A moment-matching Ferguson and Klass algorithm

Completely random measures (CRM) represent the key building block of a wide variety of popular stochastic models and play a pivotal role in modern Bayesian Nonparametrics. A popular representation of CRMs as a random series with decreasing jumps is due to Ferguson and Klass (1972). This can immediately be turned into an algorithm for sampling realizations of CRMs or more elaborate models involving transformed CRMs. However, concrete implementation requires to truncate the random series at some threshold resulting in an approximation error. The goal of this paper is to quantify the quality of the approximation by a moment-matching criterion, which consists in evaluating a measure of discrepancy between actual moments and moments based on the simulation output. Seen as a function of the truncation level, the methodology can be used to determine the truncation level needed to reach a certain level of precision. The resulting moment-matching \FK algorithm is then implemented and illustrated on several popular Bayesian nonparametric models.Comment: 24 pages, 6 figures, 5 table

Survival regression models with dependent Bayesian nonparametric priors

We present a novel Bayesian nonparametric model for regression in survival analysis. Our model builds on the classical neutral to the right model of Doksum (1974) and on the Cox proportional hazards model of Kim and Lee (2003). The use of a vector of dependent Bayesian nonparametric priors allows us to efficiently model the hazard as a function of covariates whilst allowing nonproportionality. The model can be seen as having competing latent risks. We characterize the posterior of the underlying dependent vector of completely random measures and study the asymptotic behavior of the model. We show how an MCMC scheme can provide Bayesian inference for posterior means and credible intervals. The method is illustrated using simulated and real data

Full Bayesian inference with hazard mixture models

International audienceBayesian nonparametric inferential procedures based on Markov chain Monte Carlo marginal methods typically yield point estimates in the form of posterior expectations. Though very useful and easy to implement in a variety of statistical problems, these methods may suffer from some limitations if used to estimate non-linear functionals of the posterior distribution. The main goal is to develop a novel methodology that extends a well-established marginal procedure designed for hazard mixture models, in order to draw approximate inference on survival functions that is not limited to the posterior mean but includes, as remarkable examples, credible intervals and median survival time. The proposed approach relies on a characterization of the posterior moments that, in turn, is used to approximate the posterior distribution by means of a technique based on Jacobi polynomials. The inferential performance of this methodology is analyzed by means of an extensive study of simulated data and real data consisting of leukemia remission times. Although tailored to the survival analysis context, the proposed procedure can be adapted to a range of other models for which moments of the posterior distribution can be estimated