A natural conjugate prior for the non-homogeneous poisson process with a power law intensity function

This paper develops a natural conjugate prior for the non-homogeneous Poisson process (NHPP) with a power law intensity function. This prior allows for dependence between the scale factor and the aging rate of the NHPP. The proposed prior has relatively simple closed-form expressions for its moments, facilitating the assessment of prior parameters. The use of this prior in Bayesian estimation is compared to other estimation approaches using Monte Carlo simulation. The results show that Bayesian estimation using the proposed prior generally performs at least as well as either maximum likelihood estimation or Bayesian estimation using independent prior     

A natural conjugate prior for the nonhomogeneous poisson process with an exponential intensity function

This article presents a natural conjugate prior for the nonhomogeneous Poisson process (NHPP) with an exponential intensity function, for modeling the failure rate of repairable systems. The behavior of the conjugate prior distribution with respect to its parameters is studied, and the use of this prior in Bayesian estimation is compared to two other estimation approaches (the use of independent prior distributions, and the bivariate normal distribution). The use of the conjugate prior proposed here facilitates Bayesian statistical analysis of aging. In particular, the proposed prior allows us to explicitly account for dependence between the initial failure rate and the aging rate. This is a significant improvement over the assumptions made in most prior work (either the assumption that the aging rate is known, or the assumption that the initial failure rate and the aging rate are independent). Monte Carlo simulation shows that Bayesian estimation using the proposed prior generally performs at least as well as Bayesian estimation using independent priors for the initial failure rate and the aging rate,except in the case where the prior distribution underestimates both the initial failure rate and the aging rate.     

Image reconstruction by adaptive bayesian classification with a locally proportional prior

This paper concerns the problem of reconstructing images from noisy data by means of Bayesian classification methods. In Klein and Press, 1992, the authors presented a method for reconstructing images called Adaptive Bayesian Classification (ABC). The ABC procedure was shown to preform very well in simulation experiments. The ABC procedure was multistaged; moreover, it involved selecting a prior at Stage n that was the posterior at Stage n - 1. In this paper the authors show that we can improve upon ABC for some problems by modifying the way we take the prior at each stage. The new proposal is to take the prior for the pixel label at each stage as proportional to the number of pixels with that label in a small neighborhood of the pixel. The ABC procedure with a locally proportional prior (ABC/LPP) tends to improve upon the ABC procedure for some problems because the prior in the iterative portion of ABC/LPP is contextual, while that in ABC in non- contextual.     

The Bayesian and frequentist approaches to testing a one-sided hypothesis about a multivariate mean

This paper compares the Bayesian and frequentist approaches to testing a one-sided hypothesis about a multivariate mean. First, this paper proposes a simple way to assign a Bayesian posterior probability to one-sided hypotheses about a multivariate mean. The approach is to use (almost) the exact posterior probability under the assumption that the data has multivariate normal distribution, under either a conjugate prior in large samples or under a vague Jeffreys prior. This is also approximately the Bayesian posterior probability of the hypothesis based on a suitably flat Dirichlet process prior over an unknown distribution generating the data. Then, the Bayesian approach and a frequentist approach to testing the one-sided hypothesis are compared, with results that show a major difference between Bayesian reasoning and frequentist reasoning. The Bayesian posterior probability can be substantially smaller than the frequentist p-value. A class of example is given where the Bayesian posterior probability is basically 0, while the frequentist p-value is basically 1. The Bayesian posterior probability in these examples seems to be more reasonable. Other drawbacks of the frequentist p-value as a measure of whether the one-sided hypothesis is true are also discussed.     

Random variate generation from D-distributions

Within the context of non-parametric Bayesian inference, Dykstra and Laud (1981) define an extended gamma (EG) process and use it as a prior on increasing hazard rates. The attractive features of the extended gamma (EG) process, among them its capability to index distribution functions that are absolutely continuous, are offset by the intractable nature of the computation that needs to be performed. Sampling based approaches such as the Gibbs Sampler can alleviate these difficulties but the EG processes then give rise to the problem of efficient random variate generation from a class of distributions called D-distributions. In this paper, we describe a novel technique for sampling from such distributions, thereby providing an efficient computation procedure for non-parametric Bayesian inference with a rich class of priors for hazard rates.     

Bayesian nonparametric survival analysis for grouped data

A Bayesian nonparametric estimate of the survival distribution is derived under a particular sampling scheme for grouped data that includes the possibility of censoring. The estimate uses the prior information to smooth the data, giving an estimate which is continuous. As special cases survival estimates for life tables are obtained and the estimate of Susarla and Van Ryzin (1976) is derived. As the weight of the prior information tends to zero, the Bayesian estimate reduces to a continuous version of the nonparametric maximum-likelihood estimate. An empirical Bayes modification of the procedure is illustrated on a data set from Cutler and Ederer (1958).     

