Population attributable risk (PAR) is used in epidemiology to predict the impact of removing a risk factor from the population. Until recently, no standard approach for calculating confidence intervals or the variance for PAR was available in the literature. Pirikahu et al. (2016) outlined a fully Bayesian approach to provide credible intervals for the PAR from a cross-sectional study, where the data was presented in the form of a 2 x 2 table. However, extensions to cater for other frequently used study designs were not provided. In this paper we provide methodology to calculate credible intervals for the PAR for case-control and cohort studies. Additionally, we extend the cross-sectional example to allow for the incorporation of uncertainty that arises when an imperfect diagnostic test is used. In all these situations the model becomes over-parameterised, or non-identifiable, which can result in standard "off-the-shelf" Markov chain Monte Carlo updaters taking a long time to converge or even failing altogether. We adapt an importance sampling methodology to overcome this problem, and propose some novel MCMC samplers that take into consideration the shape of the posterior ridge to aid in the convergence of the Markov chain.
Drug combination trials are increasingly common nowadays in clinical research. However, very few methods have been developed to consider toxicity attributions in the dose escalation process. We are motivated by a trial in which the clinician is able to identify certain toxicities that can be attributed to one of the agents. We present a Bayesian adaptive design in which toxicity attributions are modeled via Copula regression and the maximum tolerated dose (MTD) curve is estimated as a function of model parameters. The dose escalation algorithm uses cohorts of two patients, following the contin…
Smoking is one of the preventable threats to human health and is a major risk factor for lung cancer, upper aero-digestive cancer, and chronic obstructive pulmonary disease. Estimating and forecasting the smoking attributable fraction (SAF) of mortality can yield insights into smoking epidemics and also provide a basis for more accurate mortality and life expectancy projection. Peto et al. (1992) proposed a method to estimate the SAF using the lung cancer mortality rate as an indicator of exposure to smoking in the population of interest. Here we use the same method to estimate the all-age SAF…
This paper describes the use of flexible Bayesian regression models for estimating a partially identified probability function. Our approach permits efficient sensitivity analysis concerning the posterior impact of priors on the partially identified component of the regression model. The new methodology is illustrated on an important problem where only partially observed data is available - inferring the prevalence of accounting misconduct among publicly traded U.S. businesses.
We discuss Bayesian inference for a known-mean Gaussian model with a compound symmetric variance-covariance matrix. Since the space of such matrices is a linear subspace of that of positive definite matrices, we utilize the methods of Pisano (2022) to decompose the usual Wishart conjugate prior and derive a closed-form, three-parameter, bivariate conjugate prior distribution for the compound-symmetric half-precision matrix. The off-diagonal entry is found to have a non-central Kummer-Beta distribution conditioned on the diagonal, which is shown to have a gamma distribution generalized with Gau…
We provide a comprehensive semi-parametric study of Bayesian partially identified econometric models. While the existing literature on Bayesian partial identification has mostly focused on the structural parameter, our primary focus is on Bayesian credible sets (BCS's) of the unknown identified set and the posterior distribution of its support function. We construct a (two-sided) BCS based on the support function of the identified set. We prove the Bernstein-von Mises theorem for the posterior distribution of the support function. This powerful result in turn infers that, while the BCS and the…
We study Bayesian networks based on max-linear structural equations as introduced in Gissibl and Klüppelberg [16] and provide a summary of their independence properties. In particular we emphasize that distributions for such networks are generally not faithful to the independence model determined by their associated directed acyclic graph. In addition, we consider some of the basic issues of estimation and discuss generalized maximum likelihood estimation of the coefficients, using the concept of a generalized likelihood ratio for non-dominated families as introduced by Kiefer and Wolfowitz [2…
An accurately identified maximum tolerated dose (MTD) serves as the cornerstone of successful subsequent phases in oncology drug development. Bayesian logistic regression model (BLRM) is a popular and versatile model-based dose-finding design. However, BLRM with original overdose control strategy has been reported to be safe but "excessively conservative". In this manuscript, we investigate the reason for conservativeness and point out that a major reason could be the lack of appropriate underdose control. We propose designs that balance overdose and underdose control to improve the performanc…
The COVID-19 pandemic has caused severe public health consequences in the United States. In this study, we use a hierarchical Bayesian model to estimate the age-specific COVID-19 attributable deaths over time in the United States. The model is specified by a novel non-parametric spatial approach, a low-rank Gaussian Process (GP) projected by regularised B-splines. We show that this projection defines a new GP with attractive smoothness and computational efficiency properties, derive its kernel function, and discuss the penalty terms induced by the projected GP. Simulation analyses and benchmar…
