Nonparametric Bayes methods for directional data

A model for directional data in q dimensions is studied. The data are assumed to arise from a distribution with a density on a sphere of q — 1 dimensions. The density is unimodal and rotationally symmetric, but otherwise of unknown form. The posterior distribution of the unknown mode (mean direction) is derived, and small-sample posterior inference is discussed. The posterior mean of the density is also given. A numerical method for evaluating posterior quantities based on sampling a Markov chain is introduced. This method is generally applicable to problems involving unknown monotone functions.     

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.     

Change Point Detection in The Skew-Normal Model Parameters

Bayesian inference under the skew-normal family of distributions is discussed using an arbitrary proper prior for the skewness parameter. In particular, we review some results when a skew-normal prior distribution is considered. Considering this particular prior, we provide a stochastic representation of the posterior of the skewness parameter. Moreover, we obtain analytical expressions for the posterior mean and variance of the skewness parameter. The ultimate goal is to consider these results to one change point identification in the parameters of the location-scale skew-normal model. Some Latin American emerging market datasets are used to illustrate the methodology developed in this work.     

Bayesian multivariate normal analysis with a wishart prior

This paper considers the Bayesian analysis of the multivariate normal distribution when its covariance matrix has a Wishart prior density under the assumption of a multivariate quadratic loss function. New flexible marginal posterior distributions of the mean μ and of the covariance matrix Σ are developed and univariate cases with graphical representations are given.     

Bayesian estimates in a one-way ANOVA random effects model

Bayesian estimators of variance components are developed, based on posterior mean and posterior mode, respectively, in a one-way ANOVA random effects model with independent prior distributions. The formulas for the proposed estimators are simple. The estimators give sensible results for 'badly-behaved' datasets, where the standard unbiased estimates are negative. They are markedly robust as compared to the existing estimators such as the maximum likelihood estimators and the maximum posterior density estimators.     

Bayesian marginal inference via candidate's formula

Computing marginal probabilities is an important and fundamental issue in Bayesian inference. We present a simple method which arises from a likelihood identity for computation. The likelihood identity, called Candidate's formula, sets the marginal probability as a ratio of the prior likelihood to the posterior density. Based on Markov chain Monte Carlo output simulated from the posterior distribution, a nonparametric kernel estimate is used to estimate the posterior density contained in that ratio. This derived nonparametric Candidate's estimate requires only one evaluation of the posterior density estimate at a point. The optimal point for such evaluation can be chosen to minimize the expected mean square relative error. The results show that the best point is not necessarily the posterior mode, but rather a point compromising between high density and low Hessian. For high dimensional problems, we introduce a variance reduction approach to ease the tension caused by data sparseness. A simulation study is presented.     

Bayes Risk Consistency of Nonparametric Bayes Density Estimates2

Even though the literature on nonparametric density estimation is large, the literature on Bayesian estimation of the density function is relatively small. The reason is the lack of a suitable prior over the space of probability density functions. There have been attempts to define priors over the space of probability measures, but they have not yielded any workable prior for the purpose of density estimation. Dubins & Freedman (1963) have denned random distribution functions which are singular with probability one. Kraft (1964) has denned a class of distribution functions which have derivatives but not continuous derivatives and hence are not suitable for density estimation. The only really convenient prior is the Dirichlet process prior due to Ferguson (1973), but unfortunately this prior concentrates all its mass over the discrete distribution with a dense set of jumps. Recently Lo (1978) has overcome this difficulty by taking convolution of the Dirichlet process with a fixed continuous kernel. In Section 2, the existence of a version of the posterior distribution and the conditional expectation for arbitrary prior over the space of continuous density functions are discussed. The Bayes risk consistency of the Bayes estimator is discussed in Section 3. The Bayes estimator and its properties with respect to two specific prior distributions are discussed in Section 4. In Section 5 some negative results are presented. Finally a numerical example is given in Section 6.     

Powerful Test of Two Proportions by Assuming a Registered Prior Density

A method for constructing powerful significance tests for the equivalence of two proportions is proposed by assuming prior density values. Recent changes in the medical research environment emphasize the need for choice of a prior density in advance of any study. The proposed test is based on the posterior probability of the alternative model and preserves the significance level with minimal reduction of power. The new test performs better than the familiar mid-p test under the uniform prior density condition. In addition, the computational burden is low. Potential extensions of the proposed test to related problems are also discussed.     

