Parameter ensembles or sets of point estimates constitute one of the cornerstones of modern statistical practice. This is especially the case in Bayesian hierarchical models, where different decision-theoretic frameworks can be deployed to summarize such parameter ensembles. The estimation of these parameter ensembles may thus substantially vary depending on which inferential goals are prioritised by the modeller. In this note, we consider the problem of classifying the elements of a parameter ensemble above or below a given threshold. Two threshold classification losses (TCLs) --weighted and unweighted-- are formulated. The weighted TCL can be used to emphasize the estimation of false positives over false negatives or the converse. We prove that the weighted and unweighted TCLs are optimized by the ensembles of unit-specific posterior quantiles and posterior medians, respectively. In addition, we relate these classification loss functions on parameter ensembles to the concepts of posterior sensitivity and specificity. Finally, we find some relationships between the unweighted TCL and the absolute value loss, which explain why both functions are minimized by posterior medians.
The level set approach has proven widely successful in the study of inverse problems for interfaces, since its systematic development in the 1990s. Recently it has been employed in the context of Bayesian inversion, allowing for the quantification of uncertainty within the reconstruction of interfaces. However the Bayesian approach is very sensitive to the length and amplitude scales in the prior probabilistic model. This paper demonstrates how the scale-sensitivity can be circumvented by means of a hierarchical approach, using a single scalar parameter. Together with careful consideration of …
This paper introduces a hierarchical framework to incorporate Hellinger distance methods into Bayesian analysis. We propose to modify a prior over non-parametric densities with the exponential of twice the Hellinger distance between a candidate and a parametric density. By incorporating a prior over the parameters of the second density, we arrive at a hierarchical model in which a non-parametric model is placed between parameters and the data. The parameters of the family can then be estimated as hyperparameters in the model. In frequentist estimation, minimizing the Hellinger distance between…
We discuss the methods of Evans and Moshonov [Bayesian Analysis 1 (2006) 893--914, Bayesian Statistics and Its Applications (2007) 145--159] concerning checking for prior-data conflict and their relevance to the method proposed in this paper. [arXiv:0802.0743]
In many signal processing problems, it may be fruitful to represent the signal under study in a frame. If a probabilistic approach is adopted, it becomes then necessary to estimate the hyper-parameters characterizing the probability distribution of the frame coefficients. This problem is difficult since in general the frame synthesis operator is not bijective. Consequently, the frame coefficients are not directly observable. This paper introduces a hierarchical Bayesian model for frame representation. The posterior distribution of the frame coefficients and model hyper-parameters is derived. H…
Here we develop a method for performing nonparametric Bayesian inference on quantiles. Relying on geometric measure theory and employing a Hausdorff base measure, we are able to specify meaningful priors for the quantile while treating the distribution of the data otherwise nonparametrically. We further extend the method to a hierarchical model for quantiles of subpopulations, linking subgroups together solely through their quantiles. Our approach is computationally straightforward, allowing for censored and noisy data. We demonstrate the proposed methodology on simulated data and an applied p…
This paper introduces the parallel hierarchical sampler (PHS), a Markov chain Monte Carlo algorithm using several chains simultaneously. The connections between PHS and the parallel tempering (PT) algorithm are illustrated, convergence of PHS joint transition kernel is proved and and its practical advantages are emphasized. We illustrate the inferences obtained using PHS, parallel tempering and the Metropolis-Hastings algorithm for three Bayesian model selection problems, namely Gaussian clustering, the selection of covariates for a linear regression model and the selection of the structure of…
Department of Technology, Management and Economics, Danish Technical University（丹麦技术大学技术、管理与经济系）；Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）
