This article revisits the problem of Bayesian shape-restricted inference in the light of a recently developed approximate Gaussian process that admits an equivalent formulation of the shape constraints in terms of the basis coefficients. We propose a strategy to efficiently sample from the resulting constrained posterior by absorbing a smooth relaxation of the constraint in the likelihood and using circulant embedding techniques to sample from the unconstrained modified prior. We additionally pay careful attention to mitigate the computational complexity arising from updating hyperparameters within the covariance kernel of the Gaussian process. The developed algorithm is shown to be accurate and highly efficient in simulated and real data examples.
In this paper, we consider the multicollinearity problem in the gamma regression model when model parameters are linearly restricted. The linear restrictions are available from prior information to ensure the validity of scientific theories or structural consistency based on physical phenomena. In order to make relevant statistical inference for a model any available knowledge and prior information on the model parameters should be taken into account. This paper proposes therefore an algorithm to acquire Bayesian estimator for the parameters of a gamma regression model subjected to some linear…
Bayesian methods have proven themselves to be successful across a wide range of scientific problems and have many well-documented advantages over competing methods. However, these methods run into difficulties for two major and prevalent classes of problems: handling data sets with outliers and dealing with model misspecification. We outline the drawbacks of previous solutions to both of these problems and propose a new method as an alternative. When working with the new method, the data is summarized through a set of insufficient statistics, targeting inferential quantities of interest, and t…
This paper considers reparameterization invariant Bayesian point estimates and credible regions of model parameters for scientific inference and communication. The effect of intrinsic loss function choice in Bayesian intrinsic estimates and regions is studied with the following findings. A particular intrinsic loss function, using Kullback-Leibler divergence from the full model to the restricted model, has strong connection to a Bayesian predictive criterion, which produces point estimates with the best predictive performance. An alternative intrinsic loss function, using Kullback-Leibler dive…
Constrained learning is prevalent in many statistical tasks. Recent work proposes distance-to-set penalties to derive estimators under general constraints that can be specified as sets, but focuses on obtaining point estimates that do not come with corresponding measures of uncertainty. To remedy this, we approach distance-to-set regularization from a Bayesian lens. We consider a class of smooth distance-to-set priors, showing that they yield well-defined posteriors toward quantifying uncertainty for constrained learning problems. We discuss relationships and advantages over prior work on Baye…
In social and biomedical sciences testing in contingency tables often involves order restrictions on cell-probabilities parameters. We develop objective Bayes methods for order-constrained testing and model comparison when observations arise under product binomial or multinomial sampling. Specifically, we consider tests for monotone order of the parameters against equality of all parameters. Our strategy combines in a unified way both the intrinsic prior methodology and the encompassing prior approach in order to compute Bayes factors and posterior model probabilities. Performance of our metho…
Prior information often takes the form of parameter constraints. Bayesian methods include such information through prior distributions having constrained support. By using posterior sampling algorithms, one can quantify uncertainty without relying on asymptotic approximations. However, sharply constrained priors are (a) not necessary in some settings; and (b) tend to limit modeling scope to a narrow set of distributions that are tractable computationally. Inspired by the vast literature that replaces the slab-and-spike prior with a continuous approximation, we propose to replace the sharp indi…
This paper investigates Bayesian variable selection when there is a hierarchical dependence structure on the inclusion of predictors in the model. In particular, we study the type of dependence found in polynomial response surfaces of orders two and higher, whose model spaces are required to satisfy weak or strong heredity conditions. These conditions restrict the inclusion of higher-order terms depending upon the inclusion of lower-order parent terms. We develop classes of priors on the model space, investigate their theoretical and finite sample properties, and provide a Metropolis-Hastings …
Nonparametric Bayesian models are used routinely as flexible and powerful models of complex data. Many times, a statistician may have additional informative beliefs about data distribution of interest, e.g., its mean or subset components, that is not part of, or even compatible with, the nonparametric prior. An important challenge is then to incorporate this partial prior belief into nonparametric Bayesian models. In this paper, we are motivated by settings where practitioners have additional distributional information about a subset of the coordinates of the observations being modeled. Our ap…
Multistage ranking models, including the popular Plackett-Luce distribution (PL), rely on the assumption that the ranking process is performed sequentially, by assigning the positions from the top to the bottom one (forward order). A recent contribution to the ranking literature relaxed this assumption with the addition of the discrete-valued reference order parameter, yielding the novel Extended Plackett-Luce model (EPL). Inference on the EPL and its generalization into a finite mixture framework was originally addressed from the frequentist perspective. In this work, we propose the Bayesian …