Prentice & Pyke (1979) established that the maximum likelihood estimate of an odds-ratio in a case-control study is the same as would be found by running a logistic regression: in other words, for this specific target the incorrect prospective model is inferentially equivalent to the correct retrospective model. Similar results have been obtained for other models, and conditions have also been identified under which the corresponding Bayesian property holds, namely that the posterior distribution of the odds-ratio be the same, whether computed using the prospective or the retrospective likelih…
We propose a novel Riemannian geometric framework for variational inference in Bayesian models based on the nonparametric Fisher-Rao metric on the manifold of probability density functions. Under the square-root density representation, the manifold can be identified with the positive orthant of the unit hypersphere in L2, and the Fisher-Rao metric reduces to the standard L2 metric. Exploiting such a Riemannian structure, we formulate the task of approximating the posterior distribution as a variational problem on the hypersphere based on the alpha-divergence. This provides a tighter lower boun…
Nonresponse weighting adjustment using the response propensity score is a popular tool for handling unit nonresponse. Statistical inference after the nonresponse weighting adjustment is complicated because the effect of estimating the propensity model parameter needs to be incorporated. In this paper, we propose an approximate Bayesian approach to handle unit nonresponse with parametric model assumptions on the response probability, but without model assumptions for the outcome variable. The proposed Bayesian method is calibrated to the frequentist inference in that the credible region obtaine…
Statistical modeling can involve a tension between assumptions and statistical identification. The law of the observable data may not uniquely determine the value of a target parameter without invoking a key assumption, and, while plausible, this assumption may not be obviously true in the scientific context at hand. Moreover, there are many instances of key assumptions which are untestable, hence we cannot rely on the data to resolve the question of whether the target is legitimately identified. Working in the Bayesian paradigm, we consider the grey zone of situations where a key assumption, …
Electronic health records (EHRs) are increasingly used for clinical and comparative effectiveness research, but suffer from missing data. Motivated by health services research on diabetes care, we seek to increase the quality of EHRs by focusing on missing values of longitudinal glycosylated hemoglobin (A1c), a key risk factor for diabetes complications and adverse events. Under the framework of multiple imputation (MI), we propose an individualized Bayesian latent profiling approach to capture A1c measurement trajectories subject to missingness. The proposed method is applied to EHRs of adult…
Most estimates for penalised linear regression can be viewed as posterior modes for an appropriate choice of prior distribution. Bayesian shrinkage methods, particularly the horseshoe estimator, have recently attracted a great deal of attention in the problem of estimating sparse, high-dimensional linear models. This paper extends these ideas, and presents a Bayesian grouped model with continuous global-local shrinkage priors to handle complex group hierarchies that include overlapping and multilevel group structures. As the posterior mean is never a sparse estimate of the linear model coeffic…
Disease subtype identification (clustering) is an important problem in biomedical research. Gene expression profiles are commonly utilized to infer disease subtypes, which often lead to biologically meaningful insights into disease. Despite many successes, existing clustering methods may not perform well when genes are highly correlated and many uninformative genes are included for clustering due to the high dimensionality. In this article, we introduce a novel subtype identification method in the Bayesian setting based on gene expression profiles. This method, called BCSub, adopts an innovati…
The use of a finite mixture of normal distributions in model-based clustering allows to capture non-Gaussian data clusters. However, identifying the clusters from the normal components is challenging and in general either achieved by imposing constraints on the model or by using post-processing procedures. Within the Bayesian framework we propose a different approach based on sparse finite mixtures to achieve identifiability. We specify a hierarchical prior where the hyperparameters are carefully selected such that they are reflective of the cluster structure aimed at. In addition this prior a…
This paper presents a study of the large-sample behavior of the posterior distribution of a structural parameter which is partially identified by moment inequalities. The posterior density is derived based on the limited information likelihood. The posterior distribution converges to zero exponentially fast on any $δ$-contraction outside the identified region. Inside, it is bounded below by a positive constant if the identified region is assumed to have a nonempty interior. Our simulation evidence indicates that the Bayesian approach has advantages over frequentist methods, in the sense that, …
Vector autoregressions (VARs) with multivariate stochastic volatility are widely used for structural analysis. Often the structural model identified through economically meaningful restrictions--e.g., sign restrictions--is supposed to be independent of how the dependent variables are ordered. But since the reduced-form model is not order invariant, results from the structural analysis depend on the order of the variables. We consider a VAR based on the factor stochastic volatility that is constructed to be order invariant. We show that the presence of multivariate stochastic volatility allows …
An important problem in econometrics and marketing is to infer the causal impact that a designed market intervention has exerted on an outcome metric over time. This paper proposes to infer causal impact on the basis of a diffusion-regression state-space model that predicts the counterfactual market response in a synthetic control that would have occurred had no intervention taken place. In contrast to classical difference-in-differences schemes, state-space models make it possible to (i) infer the temporal evolution of attributable impact, (ii) incorporate empirical priors on the parameters i…
