6,104 research outputs found
Objective prior for the number of degrees of freedom of a t distribution
No full textIn this paper, we construct an objective prior for the degrees of freedom of a t distribution, when the parameter is taken to be discrete. This parameter is typically problematic to estimate and a problem in objective Bayesian inference since improper priors lead to improper posteriors, whilst proper priors may dom- inate the data likelihood. We find an objective criterion, based on loss functions, instead of trying to define objective probabilities directly. Truncating the prior on the degrees of freedom is necessary, as the t distribution, above a certain number of degrees of freedom, becomes the normal distribution. The defined prior is tested in simulation scenarios, including linear regression with t-distributed errors, and on real data: the daily returns of the closing Dow Jones index over a period of 98 days
Contributions to the Dirichlet process and related classes of random probability measures
No full textOn rates of convergence for posterior distributions in infinite-dimensional models
No full textThis paper introduces a new approach to the study of rates of convergence for posterior distributions. It is a natural extension of a recent approach to the study of Bayesian consistency. In particular, we improve on current rates of convergence for models including the mixture of Dirichlet process model and the random Bernstein polynomial model
A Nonparametric Model for Stationary Time Series
No full textStationary processes are a natural choice as statistical models for time series data, owing to their good estimating properties. In practice, however, alternative models are often proposed that sacrifice stationarity in favour of the greater modelling flexibility required by many real-life applications. We present a family of time-homogeneous Markov processes with nonparametric stationary densities, which retain the desirable statistical properties for inference, while achieving substantial modelling flexibility, matching those achievable with certain non-stationary models. A latent extension of the model enables exact inference through a trans-dimensional Markov chain Monte Carlo method. Numerical illustrations are presented
On a Gibbs sampler based random process in Bayesian nonparametrics
No full textWe define and investigate a new class of measure-valued Markov chains by resorting to ideas formulated in Bayesian nonparametrics related to the Dirichlet process and the Gibbs sampler. Dependent random prob- ability measures in this class are shown to be stationary and ergodic with respect to the law of a Dirichlet process and to converge in distribution to the neutral diffusion model
Bayesian nonparametric estimators derived from conditional Gibbs structures
No full textWe consider discrete nonparametric priors which induce Gibbs-type exchangeable random partitions and investigate their posterior behavior in detail. In particular, we deduce conditi onal distributions and the corresponding Bayesian nonparametric estimators, which can be readily exploited for predicting various features of additional samples. The results provide useful tools for genomic applications where prediction of future outcomes is require
Countable representation for infinite-dimensional diffusions derived from the two parameter Poisson Dirichlet process
No full textThis paper provides a countable representation for a class of infinite-dimensional diffusions which extends the infinitely-many-neutral-alleles model and is related to the two-parameter Poisson-Dirichlet process. By means of Gibbs sampling procedures, we define a reversible Moran-type population process. The associated process of ranked relative frequencies of types is shown to converge in distribution to the two-parameter family of diffusions, which is stationary and ergodic with respect to the two-parameter Poisson-Dirichlet distribution. The construction provides interpretation for the limiting process in terms of individual dynamics
Bayesian Estimation of the Discrepancy with Misspecified Parametric Models
No full textWe study a Bayesian model where we have made specific requests about the parameter values to be estimated. The aim is to find the parameter of a parametric family which minimizes a distance to the data generating density and then to estimate the discrepancy using nonparametric methods. We illustrate how coherent updating can proceed given that the standard Bayesian posterior from an unidentifiable model is inappropriate. Our updating is performed using Markov Chain Monte Carlo methods and in particular a novel method for dealing with intractable normalizing constants is required. Illustrations using synthetic data are provided.European Research Council (ERC) through StG "N-BNP" 306406Regione PiemonteMathematic
A nonparametric model for stationary time series
No full textWe present a family of autoregressive models with nonparametric stationary and transition densities, which achieve substantial modelling flexibility while retaining desirable statistical properties for inference. Posterior simulation involves an intractable normalizing constant; we therefore present a latent extension of the model which enables exact inference through a trans-dimensional MCMC method. We argue the capacity of this family of models to capture time homogeneous transition mechanisms, making them a powerful tool for predictive inference even when the process generating the data does not have a stationary density. Numerical illustrations are presented
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