Nonparametric survival regression using the beta-Stacy process

A novel class of hierarchical nonparametric Bayesian survival regression models for time-to-event data with uninformative right censoring is introduced. The survival curve is modeled as a random function whose prior distribution is defined using the beta-Stacy (BS) process. The prior mean of each survival probability and its prior variance are linked to a standard parametric survival regression model. This nonparametric survival regression can thus be anchored to any reference parametric form, such as a proportional hazards or an accelerated failure time model, allowing substantial departures of the predictive survival probabilities when the reference model is not supported by the data. Also, under this formulation the predictive survival probabilities will be close to the empirical survival distribution near the mode of the reference model and they will be shrunken towards its probability density in the tails of the empirical distribution.     

Consistency of Bernstein polynomial posteriors

A Bernstein prior is a probability measure on the space of all the distribution functions on [0, 1]. Under very general assumptions, it selects absolutely continuous distribution functions, whose densities are mixtures of known beta densities. The Bernstein prior is of interest in Bayesian nonparametric inference with continuous data. We study the consistency of the posterior from a Bernstein prior. We first show that, under mild assumptions, the posterior is weakly consistent for any distribution function P ₀ on [0, 1] with continuous and bounded Lebesgue density. With slightly stronger assumptions on the prior, the posterior is also Hellinger consistent. This implies that the predictive density from a Bernstein prior, which is a Bayesian density estimate, converges in the Hellinger sense to the true density (assuming that it is continuous and bounded). We also study a sieve maximum likelihood version of the density estimator and show that it is also Hellinger consistent under weak assumptions. When the order of the Bernstein polynomial, i.e. the number of components in the beta distribution mixture, is truncated, we show that under mild restrictions the posterior concentrates on the set of pseudotrue densities. Finally, we study the behaviour of the predictive density numerically and we also study a hybrid Bayes–maximum likelihood density estimator.     

On Bayesian estimators in multistage binomial designs

A new class of Bayesian estimators for a proportion in multistage binomial designs is considered. Priors belong to the beta-J distribution family, which is derived from the Fisher information associated with the design. The transposition of the beta parameters of the Haldane and the uniform priors in fixed binomial experiments into the beta-J distribution yields bias-corrected versions of these priors in multistage designs. We show that the estimator of the posterior mean based on the corrected Haldane prior and the estimator of the posterior mode based on the corrected uniform prior have good frequentist properties. An easy-to-use approximation of the estimator of the posterior mode is provided. The new Bayesian estimators are compared to Whitehead's and the uniformly minimum variance estimators through several multistage designs. Last, the bias of the estimator of the posterior mode is derived for a particular case.     

A Direct Approach to Understanding Posterior Consistency of Bayesian Regression Problems

Previous approaches to establishing posterior consistency of Bayesian regression problems have used general theorems that involve verifying sufficient conditions for posterior consistency. In this article, we consider a direct approach by computing the posterior density explicitly and evaluating its asymptotic behavior. For this purpose, we deal with a sample size dependent prior based on a truncated regression function with increasing sample size, and evaluate the asymptotic properties of the resulting posterior. Based on a concept called posterior density consistency, we attempt to understand posterior consistency. As an application, we illustrate that the posterior density of an orthogonal semiparametric regression model is consistent.     

Computing some bayes estimates for the mean of a normal distribution

For sampling from a normal population with unknown mean, two families of prior densities for the mean are discussed. The corresponding posterior densities are found. A data analyst may choose a prior from these families to represent prior beliefs and then compute the corresponding Bayes estimator, using the techniques discussed.     

On Optimality of Bayesian Wavelet Estimators

Abstract.  We investigate the asymptotic optimality of several Bayesian wavelet estimators, namely, posterior mean, posterior median and Bayes Factor, where the prior imposed on wavelet coefficients is a mixture of a mass function at zero and a Gaussian density. We show that in terms of the mean squared error, for the properly chosen hyperparameters of the prior, all the three resulting Bayesian wavelet estimators achieve optimal minimax rates within any prescribed Besov space     for p  ≥ 2. For 1 ≤  p  < 2, the Bayes Factor is still optimal for (2 s +2)/(2 s +1) ≤  p  < 2 and always outperforms the posterior mean and the posterior median that can achieve only the best possible rates for linear estimators in this case.     