This article tackles the old problem of prediction via a nonparametric transformation model (NTM) in a new Bayesian way. Estimation of NTMs is known challenging due to model unidentifiability though appealing because of its robust prediction capability in survival analysis. Inspired by the uniqueness of the posterior predictive distribution, we achieve efficient prediction via the NTM aforementioned under the Bayesian paradigm. Our strategy is to assign weakly informative priors to nonparametric components rather than identify the model by adding complicated constraints in the existing literat…
The stratified proportional hazards model represents a simple solution to account for heterogeneity within the data while keeping the multiplicative effect on the hazard function. Strata are typically defined a priori by resorting to the values taken by a categorical covariate. A general framework is proposed, which allows for the stratification of a generic accelerated life time model, including as a special case the Weibull proportional hazard model. The stratification is determined a posteriori by taking into account that strata might be characterized by different baseline survivals as well…
In (exploratory) factor analysis, the loading matrix is identified only up to orthogonal rotation. For identifiability, one thus often takes the loading matrix to be lower triangular with positive diagonal entries. In Bayesian inference, a standard practice is then to specify a prior under which the loadings are independent, the off-diagonal loadings are normally distributed, and the diagonal loadings follow a truncated normal distribution. This prior specification, however, depends in an important way on how the variables and associated rows of the loading matrix are ordered. We show how a mi…
Longitudinal fMRI datasets hold great promise for the study of neurodegenerative diseases, but realizing their potential depends on extracting accurate fMRI-based brain measures in individuals over time. This is especially true for rare, heterogeneous and/or rapidly progressing diseases, which often involve small samples whose functional features may vary dramatically across subjects and over time, making traditional group-difference analyses of limited utility. One such disease is ALS, which results in extreme motor function loss and eventual death. Here, we analyze a rich longitudinal datase…
The structure of a Bayesian network encodes most of the information about the probability distribution of the data, which is uniquely identified given some general distributional assumptions. Therefore it's important to study the variability of its network structure, which can be used to compare the performance of different learning algorithms and to measure the strength of any arbitrary subset of arcs. In this paper we will introduce some descriptive statistics and the corresponding parametric and Monte Carlo tests on the undirected graph underlying the structure of a Bayesian network, modele…
This study presents a Bayesian spectral density approach for identification and uncertainty quantification of flutter derivatives of bridge sections utilizing buffeting displacement responses, where the wind tunnel test is conducted in turbulent flow. Different from traditional time-domain approaches (e.g., least square method and stochastic subspace identification), the newly-proposed approach is operated in frequency domain. Based on the affine invariant ensemble sampler algorithm, Markov chain Monte-Carlo sampling is employed to accomplish the Bayesian inference. The probability density fun…
The structure of a Bayesian network includes a great deal of information about the probability distribution of the data, which is uniquely identified given some general distributional assumptions. Therefore it's important to study its variability, which can be used to compare the performance of different learning algorithms and to measure the strength of any arbitrary subset of arcs. In this paper we will introduce some descriptive statistics and the corresponding parametric and Monte Carlo tests on the undirected graph underlying the structure of a Bayesian network, modeled as a multivariate …
Smoking is one of the main risk factors that has affected human mortality and life expectancy over the past century. Smoking accounts for a large part of the nonlinearities in the growth of life expectancy and of the geographic and sex differences in mortality. As Bongaarts (2006) and Janssen (2018) suggested, accounting for smoking could improve the quality of mortality forecasts due to the predictable nature of the smoking epidemic. We propose a new Bayesian hierarchical model to forecast life expectancy at birth for both sexes and for 69 countries with good data on smoking-related mortality…
We address the problem of static OD matrix estimation from a formal statistical viewpoint. We adopt a novel Bayesian framework to develop a class of models that explicitly cast trip configurations in the study region as random variables. As a consequence, classical solutions from growth factor, gravity, and maximum entropy models are identified to specific estimators under the proposed models. We show that each of these solutions usually account for only a small fraction of the posterior probability mass in the ensemble and we then contend that the uncertainty in the inference should be propag…
For log-linear analysis, the hyper Dirichlet conjugate prior is available to work in the Bayesian paradigm. With this prior, the MC3 algorithm allows for exploration of the space of models to try to find those with the highest posterior probability. Once top models have been identified, a block Gibbs sampler can be constructed to sample from the posterior distribution and to estimate parameters of interest. Our aim in this paper, is to introduce the bayesloglin R package \citep{R} which contains functions to carry out these tasks.
