Human fecundity is an issue of considerable interest for both epidemiological and clinical audiences, and is dependent upon a couple’s biologic capacity for reproduction coupled with behaviors that place a couple at risk for pregnancy. Bayesian hierarchical models have been proposed to better model the conception probabilities by accounting for the acts of intercourse around the day of ovulation, i.e., during the fertile window. These models can be viewed in the framework of a generalized nonlinear model with an exponential link. However, a fixed choice of link function may not always provide the best fit, leading to potentially biased estimates for probability of conception. Motivated by this, we propose a general class of models for fecundity by relaxing the choice of the link function under the generalized nonlinear model framework. We use a sample from the Oxford Conception Study (OCS) to illustrate the utility and fit of this general class of models for estimating human conception. Our findings reinforce the need for attention to be paid to the choice of link function in modeling conception, as it may bias the estimation of conception probabilities. Various properties of the proposed models are examined and a Markov chain Monte Carlo sampling algorithm was developed for implementing the Bayesian computations. The deviance information criterion measure and logarithm of pseudo marginal likelihood are used for guiding the choice of links. The supplemental material section contains technical details of the proof of the theorem stated in the paper, and contains further simulation results and analysis.

A nonparametric Bayesian model is proposed for segmenting time-evolving multivariate spatial point process data. An inhomogeneous Poisson process is assumed, with a logistic stick-breaking process (LSBP) used to encourage piecewise-constant spatial Poisson intensities. The LSBP explicitly favors spatially contiguous segments, and infers the number of segments based on the observed data. The temporal dynamics of the segmentation and of the Poisson intensities are modeled with exponential correlation in time, implemented in the form of a first-order autoregressive model for uniformly sampled discrete data, and via a Gaussian process with an exponential kernel for general temporal sampling. We consider and compare two different inference techniques: a Markov chain Monte Carlo sampler, which has relatively high computational complexity; and an approximate and efficient variational Bayesian analysis. The model is demonstrated with a simulated example and a real example of space-time crime events in Cincinnati, Ohio, USA.

A new regression model for proportions is presented by considering the Beta rectangular distribution proposed by Hahn (2008). This new model includes the Beta regression model introduced by Ferrari and Cribari-Neto (2004) and the variable dispersion Beta regression model introduced by Smithson and Verkuilen (2006) as particular cases. Like Branscum, Johnson, and Thurmond (2007), a Bayesian inference approach is adopted using Markov Chain Monte Carlo (MCMC) algorithms. Simulation studies on the influence of outliers by considering contaminated data under four perturbation patterns to generate outliers were carried out and confirm that the Beta rectangular regression model seems to be a new robust alternative for modeling proportion data and that the Beta regression model shows sensitivity to the estimation of regression coefficients, to the posterior distribution of all parameters and to the model comparison criteria considered. Furthermore, two applications are presented to illustrate the robustness of the Beta rectangular model.

Recently, the graphical lasso procedure has become popular in estimating Gaussian graphical models. In this paper, we introduce a fully Bayesian treatment of graphical lasso models. We first investigate the graphical lasso prior that has been relatively unexplored. Using data augmentation, we develop a simple but highly efficient block Gibbs sampler for simulating covariance matrices. We then generalize the Bayesian graphical lasso to the Bayesian adaptive graphical lasso. Finally, we illustrate and compare the results from our approach to those obtained using the standard graphical lasso procedures for real and simulated data. In terms of both covariance matrix estimation and graphical structure learning, the Bayesian adaptive graphical lasso appears to be the top overall performer among a range of frequentist and Bayesian methods.

This paper argues that the half-Cauchy distribution should replace the inverse-Gamma distribution as a default prior for a top-level scale parameter in Bayesian hierarchical models, at least for cases where a proper prior is necessary. Our arguments involve a blend of Bayesian and frequentist reasoning, and are intended to complement the case made by Gelman (2006) in support of folded-$t$ priors. First, we generalize the half-Cauchy prior to the wider class of hypergeometric inverted-beta priors. We derive expressions for posterior moments and marginal densities when these priors are used for a top-level normal variance in a Bayesian hierarchical model. We go on to prove a proposition that, together with the results for moments and marginals, allows us to characterize the frequentist risk of the Bayes estimators under all global-shrinkage priors in the class. These results, in turn, allow us to study the frequentist properties of the half-Cauchy prior versus a wide class of alternatives. The half-Cauchy occupies a sensible middle ground within this class: it performs well near the origin, but does not lead to drastic compromises in other parts of the parameter space. This provides an alternative, classical justification for the routine use of this prior. We also consider situations where the underlying mean vector is sparse, where we argue that the usual conjugate choice of an inverse-gamma prior is particularly inappropriate, and can severely distort inference. Finally, we summarize some open issues in the specification of default priors for scale terms in hierarchical models.