Identifying genetic interactions for a given microorganism such as yeast is difficult. Quantitative Fitness Analysis (QFA) is a high-throughput experimental and computational methodology for quantifying the fitness of microbial cultures. QFA can be used to compare between fitness observations for different genotypes and thereby infer genetic interaction strengths. Current "naive" frequentist statistical approaches used in QFA do not model between-genotype variation or difference in genotype variation under different conditions. In this thesis, a Bayesian approach is introduced to evaluate hier…
Bayesian hierarchical modeling is a popular approach to capturing unobserved heterogeneity across individual units. However, standard estimation methods such as Markov chain Monte Carlo (MCMC) can be impracticable for modeling outcomes from a large number of units. We develop a new method to sample from posterior distributions of Bayesian models, without using MCMC. Samples are independent, so they can be collected in parallel, and we do not need to be concerned with issues like chain convergence and autocorrelation. The algorithm is scalable under the weak assumption that individual units are…
Considering the flexibility and applicability of Bayesian modeling, in this work we revise the main characteristics of two hierarchical models in a regression setting. We study the full probabilistic structure of the models along with the full conditional distribution for each model parameter. Under our hierarchical extensions, we allow the mean of the second stage of the model to have a linear dependency on a set of covariates. The Gibbs sampling algorithms used to obtain samples when fitting the models are fully described and derived. In addition, we consider a case study in which the plant …
We develop a new method to sample from posterior distributions in hierarchical models without using Markov chain Monte Carlo. This method, which is a variant of importance sampling ideas, is generally applicable to high-dimensional models involving large data sets. Samples are independent, so they can be collected in parallel, and we do not need to be concerned with issues like chain convergence and autocorrelation. Additionally, the method can be used to compute marginal likelihoods.
Columbia University（哥伦比亚大学）；University of Ljubljana（卢布尔雅那大学）；Aalto University（阿尔托大学）
Stacking is a widely used model averaging technique that asymptotically yields optimal predictions among linear averages. We show that stacking is most effective when model predictive performance is heterogeneous in inputs, and we can further improve the stacked mixture with a hierarchical model. We generalize stacking to Bayesian hierarchical stacking. The model weights are varying as a function of data, partially-pooled, and inferred using Bayesian inference. We further incorporate discrete and continuous inputs, other structured priors, and time series and longitudinal data. To verify the p…
Collected data, which is used for analysis or prediction tasks, often have a hierarchical structure, for example, data from various people performing the same task. Modeling the data's structure can improve the reliability of the derived results and prediction performance of newly unobserved data. Bayesian modeling provides a tool-kit for designing hierarchical models. However, Markov Chain Monte Carlo methods which are commonly used for parameter estimation are computationally expensive. This often renders its use for many applications not applicable. However, variational Bayesian methods all…
This paper presents the Bayesian Hierarchical Words Representation (BHWR) learning algorithm. BHWR facilitates Variational Bayes word representation learning combined with semantic taxonomy modeling via hierarchical priors. By propagating relevant information between related words, BHWR utilizes the taxonomy to improve the quality of such representations. Evaluation of several linguistic datasets demonstrates the advantages of BHWR over suitable alternatives that facilitate Bayesian modeling with or without semantic priors. Finally, we further show that BHWR produces better representations for…
This paper introduces a computational framework to incorporate flexible regularization techniques in ensemble Kalman methods for nonlinear inverse problems. The proposed methodology approximates the maximum a posteriori (MAP) estimate of a hierarchical Bayesian model characterized by a conditionally Gaussian prior and generalized gamma hyperpriors. Suitable choices of hyperparameters yield sparsity-promoting regularization. We propose an iterative algorithm for MAP estimation, which alternates between updating the unknown with an ensemble Kalman method and updating the hyperparameters in the r…
Istituto Dalle Molle di studi sull’Intelligenza Artificiale (IDSIA)（达勒莫勒人工智能研究所 (IDSIA)）；University of Ljubljana（卢布尔雅那大学）