In this paper, we propose a general framework for combining evidence of varying quality to estimate underlying binary latent variables in the presence of restrictions imposed to respect the scientific context. The resulting algorithms cluster the multivariate binary data in a manner partly guided by prior knowledge. The primary model assumptions are that 1) subjects belong to classes defined by unobserved binary states, such as the true presence or absence of pathogens in epidemiology, or of antibodies in medicine, or the "ability" to correctly answer test questions in psychology, 2) a binary …
Squared error loss remains the most commonly used loss function for constructing a Bayes estimator of the parameter of interest. However, it can lead to sub-optimal solutions when a parameter is defined in a restricted space. It can also be an inappropriate choice in the context when an extreme overestimation and/or underestimation results in severe consequences and a more conservative estimator is preferred. We advocate a class of loss functions for parameters defined on restricted spaces which infinitely penalize boundary decisions like the squared error loss does on the real line. We also r…
For estimating a lower bounded parametric function in the framework of Marchand and Strawderman (2006), we provide through a unified approach a class of Bayesian confidence intervals with credibility $1-α$ and frequentist coverage probability bounded below by $\frac{1-α}{1+α}$. In cases where the underlying pivotal distribution is symmetric, the findings represent extensions with respect to the specification of the credible set achieved through the choice of a {\it spending function}, and include Marchand and Strawderman's HPD procedure result. For non-symmetric cases, the determination of a s…
Shape constrained regression analysis has applications in dose-response modeling, environmental risk assessment, disease screening and many other areas. Incorporating the shape constraints can improve estimation efficiency and avoid implausible results. We propose two novel methods focusing on Bayesian monotone curve and surface estimation using Gaussian process projections. The first projects samples from an unconstrained prior, while the second projects samples from the Gaussian process posterior. Theory is developed on continuity of the projection, posterior consistency and rates of contrac…
Clustering is a well-known and studied problem, one of its variants, called contiguity-constrained clustering, accepts as a second input a graph used to encode prior information about cluster structure by means of contiguity constraints i.e. clusters must form connected subgraphs of this graph. This paper discusses the interest of such a setting and proposes a new way to formalise it in a Bayesian setting, using results on spanning trees to compute exactly a posteriori probabilities of candidate partitions. An algorithmic solution is then investigated to find a maximum a posteriori (MAP) parti…
In this article we consider a simple step stress set up under the cumulative exposure model assumption. At each stress level the lifetime distribution of the experimental units are assumed to follow the generalized exponential distribution. We provide the order restricted Bayesian inference of the model parameters by considering the fact that the expected lifetime of the experimental units are larger in lower stress level. Analysis and the related results are extended to different censoring schemes also. The Bayes estimates and the associated credible intervals of the unknown parameters are co…
Consider the setting where there are B>1 candidate statistical models, and one is interested in model selection. Two common approaches to solve this problem are to select a single model or to combine the candidate models through model averaging. Instead, we select a subset of the combined parameter space associated with the models. Specifically, a model averaging perspective is used to increase the parameter space, and a model selection criterion is used to select a subset of this expanded parameter space. We account for the variability of the criterion by adapting Yekutieli (2012)'s method to…
In this paper, we investigate the statistical convergence rate of a Bayesian low-rank tensor estimator. Our problem setting is the regression problem where a tensor structure underlying the data is estimated. This problem setting occurs in many practical applications, such as collaborative filtering, multi-task learning, and spatio-temporal data analysis. The convergence rate is analyzed in terms of both in-sample and out-of-sample predictive accuracies. It is shown that a near optimal rate is achieved without any strong convexity of the observation. Moreover, we show that the method has adapt…
The multivariate normal linear model is one of the most widely employed models for statistical inference in applied research. Special cases include (multivariate) t testing, (M)AN(C)OVA, (multivariate) multiple regression, and repeated measures analysis. Statistical procedures for model selection where the models may have equality and order constraints on the model parameters of interest are limited however. This paper presents a default Bayes factor for this model selection problem. The default Bayes factor is based on generalized fractional Bayes methodology where different fractions are use…
A major problem for the learning of Bayesian networks (BNs) is the exponential number of parameters needed for conditional probability tables. Recent research reduces this complexity by modeling local structure in the probability tables. We examine the use of log-linear local models. While log-linear models in this context are not new (Whittaker, 1990; Buntine, 1991; Neal, 1992; Heckerman and Meek, 1997), for structure learning they are generally subsumed under a naive Bayes model. We describe an alternative interpretation, and use a Minimum Message Length (MML) (Wallace, 1987) metric for stru…
