both compound random measures and normalized compound random measures to be derived. In particular, we show how compound random measures can be constructed with gamma, $σ$-stable and generalized gamma process marginals. We also derive several forms of the Laplace exponent and characterize dependence through both the Lévy copula and correlation function. A slice sampler and an augmented Pólya urn scheme sampler are described for posterior inference when a normalized compound random measure is used as the mixing measure in a nonparametric mixture model and a data example is discussed.

stat.ME↗

arXiv2014

Merging exchangeable occupancy models: $\mathcal{M}^{(a)}$- models and relation with the maximum entropy principle

In this paper a new transformation of occupancy models, called merging, is introduced. In particular, it will be studied the effect of merging on a class of occupancy models that was recently introduced in Collet et al (2013). These results have an interesting interpretation in the so-called entropy maximization inference. The last part of the paper is devoted to highlight the impact of our findings in this research area.

math.PR↗

arXiv2014

A Bayesian Beta Markov Random Field Calibration of the Term Structure of Implied Risk Neutral Densities

We build on the work in Fackler and King 1990, and propose a more general calibration model for implied risk neutral densities. Our model allows for the joint calibration of a set of densities at different maturities and dates through a Bayesian dynamic Beta Markov Random Field. Our approach allows for possible time dependence between densities with the same maturity, and for dependence across maturities at the same point in time. This approach to the problem encompasses model flexibility, parameter parsimony and, more importantly, information pooling across densities.

stat.AP↗

arXiv2014

Generalized Species Sampling Priors with Latent Beta reinforcements

Many popular Bayesian nonparametric priors can be characterized in terms of exchangeable species sampling sequences. However, in some applications, exchangeability may not be appropriate. We introduce a {novel and probabilistically coherent family of non-exchangeable species sampling sequences characterized by a tractable predictive probability function with weights driven by a sequence of independent Beta random variables. We compare their theoretical clustering properties with those of the Dirichlet Process and the two parameters Poisson-Dirichlet process. The proposed construction provides a complete characterization of the joint process, differently from existing work. We then propose the use of such process as prior distribution in a hierarchical Bayes modeling framework, and we describe a Markov Chain Monte Carlo sampler for posterior inference. We evaluate the performance of the prior and the robustness of the resulting inference in a simulation study, providing a comparison with popular Dirichlet Processes mixtures and Hidden Markov Models. Finally, we develop an application to the detection of chromosomal aberrations in breast cancer by leveraging array CGH data.

math.ST↗

arXiv2013

New isometry of Krall-Laguerre orthogonal polynomials in martingale spaces

Sets of orthogonal martingales are importants because they can be used as stochastic integrators in a kind of chaotic representation property, see [20]. In this paper, we revisited the problem studied by W. Schoutens in [21], investigating how an inner product derived from an Uvarov transformation of the Laguerre weight function is used in the orthogonalization procedure of a sequence of martingales related to a certain Lévy process, called Teugels Martingales. Since the Uvarov transformation depends by a c<0, we are able to provide infinite sets of strongly orthogonal martingales, each one for every c in (-infty,0). In a similar fashion of [21], we introduce a suitable isometry between the space of polynomials and the space of linear combinations of Teugels martingales as well as the general orthogonalization procedure. Finally, the new construction is applied to the Gamma process.

math.PR↗

arXiv2013

Exchangeable Occupancy Models and Discrete Processes with the Generalized Uniform Order Statistics Property

This work focuses on Exchangeable Occupancy Models (EOM) and their relations with the Uniform Order Statistics Property (UOSP) for point processes in discrete time. As our main purpose, we show how definitions and results presented in Shaked, Spizzichino and Suter (2004) can be unified and generalized in the frame of occupancy models. We first show some general facts about EOM's. Then we introduce a class of EOM's, called $\mathcal{M}^{(a)}$-models, and a concept of generalized Uniform Order Statistics Property in discrete time. For processes with this property, we prove a general characterization result in terms of $\mathcal{M}^{(a)}$-models. Our interest is also focused on properties of closure w.r.t. some natural transformations of EOM's.

math.PR↗

arXiv2011

Beta-Product Poisson-Dirichlet Processes

Time series data may exhibit clustering over time and, in a multiple time series context, the clustering behavior may differ across the series. This paper is motivated by the Bayesian non--parametric modeling of the dependence between the clustering structures and the distributions of different time series. We follow a Dirichlet process mixture approach and introduce a new class of multivariate dependent Dirichlet processes (DDP). The proposed DDP are represented in terms of vector of stick-breaking processes with dependent weights. The weights are beta random vectors that determine different and dependent clustering effects along the dimension of the DDP vector. We discuss some theoretical properties and provide an efficient Monte Carlo Markov Chain algorithm for posterior computation. The effectiveness of the method is illustrated with a simulation study and an application to the United States and the European Union industrial production indexes.

math.ST↗

arXiv2010

Interacting Multiple Try Algorithms with Different Proposal Distributions

We propose a new class of interacting Markov chain Monte Carlo (MCMC) algorithms designed for increasing the efficiency of a modified multiple-try Metropolis (MTM) algorithm. The extension with respect to the existing MCMC literature is twofold. The sampler proposed extends the basic MTM algorithm by allowing different proposal distributions in the multiple-try generation step. We exploit the structure of the MTM algorithm with different proposal distributions to naturally introduce an interacting MTM mechanism (IMTM) that expands the class of population Monte Carlo methods. We show the validity of the algorithm and discuss the choice of the selection weights and of the different proposals. We provide numerical studies which show that the new algorithm can perform better than the basic MTM algorithm and that the interaction mechanism allows the IMTM to efficiently explore the state space.

stat.CO↗

arXiv2010

Limiting behavior of the search cost distribution for the move-to-front rule in the stable case

Move-to-front rule is a heuristic updating a list of n items according to requests. Items are required with unknown probabilities (or popularities). The induced Markov chain is known to be ergodic. One main problem is the study of the distribution of the search cost dened as the position of the required item. Here we first establish the link between two recent papers that both extend results proved by Kingman on the expected stationary search cost. Combining results contained in these papers, we obtain the limiting behavior for any moments of the stationary seach cost as n tends to innity.

math.PR↗

arXiv2008

Conditionally identically distributed species sampling sequences

Conditional identity in distribution (Berti et al. (2004)) is a new type of dependence for random variables, which generalizes the well-known notion of exchangeability. In this paper, a class of random sequences, called Generalized Species Sampling Sequences, is defined and a condition to have conditional identity in distribution is given. Moreover, a class of generalized species sampling sequences that are conditionally identically distributed is introduced and studied: the Generalized Ottawa sequences (GOS). This class contains a '`randomly reinforced'' version of the Pólya urn and of the Blackwell-MacQueen urn scheme. For the empirical means and the predictive means of a GOS, we prove two convergence results toward suitable mixtures of Gaussian distributions. The first one is in the sense of stable convergence and the second one in the sense of almost sure conditional convergence. In the last part of the paper we study the length of the partition induced by a GOS at time $n$, i.e. the random number of distinct values of a GOS until time $n$. Under suitable conditions, we prove a strong law of large numbers and a central limit theorem in the sense of stable convergence. All the given results in the paper are accompanied by some examples.

math.PR↗