Usually one compares the accuracy of two competing classifiers via null hypothesis significance tests (nhst). Yet the nhst tests suffer from important shortcomings, which can be overcome by switching to Bayesian hypothesis testing. We propose a Bayesian hierarchical model which jointly analyzes the cross-validation results obtained by two classifiers on multiple data sets. It returns the posterior probability of the accuracies of the two classifiers being practically equivalent or significantly different. A further strength of the hierarchical model is that, by jointly analyzing the results ob…
Methods based on Bayesian decision tree ensembles have proven valuable in constructing high-quality predictions, and are particularly attractive in certain settings because they encourage low-order interaction effects. Despite adapting to the presence of low-order interactions for prediction purpose, we show that Bayesian decision tree ensembles are generally anti-conservative for the purpose of conducting interaction detection. We address this problem by introducing Dirichlet process forests (DP-Forests), which leverage the presence of low-order interactions by clustering the trees so that tr…
Deep ensembles can be considered as the current state-of-the-art for uncertainty quantification in deep learning. While the approach was originally proposed as a non-Bayesian technique, arguments supporting its Bayesian footing have been put forward as well. We show that deep ensembles can be viewed as an approximate Bayesian method by specifying the corresponding assumptions. Our findings lead to an improved approximation which results in an enlarged epistemic part of the uncertainty. Numerical examples suggest that the improved approximation can lead to more reliable uncertainties. Analytica…
Hierarchical models are increasingly used in many applications. Along with this increased use comes a desire to investigate whether the model is compatible with the observed data. Bayesian methods are well suited to eliminate the many (nuisance) parameters in these complicated models; in this paper we investigate Bayesian methods for model checking. Since we contemplate model checking as a preliminary, exploratory analysis, we concentrate on objective Bayesian methods in which careful specification of an informative prior distribution is avoided. Numerous examples are given and different propo…
Quantitative Fitness Analysis (QFA) is a high-throughput experimental and computational methodology for measuring the growth of microbial populations. QFA screens can be used to compare the health of cell populations with and without a mutation in a query gene in order to infer genetic interaction strengths genome-wide, examining thousands of separate genotypes. We introduce Bayesian, hierarchical models of population growth rates and genetic interactions that better reflect QFA experimental design than current approaches. Our new approach models population dynamics and genetic interaction sim…
We characterise the convergence of the Gibbs sampler which samples from the joint posterior distribution of parameters and missing data in hierarchical linear models with arbitrary symmetric error distributions. We show that the convergence can be uniform, geometric or sub-geometric depending on the relative tail behaviour of the error distributions, and on the parametrisation chosen. Our theory is applied to characterise the convergence of the Gibbs sampler on latent Gaussian process models. We indicate how the theoretical framework we introduce will be useful in analyzing more complex models…
We combine conditioning techniques with sparse grid quadrature rules to develop a computationally efficient method to approximate marginal, but not necessarily univariate, posterior quantities, yielding approximate Bayesian inference via Sparse grid Quadrature Evaluation (BISQuE) for hierarchical models. BISQuE reformulates posterior quantities as weighted integrals of conditional quantities, such as densities and expectations. Sparse grid quadrature rules allow computationally efficient approximation of high dimensional integrals, which appear in hierarchical models with many hyperparameters.…
We have developed a sophisticated statistical model for predicting the hitting performance of Major League baseball players. The Bayesian paradigm provides a principled method for balancing past performance with crucial covariates, such as player age and position. We share information across time and across players by using mixture distributions to control shrinkage for improved accuracy. We compare the performance of our model to current sabermetric methods on a held-out season (2006), and discuss both successes and limitations.