Bayesian estimation and prediction intervals for a Rayleigh distribution under a conjugate prior

This paper is an effort to obtain Bayes estimators of Rayleigh parameter and its associated risk based on a conjugate prior (square root inverted gamma prior) with respect to both symmetric loss function (squared error loss), and asymmetric loss function (precautionary loss function). We also derive the highest posterior density (HPD) interval for the Rayleigh parameter as well as the HPD prediction intervals for a future observation from this distribution. An illustrative example to test how the Rayleigh distribution fits a real data set is presented. Finally, Monte Carlo simulations are performed to compare the performances of the Bayes estimates under different conditions.     

Posterior consistency for the spectral density of non-Gaussian stationary time series

Various nonparametric approaches for Bayesian spectral density estimation of stationary time series have been suggested in the literature, mostly based on the Whittle likelihood approximation. A generalization of this approximation involving a nonparametric correction of a parametric likelihood has been proposed in the literature with a proof of posterior consistency for spectral density estimation in combination with the Bernstein–Dirichlet process prior for Gaussian time series. In this article, we will extend the posterior consistency result to non-Gaussian time series by employing a general consistency theorem for dependent data and misspecified models. As a special case, posterior consistency for the spectral density under the Whittle likelihood is also extended to non-Gaussian time series. Small sample properties of this approach are illustrated with several examples of non-Gaussian time series.     

Bayesian Confidence Interval for the Ratio of Marginal Probabilities in the Matched-Pair Design

This article studies the construction of a Bayesian confidence interval for the ratio of marginal probabilities in matched-pair designs. Under a Dirichlet prior distribution, the exact posterior distribution of the ratio is derived. The tail confidence interval and the highest posterior density (HPD) interval are studied, and their frequentist performances are investigated by simulation in terms of mean coverage probability and mean expected length of the interval. An advantage of Bayesian confidence interval is that it is always well defined for any data structure and has shorter mean expected width. We also find that the Bayesian tail interval at Jeffreys prior performs as well as or better than the frequentist confidence intervals.     

Bayesian estimation of the dispersion matrix of a multivariate normal distribution

The estimation of the dispersion matrix of a multivariate normal distribution with zero mean on the basis of a random sample is discussed from a Bayesian view. An inverted-Wishart distribu- tion for the dispersion is taken, with its defining matrix of intraclass form. Some consistency properties are described. The posterior distribution is found and its mode investigated as a possible estimate in preference to that of maximum likelihood     

A prior for the variance in hierarchical models

The choice of prior distributions for the variances can be important and quite difficult in Bayesian hierarchical and variance component models. For situations where little prior information is available, a ‘nonin-formative’ type prior is usually chosen. ‘Noninformative’ priors have been discussed by many authors and used in many contexts. However, care must be taken using these prior distributions as many are improper and thus, can lead to improper posterior distributions. Additionally, in small samples, these priors can be ‘informative’. In this paper, we investigate a proper ‘vague’ prior, the uniform shrinkage prior (Strawder-man 1971; Christiansen & Morris 1997). We discuss its properties and show how posterior distributions for common hierarchical models using this prior lead to proper posterior distributions. We also illustrate the attractive frequentist properties of this prior for a normal hierarchical model including testing and estimation. To conclude, we generalize this prior to the multivariate situation of a covariance matrix.     

Wavelet Shrinkage with Double Weibull Prior

In this article, we propose a denoising methodology in the wavelet domain based on a Bayesian hierarchical model using Double Weibull prior. We propose two estimators, one based on posterior mean (Double Weibull Wavelet Shrinker, DWWS) and the other based on larger posterior mode (DWWS-LPM), and show how to calculate them efficiently. Traditionally, mixture priors have been used for modeling sparse wavelet coefficients. The interesting feature of this article is the use of non-mixture prior. We show that the methodology provides good denoising performance, comparable even to state-of-the-art methods that use mixture priors and empirical Bayes setting of hyperparameters, which is demonstrated by extensive simulations on standardly used test functions. An application to real-word dataset is also considered.