We propose a Bayesian nonparametric method for low-pass filtering that can naturally handle unevenly-sampled and noise-corrupted observations. The proposed model is constructed as a latent-factor model for time series, where the latent factors are Gaussian processes with non-overlapping spectra. With this construction, the low-pass version of the time series can be identified as the low-frequency latent component, and therefore it can be found by means of Bayesian inference. We show that the model admits exact training and can be implemented with minimal numerical approximations. Finally, the …
Frequentist coverage of $(1-α)$-highest posterior density (HPD) credible sets is studied in a signal plus noise model under a large class of noise distributions. We consider a specific class of spike-and-slab prior distributions. Different regimes are identified and we derive closed form expressions for the $(1-α)$-HPD on each of these regimes. Similar to the earlier work by Marchand and Strawderman, it is shown that under suitable conditions, the frequentist coverage can drop to $1-3α/2.$
A lower bound is an important tool for predicting the performance that an estimator can achieve under a particular statistical model. Bayesian bounds are a kind of such bounds which not only utilizes the observation statistics but also includes the prior model information. In reality, however, the true model generating the data is either unknown or simplified when deriving estimators, which motivates the works to derive estimation bounds under modeling mismatch situations. This paper provides a derivation of a Bayesian Cramér-Rao bound under model misspecification, defining important concepts …
In this paper we propose and study local linear and polynomial based estimators for implementing Approximate Bayesian Computation (ABC) style indirect inference and GMM estimators. This method makes use of nonparametric regression in the computation of GMM and Indirect Inference models. We provide formal conditions under which frequentist inference is asymptotically valid and demonstrate the validity of the estimated posterior quantiles for confidence interval construction. We also show that in this setting, local linear kernel regression methods have theoretical advantages over local constant…
For civil structures, structural damage due to severe loading events such as earthquakes, or due to long-term environmental degradation, usually occurs in localized areas of a structure. A new sparse Bayesian probabilistic framework for computing the probability of localized stiffness reductions induced by damage is presented that uses noisy incomplete modal data from before and after possible damage. This new approach employs system modal parameters of the structure as extra variables for Bayesian model updating with incomplete modal data. A specific hierarchical Bayesian model is constructed…
The Power Law Process, also known as Non-Homogeneous Poisson Process, has been used in various aspects, one of which is the software reliability assessment. Specifically, by using its intensity function to compute the rate of change of a software reliability as time-varying function. Justification of Bayesian analysis applicability to the Power Law Process was shown using real data. The probability distribution that best characterizes the behavior of the key parameter of the intensity function was first identified, then the likelihood-based Bayesian reliability estimate of the Power Law Proces…
Correlated component analysis as proposed by Dmochowski et al. (2012) is a tool for investigating brain process similarity in the responses to multiple views of a given stimulus. Correlated components are identified under the assumption that the involved spatial networks are identical. Here we propose a hierarchical probabilistic model that can infer the level of universality in such multi-view data, from completely unrelated representations, corresponding to canonical correlation analysis, to identical representations as in correlated component analysis. This new model, which we denote Bayesi…
The focus in this paper is Bayesian system identification based on noisy incomplete modal data where we can impose spatially-sparse stiffness changes when updating a structural model. To this end, based on a similar hierarchical sparse Bayesian learning model from our previous work, we propose two Gibbs sampling algorithms. The algorithms differ in their strategies to deal with the posterior uncertainty of the equation-error precision parameter, but both sample from the conditional posterior probability density functions (PDFs) for the structural stiffness parameters and system modal parameter…
In this work we perform a meta-analysis of neuroimaging data, consisting of locations of peak activations identified in 162 separate studies on emotion. Neuroimaging meta-analyses are typically performed using kernel-based methods. However, these methods require the width of the kernel to be set a priori and to be constant across the brain. To address these issues, we propose a fully Bayesian nonparametric binary regression method to perform neuroimaging meta-analyses. In our method, each location (or voxel) has a probability of being a peak activation, and the corresponding probability functi…
The evaluation of a match between the DNA profile of a stain found on a crime scene and that of a suspect (previously identified) involves the use of the unknown parameter $p=(p_1, p_2, ...)$, (the ordered vector which represents the proportions of the different DNA profiles in the population of potential donors) and the names of the different DNA types. We propose a Bayesian non parametric method which considers $P$ as a random variable distributed according to the two-parameter Poisson Dirichlet distribution, and discard information about names of DNA types. The ultimate goal of this model i…
