Abstract not found

In Bayesian inference, a Bayes factor is defined as the ratio of posterior odds versus prior odds where posterior odds is simply a ratio of the normalizing constants of two posterior densities. In many practical problems, the two posteriors have different dimensions. For such cases, the current Monte Carlo methods such as the bridge sampling method (Meng and Wong 1996), the path sampling method (Gelman and Meng 1994), and the ratio importance sampling method (Chen and Shao 1994) cannot directly be applied. In this article, we extend importance sampling, bridge sampling, and ratio importance sampling to problems of different dimensions. Then we find global optimal importance sampling, bridge sampling, and ratio importance sampling in the sense of minimizing asymptotic relative mean-square errors of estimators. Implementation algorithms, which can asymptotically achieve the optimal simulation errors, are developed and two illustrative examples are also provided.

We develop computational procedures for a class of Bayesian nonparametric and semiparametric multiplicative intensity models incorporating kernel mixtures of spatial weighted gamma measures. A key feature of our approach is that explicit expressions for posterior distributions of these models share many common structural features with the posterior distributions of Bayesian hierarchical models using the Dirichlet process. Using this fact, along with an approximation for the weighted gamma process, we show that with some care, one can adapt efficient algorithms used for the Dirichlet process to this setting. We discuss blocked Gibbs sampling procedures and Pólya urn Gibbs samplers. We illustrate our methods with applications to proportional hazard models, Poisson spatial regression models, recurrent events, and panel count data.

... This article proposes a Bayesian approach for addressing hypotheses of this type. Ve reparameterize the Cox model in terms of a cumulative product of parameters having conjugate prior densities, consisting of mixtures of point masses at one and truncated garnma densities. Due to the structure of the model, posterior computation can proceed via a simple and efficient Gibbs sampling algorithm. Posterior probabilities for the global null hypothesis and sub-hypotheses coinparing the hazards fbr specific groups eau be calcnlated directly from the output of a single Gibbs ('.hain. The approach allows for level sets across which a predictor has no effect. Generalizations to multiple predictors are described, and the method is applied to a study of emergency medical treatment for stroke.

We investigate the variable selection problem for Cox’s proportional hazards model, and propose a unified model selection and estimation procedure with desired theoretical properties and computational convenience. The new method is based on a penalized log partial likelihood with the adaptively-weighted L1 penalty on regression coefficients, and is named adaptive-LASSO (ALASSO) estimator. Instead of applying the same penalty to all the coefficients as other shrinkage methods, the ALASSO advocates different penalties for different coefficients: unimportant variables receive larger penalties than important variables. In this way, important variables can be protectively preserved in the model selection process, while unimportant ones are shrunk more towards zero and thus more likely to be dropped from the model. We study the consistency and rate of convergence of the proposed estimator. Further, with proper choice of regularization parameters, we have shown that the ALASSO perform as well as the oracle procedure in variable selection; namely, it works as well as if the correct submodel were known. Another advantage of the ALASSO is its convex optimization form and convenience in implementation. Simulated and real examples show that the ALASSO estimator compares favorably with the LASSO.

Developing regression relationships is a primary inferential activity. We consider such relationships in the context of hierarchical models incorporating linear structure at each stage. Modern statistical work encourages less presumptive, i.e., nonparametric specifications for at least a portion of the modeling. That is, we seek to enrich the class of standard parametric hierarchical models by wandering nonparametrically near (in some sense) the standard class but retaining the linear structure. This enterprise falls within what is referred to as semiparametric modeling. We focus here on nonparametric modeling of monotone functions associated with the model. Such monotone functions arise, for example, as the stochastic mechanism itself using the cumulative distribution function, as the link function in a generalized linear model, as the cumulative hazard function in survival analysis models, and as the calibration function in errors-in-variables models. Nonparametric approaches for mod...

A class of random hazard rates, that is defined as a mixture of an indicator kernel convoluted with a completely random measure, is of interest. We provide an explicit characterization of the posterior distribution of this mixture hazard rate model via a finite mixture of S-paths. A closed and tractable Bayes estimator for the hazard rate is derived to be a finite sum over S-paths. The path characterization or the estimator is proved to be a Rao-Blackwellization of an existing partition characterization or partition-sum estimator. This accentuates the importance of S-path in Bayesian modeling of monotone hazard rates. An efficient Markov chain Monte Carlo (MCMC) method is proposed to approximate this class of estimates. It is shown that S-path characterization also exists in modeling with covariates by a proportional hazard model, and the proposed algorithm again applies. Numerical results of the method are given to demonstrate its practicality and effectiveness.

SUMMARY. Although Cox proportional hazards regression is the default analysis for time to event data, there is typically uncertainty about whether the effects of a predictor are more appropriately characterized by a multiplicative or additive model. To accommodate this uncertainty, we place a model selection prior on the coefficients in an additive-multiplicative hazards model. This prior assigns positive probability, not only to the model that has both additive and multiplicative effects for each predictor, but also to sub-models corresponding to no association, to only additive effects, and to only proportional effects. The additive component of the model is constrained to ensure non-negative hazards, a condition often violated by current methods. After augmenting the data with Poisson latent variables, the prior is conditionally conjugate, and posterior computation can proceed via an efficient Gibbs sampling algorithm. Simulation study results are presented, and the methodology is illustrated using data from the Framingham heart study.

of random hazard rates: an asymptotic