We classify two types of Hierarchical Bayesian Model found in the literature as Hierarchical Prior Model (HPM) and Hierarchical Stochastic Model (HSM). Then, we focus on studying the theoretical implications of the HSM. Using examples of polynomial functions, we show that the HSM is capable of separating different types of uncertainties in a system and quantifying uncertainty of reduced order models under the Bayesian model class selection framework. To tackle the huge computational cost for analyzing HSM, we propose an efficient approximation scheme based on Importance Sampling and Empirical …
This paper devises a fully Bayesian sample size determination method for hierarchical model-based small area estimation with a decision risk approach. A new loss function specified around a desired maximum posterior variance target implements conventional official statistics criteria of estimator reliability (coefficient of variation of up to 20 per cent). This approach comes with an efficient binary search algorithm identifying the minimum effective sample size needed to produce small area estimates under this threshold constraint. Traditional survey sampling design tools can then be used to …
This paper presents a hierarchical Bayesian model to reconstruct sparse images when the observations are obtained from linear transformations and corrupted by an additive white Gaussian noise. Our hierarchical Bayes model is well suited to such naturally sparse image applications as it seamlessly accounts for properties such as sparsity and positivity of the image via appropriate Bayes priors. We propose a prior that is based on a weighted mixture of a positive exponential distribution and a mass at zero. The prior has hyperparameters that are tuned automatically by marginalization over the hi…
Demographic studies of cosmic populations must contend with measurement errors and selection effects. We survey some of the key ideas astronomers have developed to deal with these complications, in the context of galaxy surveys and the literature on corrections for Malmquist and Eddington bias. From the perspective of modern statistics, such corrections arise naturally in the context of multilevel models, particularly in Bayesian treatments of such models: hierarchical Bayesian models. We survey some key lessons from hierarchical Bayesian modeling, including shrinkage estimation, which is clos…
In this study, we present a Bayesian hierarchical framework to model fluid-induced seismicity. The framework is based on a non-homogeneous Poisson process (NHPP) with a fluid-induced seismicity rate proportional to the rate of injected fluid. The fluid-induced seismicity rate model depends upon a set of physically meaningful parameters, and has been validated for six fluid-induced case studies. In line with the vision of hierarchical Bayesian modeling, the rate parameters are considered as random variables. We develop both the Bayesian inference and updating rules, which are used to develop a …
In this work we perform outlier detection using ensembles of neural networks obtained by variational approximation of the posterior in a Bayesian neural network setting. The variational parameters are obtained by sampling from the true posterior by gradient descent. We show our outlier detection results are comparable to those obtained using other efficient ensembling methods.
Bayesian hierarchical models are used to share information between related samples and obtain more accurate estimates of sample-level parameters, common structure, and variation between samples. When the parameter of interest is the distribution or density of a continuous variable, a hierarchical model for continuous distributions is required. A number of such models have been described in the literature using extensions of the Dirichlet process and related processes, typically as a distribution on the parameters of a mixing kernel. We propose a new hierarchical model based on the Pólya tree, …
Analyzing and understanding the structure of complex relational data is important in many applications including analysis of the connectivity in the human brain. Such networks can have prominent patterns on different scales, calling for a hierarchically structured model. We propose two non-parametric Bayesian hierarchical network models based on Gibbs fragmentation tree priors, and demonstrate their ability to capture nested patterns in simulated networks. On real networks we demonstrate detection of hierarchical structure and show predictive performance on par with the state of the art. We en…
Tree ensembles, such as random forests and boosted trees, are renowned for their high prediction performance. However, their interpretability is critically limited due to the enormous complexity. In this study, we present a method to make a complex tree ensemble interpretable by simplifying the model. Specifically, we formalize the simplification of tree ensembles as a model selection problem. Given a complex tree ensemble, we aim at obtaining the simplest representation that is essentially equivalent to the original one. To this end, we derive a Bayesian model selection algorithm that optimiz…
Prior sensitivity examination plays an important role in applied Bayesian analyses. This is especially true for Bayesian hierarchical models, where interpretability of the parameters within deeper layers in the hierarchy becomes challenging. In addition, lack of information together with identifiability issues may imply that the prior distributions for such models have an undesired influence on the posterior inference. Despite its relevance, informal approaches to prior sensitivity analysis are currently used. They require repetitive re-runs of the model with ad-hoc modified base prior paramet…