There are several ways to parameterize a distribution belonging to an exponential family, each one leading to a different Bayesian analysis of the data under standard conjugate priors. To overcome this problem, we propose a new class of conjugate priors which is invariant with respect to smooth reparameterization. This class of priors contains the Jeffreys prior as a special case, according to the value of the hyperparameters. Moreover, these conjugate distributions coincide with the posterior distributions resulting from a Jeffreys prior. Then these priors appear naturally when several datasets are analyzed sequentially and when the Jeffreys prior is chosen for the first dataset. We apply our approach to inverse Gaussian models and propose full invariant analyses of three datasets.

Phylogeography is the study of evolutionary history among populations in a species associated with geographic genetic variation. This paper examines the phylogeography of three African gorilla subspecies based on two types of DNA sequence data. One type is HV1, the first hyper-variable region in the control region of the mitochondrial genome. The other type is nuclear mitochondrial DNA (Numt DNA), which results from the introgression of a copy of HV1 from the mitochondrial genome into the nuclear genome. Numt and HV1 sequences evolve independently when in different organelles, but they share a common evolutionary history at the same locus in the mitochondrial genome prior to introgression. This study estimates the evolutionary history of gorilla populations in terms of population divergence times and effective population sizes. Also, this study estimates the number of introgression events. The estimates are obtained in a Bayesian framework using novel Markov chain Monte Carlo methods. The method is based on a hybrid coalescent process that combines separate coalescent processes for HV1 and Numt sequences along with a transfer model for introgression events within a single population tree. This Bayesian method for the analysis of Numt and HV1 sequences is the first approach specifically designed to model the evolutionary history of homologous multi-locus sequences within a population tree framework. The data analysis reveals highly discordant estimates of the divergence time between eastern and western gorilla populations for HV1 and Numt sequences. The discordant east-west split times are evidence of male-mediated gene flow between east and west long after female gorillas stopped this migration. In addition, the analysis estimates multiple independent introgression events.

The log-normal distribution is a popular model in biostatistics and other fields of statistics. Bayesian inference on the mean and median of the distribution is problematic because, for many popular choices of the prior for the variance (on the log-scale) parameter, the posterior distribution has no finite moments, leading to Bayes estimators with infinite expected loss for the most common choices of the loss function. We propose a generalized inverse Gaussian prior for the variance parameter, that leads to a log-generalized hyperbolic posterior, for which it is easy to calculate quantiles and moments, provided that they exist. We derive the constraints on the prior parameters that yield finite posterior moments of order $r$. We investigate the choice of prior parameters leading to Bayes estimators with optimal frequentist mean square error. For the estimation of the lognormal mean we show, using simulation, that the Bayes estimator under quadratic loss compares favorably in terms of frequentist mean square error to known estimators.

We present the discrete infinite logistic normal distribution (DILN), a Bayesian nonparametric prior for mixed membership models. DILN generalizes the hierarchical Dirichlet process (HDP) to model correlation structure between the weights of the atoms at the group level. We derive a representation of DILN as a normalized collection of gamma-distributed random variables and study its statistical properties. We derive a variational inference algorithm for approximate posterior inference. We apply DILN to topic modeling of documents and study its empirical performance on four corpora, comparing performance with the HDP and the correlated topic model (CTM). To compute with large-scale data, we develop a stochastic variational inference algorithm for DILN and compare with similar algorithms for HDP and latent Dirichlet allocation (LDA) on a collection of 350,000 articles from Nature.

The trend of treating patients with combined drugs has grown in cancer clinical trials. Often, evaluating the synergism of multiple drugs is the primary motivation for such drug-combination studies. To enhance patient response, a new agent is often investigated together with an existing standard of care (SOC) agent. Often, a certain amount of dosage of the SOC is administered in order to maintain at least some therapeutic effects in patients. For clinical trials involving a continuous-dose SOC and a discrete-dose agent, we propose a two-stage Bayesian adaptive dose-finding design. The first stage takes a continual reassessment method to locate the appropriate dose for the discrete-dose agent while fixing the continuous-dose SOC at the minimal therapeutic dose. In the second stage, we make a fine dose adjustment by calibrating the continuous dose to achieve the target toxicity rate as closely as possible. Dose escalation or de-escalation is based on the posterior estimates of the joint toxicity probabilities of combined doses. As the toxicity data accumulate during the trial, we adaptively assign each cohort of patients to the most appropriate dose combination. We conduct extensive simulation studies to examine the operating characteristics of the proposed two-stage design and demonstrate the design’s good performance with practical scenarios.