A note on the finite-dimensional Dirichlet prior

As an approximation to the Dirichlet process which involves the infinite-dimensional distribution, finite-dimensional Dirichlet prior is a widely appreciated method to model the underlying distribution in non parametric Bayesian analysis. In this short note, we present some key characteristics of finite-dimensional Dirichlet process and exploit some important sampling properties which are very useful in Bayesian non parametric/semiparametric analysis.     

Reanalyzing Ultimatum Bargaining—Comparing Nondecreasing Curves Without Shape Constraints

We employ a hierarchical Bayesian method with exchangeable prior distributions to estimate and compare similar nondecreasing response curves. A Dirichlet process distribution is assigned to each of the response curves as a first stage prior. A second stage prior is then used to model the hyperparameters. We define parameters which will be used to compare the response curves. A Markov chain Monte Carlo method is applied to compute the resulting Bayesian estimates. To illustrate the methodology, we re-examine data from an experiment designed to test whether experimenter observation influences the ultimatum game. A major restriction of the original analysis was the shape constraint that the present technique allows us to greatly relax. We also consider independent priors and use Bayes factors to compare various models.     

Bayesian analysis for monotone hazard ratio

We propose a Bayesian approach for estimating the hazard functions under the constraint of a monotone hazard ratio. We construct a model for the monotone hazard ratio utilizing the Cox’s proportional hazards model with a monotone time-dependent coefficient. To reduce computational complexity, we use a signed gamma process prior for the time-dependent coefficient and the Bayesian bootstrap prior for the baseline hazard function. We develope an efficient MCMC algorithm and illustrate the proposed method on simulated and real data sets.     

Characterizing variation of nonparametric random probability measures using the Kullback–Leibler divergence

This work characterizes the dispersion of some popular random probability measures, including the bootstrap, the Bayesian bootstrap, and the Pólya tree prior. This dispersion is measured in terms of the variation of the Kullback–Leibler divergence of a random draw from the process to that of its baseline centring measure. By providing a quantitative expression of this dispersion around the baseline distribution, our work provides insight for comparing different parameterizations of the models and for the setting of prior parameters in applied Bayesian settings. This highlights some limitations of the existing canonical choice of parameter settings in the Pólya tree process.     

Objective Bayesian Analysis for Log-logistic Distribution

In this article, Bayesian approach is applied to estimate the parameters of Log-logistic distribution under reference prior and Jeffreys’ prior. The reference prior is derived and it is found that the reference prior is also a second-order matching priors as for the case of any parameter of interest. The Bayesian estimators cannot be obtained in explicit forms. Metropolis within Gibbs sampling algorithm is used to obtain the Bayesian estimators. The Bayesian estimates are compared with the maximum likelihood estimates via simulation study. A real dataset is considered for illustrative purposes.     

Adaptive Bayesian Procedures Using Random Series Priors

We consider a general class of prior distributions for nonparametric Bayesian estimation which uses finite random series with a random number of terms. A prior is constructed through distributions on the number of basis functions and the associated coefficients. We derive a general result on adaptive posterior contraction rates for all smoothness levels of the target function in the true model by constructing an appropriate ‘sieve’ and applying the general theory of posterior contraction rates. We apply this general result on several statistical problems such as density estimation, various nonparametric regressions, classification, spectral density estimation and functional regression. The prior can be viewed as an alternative to the commonly used Gaussian process prior, but properties of the posterior distribution can be analysed by relatively simpler techniques. An interesting approximation property of B‐spline basis expansion established in this paper allows a canonical choice of prior on coefficients in a random series and allows a simple computational approach without using Markov chain Monte Carlo methods. A simulation study is conducted to show that the accuracy of the Bayesian estimators based on the random series prior and the Gaussian process prior are comparable. We apply the method on Tecator data using functional regression models.     

A Bayesian nonparametric estimator based on left censored data

Summary This paper introduces a Bayesian nonparametric estimator for an unknown distribution function based on left censored observations. Hjort (1990)/Lo (1993) introduced Bayesian nonparametric estimators derived from beta/beta-neutral processes which allow for right censoring. These processes are taken as priors from the class ofneutral to the right processes (Doksum, 1974). The Kaplan-Meier nonparametric product limit estimator can be obtained from these Bayesian nonparametric estimators in the limiting case of a vague prior. The present paper introduces what can be seen as the correspondingleft beta/beta-neutral process prior which allow for left censoring. The Bayesian nonparametyric estimator is obtained as in the corresponding product limit estimator based on left censored data.     