We propose a generalised framework for the updating of a prior ensemble to a posterior ensemble, an essential yet challenging part in ensemble-based filtering methods. The proposed framework is based on a generalised and fully Bayesian view on the traditional ensemble Kalman filter (EnKF). In the EnKF, the updating of the ensemble is based on Gaussian assumptions, whereas in our general setup the updating may be based on another parametric family. In addition, we propose to formulate an optimality criterion and to find the optimal update with respect to this criterion. The framework is fully B…
This paper proposes new methodology for sequential state and parameter estimation within the ensemble Kalman filter. The method is fully Bayesian and propagates the joint posterior density of states and parameters over time. In order to implement the method we consider two representations of the marginal posterior distribution of the parameters: a grid-based approach and a Gaussian approximation. Contrary to existing algorithms, the new method explicitly accounts for parameter uncertainty and provides a formal way to combine information about the parameters from data at different time periods.…
Hierarchical models are versatile tools for joint modeling of data sets arising from different, but related, sources. Fully Bayesian inference may, however, become computationally prohibitive if the source-specific data models are complex, or if the number of sources is very large. To facilitate computation, we propose an approach, where inference is first made independently for the parameters of each data set, whereupon the obtained posterior samples are used as observed data in a substitute hierarchical model, based on a scaled likelihood function. Compared to direct inference in a full hier…
Deployments of Bayesian Optimization (BO) for functions with stochastic evaluations, such as parameter tuning via cross validation and simulation optimization, typically optimize an average of a fixed set of noisy realizations of the objective function. However, disregarding the true objective function in this manner finds a high-precision optimum of the wrong function. To solve this problem, we propose Bayesian Optimization by Sampling Hierarchically (BOSH), a novel BO routine pairing a hierarchical Gaussian process with an information-theoretic framework to generate a growing pool of realiza…
We explore the link between deep ensembles and Gaussian processes (GPs) through the lens of the Neural Tangent Kernel (NTK): a recent development in understanding the training dynamics of wide neural networks (NNs). Previous work has shown that even in the infinite width limit, when NNs become GPs, there is no GP posterior interpretation to a deep ensemble trained with squared error loss. We introduce a simple modification to standard deep ensembles training, through addition of a computationally-tractable, randomised and untrainable function to each ensemble member, that enables a posterior i…
It is often of interest to combine available estimates of a similar quantity from multiple data sources. When the corresponding variances of each estimate are also available, a model should take into account the uncertainty of the estimates themselves as well as the uncertainty in the estimation of variances. In addition, if there exists a strong association between estimates and their variances, the correlation between these two quantities should also be considered. In this paper, we propose a bivariate hierarchical Bayesian model that jointly models the estimates and their estimated variance…
Massachusetts Institute of Technology（麻省理工学院）；Harvard University（哈佛大学）
In this note we provide detailed derivations of two versions of small-variance asymptotics for hierarchical Dirichlet process (HDP) mixture models and the HDP hidden Markov model (HDP-HMM, a.k.a. the infinite HMM). We include derivations for the probabilities of certain CRP and CRF partitions, which are of more general interest.
In this paper, we employ variational arguments to establish a connection between ensemble methods for Neural Networks and Bayesian inference. We consider an ensemble-based scheme where each model/particle corresponds to a perturbation of the data by means of parametric bootstrap and a perturbation of the prior. We derive conditions under which any optimization steps of the particles makes the associated distribution reduce its divergence to the posterior over model parameters. Such conditions do not require any particular form for the approximation and they are purely geometrical, giving insig…
We propose a Bayesian hierarchical model for spatial extremes on a large domain. In the data layer a Gaussian elliptical copula having generalized extreme value (GEV) marginals is applied. Spatial dependence in the GEV parameters are captured with a latent spatial regression with spatially varying coefficients. Using a composite likelihood approach, we are able to efficiently incorporate a large precipitation dataset, which includes stations with missing data. The model is demonstrated by application to fall precipitation extremes at approximately 2600 stations covering the western United Stat…
We present a case study and methodological developments in large-scale hierarchical dynamic modeling for personalized prediction in commerce. The context is supermarket sales, where improved forecasting of customer/household-specific purchasing behavior informs decisions about personalized pricing and promotions on a continuing basis. This is a big data, big modeling and forecasting setting involving many thousands of customers and items on sale, requiring sequential analysis, addressing information flows at multiple levels over time, and with heterogeneity of customer profiles and item catego…
