We describe and illustrate Bayesian inference in models for density estimation using mixtures of Dirichlet processes. These models provide natural settings for density estimation and are exemplified by special cases where data are modeled as a sample from mixtures of normal distributions. Efficient simulation methods are used to approximate various prior, posterior, and predictive distributions. This allows for direct inference on a variety of practical issues, including problems of local versus global smoothing, uncertainty about density estimates, assessment of modality, and the inference on the numbers of components. Also, convergence results are established for a general class of normal mixture models.

https://stat.duke.edu/~scs/Courses/Stat376/Papers/DirichletProc/EscobarWestJASA1995.pdf

Bayesian Density Estimation and Inference Using Mixtures

Great strides in the analysis of survival data using Bayesian methods have been made in the past ten years due to advances in Bayesian computation and the feasibility of such methods. In this chapter, we review Bayesian advances in survival analysis and discuss the various semiparametric modeling techniques that are now commonly used. We review parametric and semiparametric approaches to Bayesian survival analysis, with a focus on proportional hazards models. Reference to other types of models are also given.


Keywords:

beta process;
Cox model;
Dirichlet process;
gamma process;
Gibbs sampling;
piecewise exponential model;
Weibull model

/pdf/bayesian-survival-analysis-4yeebbv71t.pdf

Bayesian Survival Analysis

In this article, the Dirichlet process prior is used to provide a nonparametric Bayesian estimate of a vector of normal means. In the past there have been computational difficulties with this model. This article solves the computational difficulties by developing a “Gibbs sampler” algorithm. The estimator developed in this article is then compared to parametric empirical Bayes estimators (PEB) and nonparametric empirical Bayes estimators (NPEB) in a Monte Carlo study. The Monte Carlo study demonstrates that in some conditions the PEB is better than the NPEB and in other conditions the NPEB is better than the PEB. The Monte Carlo study also shows that the estimator developed in this article produces estimates that are about as good as the PEB when the PEB is better and produces estimates that are as good as the NPEB estimator when that method is better.

https://www.jstor.org/stable/pdfplus/2291223.pdf

Estimating Normal Means with a Dirichlet Process Prior

Weak Convergence and Empirical Processes - A. W. van der Vaart; J. A. Wellner.

This important reference/text focuses on the theory of reduction of a Bayesian experiment considered as a (unique) probability measure on a product space (parameter space x sample space).Treating the basic model and its essential properties, it comprehensively examines sufficiency, including its applications to identification and to comparison of models, as xwll as ancillarity, with its application to exogeneity.

Elements of Bayesian Statistics

The authors give the exact asymptotic behaviour of the expected average absolute error of a beta kernel density estimator proposed by Chen (1999). They also prove the uniform weak consistency of this estimator for the class of continuous densities.

Consistency of the beta kernel density function estimator

Nonparametric estimation for an unknown probability density function f with a known compact support [0, 1] not necessarily bounded at x=0 is considered. For such class of density functions, we consider the Bernstein estimator. The uniform weak consistency and the uniform strong consistency on each compact I in (0, 1) are established for the Bernstein estimator. We prove also the almost sure convergence to infinity at x=0 of the Bernstein estimator when the density function f is unbounded at x=0. To select the optimal bandwidth parameter of the Bernstein estimator, the least squares cross-validation and the likelihood cross-validation methods are developed.

Bernstein estimator for unbounded density function

A model for competing (resp complementary) risks survival data where the failure time can be left (resp right) censored is proposed Product-limit estimators for the survival functions of the individual risks are derived We deduce the strong convergence of our estimators on the whole real half-line without any additional assumptions and their asymptotic normality under conditions concerning only the observed distribution When the observations are generated according to the double censoring model introduced by Turnbull, the product-limit estimators represent upper and lower bounds for Turnbull's estimator

/pdf/product-limit-estimators-of-the-survival-function-with-twice-4zh1jwsomi.pdf

Product-limit estimators of the survival function with twice censored data

In this paper, we study the identification of a particular case of the 3PL model, namely when the discrimination parameters are all constant and equal to 1. We term this model, 1PL-G model. The identification analysis is performed under three different specifications. The first specification considers the abilities as unknown parameters. It is proved that the item parameters and the abilities are identified if a difficulty parameter and a guessing parameter are fixed at zero. The second specification assumes that the abilities are mutually independent and identically distributed according to a distribution known up to the scale parameter. It is shown that the item parameters and the scale parameter are identified if a guessing parameter is fixed at zero. The third specification corresponds to a semi-parametric 1PL-G model, where the distribution G generating the abilities is a parameter of interest. It is not only shown that, after fixing a difficulty parameter and a guessing parameter at zero, the item parameters are identified, but also that under those restrictions the distribution G is not identified. It is finally shown that, after introducing two identification restrictions, either on the distribution G or on the item parameters, the distribution G and the item parameters are identified provided an infinite quantity of items is available.

Jean-Marie Rolin

Papers

Elements of Bayesian Statistics

Consistency of the beta kernel density function estimator

Bernstein estimator for unbounded density function

Product-limit estimators of the survival function with twice censored data

Identification of the 1PL model with guessing parameter: parametric and semi-parametric results.