On forecasting with univariate autoregressive processes: a bayesian approach

Using a normal-gamma prior density for the parameters of a p-th order autoregressive process, the Bayesian predictive density of k future observations is derived and it is shown that it is the product of k univariate t densities. Our results are illustrated with one step ahead forecasts employing AR(1) and AR(2) models with a vague prior density for the parameters.     

On a Posterior Predictive Density Sample Size Criterion

Abstract.  Let Ω be a space of densities with respect to some σ -finite measure μ and let Π be a prior distribution having support Ω with respect to some suitable topology. Conditional on f , let X ⁿ  = ( X ₁ ,…, X _n ) be an independent and identically distributed sample of size n from f . This paper introduces a Bayesian non-parametric criterion for sample size determination which is based on the integrated squared distance between posterior predictive densities. An expression for the sample size is obtained when the prior is a Dirichlet mixture of normal densities.     

Reinforced urns and the subdistribution beta‐Stacy process prior for competing risks analysis

In this paper, we introduce the subdistribution beta‐Stacy process, a novel Bayesian nonparametric process prior for subdistribution functions useful for the analysis of competing risks data. In particular, we (i) characterize this process from a predictive perspective by means of an urn model with reinforcement, (ii) show that it is conjugate with respect to right‐censored data, and (iii) highlight its relations with other prior processes for competing risks data. Additionally, we consider the subdistribution beta‐Stacy process prior in a nonparametric regression model for competing risks data, which, contrary to most others available in the literature, is not based on the proportional hazards assumption.     

Bayesian and frequentist prediction limits for the Poisson distribution

Prediction limits for Poisson distribution are useful in real life when predicting the occurrences of some phenomena, for example, the number of infections from a disease per year among school children, or the number of hospitalizations per year among patients with cardiovascular disease. In order to allocate the right resources and to estimate the associated cost, one would want to know the worst (i.e., an upper limit) and the best (i.e., the lower limit) scenarios. Under the Poisson distribution, we construct the optimal frequentist and Bayesian prediction limits, and assess frequentist properties of the Bayesian prediction limits. We show that Bayesian upper prediction limit derived from uniform prior distribution and Bayesian lower prediction limit derived from modified Jeffreys non informative prior coincide with their respective frequentist limits. This is not the case for the Bayesian lower prediction limit derived from a uniform prior and the Bayesian upper prediction limit derived from a modified Jeffreys prior distribution. Furthermore, it is shown that not all Bayesian prediction limits derived from a proper prior can be interpreted in a frequentist context. Using a counterexample, we state a sufficient condition and show that Bayesian prediction limits derived from proper priors satisfying our condition cannot be interpreted in a frequentist context. Analysis of simulated data and data on Atlantic tropical storm occurrences are presented.     

Hybrid Dirichlet mixture models for functional data

Summary.  In functional data analysis, curves or surfaces are observed, up to measurement error, at a finite set of locations, for, say, a sample of n individuals. Often, the curves are homogeneous, except perhaps for individual-specific regions that provide heterogeneous behaviour (e.g. 'damaged' areas of irregular shape on an otherwise smooth surface). Motivated by applications with functional data of this nature, we propose a Bayesian mixture model, with the aim of dimension reduction, by representing the sample of n curves through a smaller set of canonical curves. We propose a novel prior on the space of probability measures for a random curve which extends the popular Dirichlet priors by allowing local clustering: non-homogeneous portions of a curve can be allocated to different clusters and the n individual curves can be represented as recombinations (hybrids) of a few canonical curves. More precisely, the prior proposed envisions a conceptual hidden factor with k -levels that acts locally on each curve. We discuss several models incorporating this prior and illustrate its performance with simulated and real data sets. We examine theoretical properties of the proposed finite hybrid Dirichlet mixtures, specifically, their behaviour as the number of the mixture components goes to ∞ and their connection with Dirichlet process mixtures.     

Bayesian Estimation of the Change Point Using X¯ Control Chart

The process personnel always seek the opportunity to improve the processes. One of the essential steps for process improvement is to quickly recognize the starting time or the change point of a process disturbance. The proposed approach combines the X¯ control chart with the Bayesian estimation technique. We show that the control chart has some information about the change point and this information can be used to make an informative prior. Then two Bayes estimators corresponding to the informative and a non informative prior along with MLE are considered. Their efficiencies are compared through a series of simulations. The results show that the Bayes estimator with the informative prior is more accurate and more precise when the means of the process before and after the change point time are not too closed. In addition, the efficiency of the Bayes estimator with the informative prior increases as the change point goes away from the origin.