We consider the problem of shape restricted nonparametric regression on a closed set X ?\in R; where it is reasonable to assume the function has no more than H local extrema interior to X: Following a Bayesian approach we develop a nonparametric prior over a novel class of local extrema splines. This approach is shown to be consistent when modeling any continuously differentiable function within the class of functions considered, and is used to develop methods for hypothesis testing on the shape of the curve. Sampling algorithms are developed, and the method is applied in simulation studies an…
RIKEN Center for Advanced Intelligence Project（日本理化学研究所先进智能项目中心）；Institute for Physics of Intelligence（智能物理研究所）；The University of Tokyo（东京大学）
Restricted Boltzmann machines (RBMs) with low-precision synapses are much appealing with high energy efficiency. However, training RBMs with binary synapses is challenging due to the discrete nature of synapses. Recently Huang proposed one efficient method to train RBMs with binary synapses by using a combination of gradient ascent and the message passing algorithm under the variational inference framework. However, additional heuristic clipping operation is needed. In this technical note, inspired from Huang's work , we propose one alternative optimization method using the Bayesian learning r…
Size-constrained clustering (SCC) refers to the dual problem of using observations to determine latent cluster structure while at the same time assigning observations to the unknown clusters subject to an analyst defined constraint on cluster sizes. While several approaches have been proposed, SCC remains a difficult problem due to the combinatorial dependency between observations introduced by the size-constraints. Here we reformulate SCC as a decision problem and introduce a novel loss function to capture various types of size constraints. As opposed to prior work, our approach is uniquely s…
This paper develops meshless methods for probabilistically describing discretisation error in the numerical solution of partial differential equations. This construction enables the solution of Bayesian inverse problems while accounting for the impact of the discretisation of the forward problem. In particular, this drives statistical inferences to be more conservative in the presence of significant solver error. Theoretical results are presented describing rates of convergence for the posteriors in both the forward and inverse problems. This method is tested on a challenging inverse problem w…
Functional BART with Shape Priors: A Bayesian Tree Approach to Constrained Functional Regression
The University of Texas Health Science Center at Houston（德克萨斯大学健康科学中心休斯顿分校）；Beijing Technology and Business University（北京工商大学）；Beijing Normal-Hong Kong Baptist University（香港中文大学（深圳））
Automatic Chemical Design is a framework for generating novel molecules with optimized properties. The original scheme, featuring Bayesian optimization over the latent space of a variational autoencoder, suffers from the pathology that it tends to produce invalid molecular structures. First, we demonstrate empirically that this pathology arises when the Bayesian optimization scheme queries latent points far away from the data on which the variational autoencoder has been trained. Secondly, by reformulating the search procedure as a constrained Bayesian optimization problem, we show that the ef…
In professional tennis, it is often acknowledged that the server has an initial advantage. Indeed, the majority of points are won by the server, making the serve one of the most important elements in this sport. In this paper, we focus on the role of the serve advantage in winning a point as a function of the rally length. We propose a Bayesian isotonic logistic regression model for the probability of winning a point on serve. In particular, we decompose the logit of the probability of winning via a linear combination of B-splines basis functions, with athlete-specific basis function coefficie…
Many real-world optimisation problems such as hyperparameter tuning in machine learning or simulation-based optimisation can be formulated as expensive-to-evaluate black-box functions. A popular approach to tackle such problems is Bayesian optimisation (BO), which builds a response surface model based on the data collected so far, and uses the mean and uncertainty predicted by the model to decide what information to collect next. In this paper, we propose a novel variant of the well-known Knowledge Gradient acquisition function that allows it to handle constraints. We empirically compare the n…
Duke University（杜克大学）；Iowa State University（爱荷华州立大学）
A restricted Boltzmann machine (RBM) is an undirected graphical model constructed for discrete or continuous random variables, with two layers, one hidden and one visible, and no conditional dependency within a layer. In recent years, RBMs have risen to prominence due to their connection to deep learning. By treating a hidden layer of one RBM as the visible layer in a second RBM, a deep architecture can be created. RBMs are thought to thereby have the ability to encode very complex and rich structures in data, making them attractive for supervised learning. However, the generative behavior of …
We present a novel approach for constrained Bayesian inference. Unlike current methods, our approach does not require convexity of the constraint set. We reduce the constrained variational inference to a parametric optimization over the feasible set of densities and propose a general recipe for such problems. We apply the proposed constrained Bayesian inference approach to multitask learning subject to rank constraints on the weight matrix. Further, constrained parameter estimation is applied to recover the sparse conditional independence structure encoded by prior precision matrices. Our appr…
Southern University of Science and Technology（南方科技大学）
Black-box problems are common in real life like structural design, drug experiments, and machine learning. When optimizing black-box systems, decision-makers always consider multiple performances and give the final decision by comprehensive evaluations. Motivated by such practical needs, we focus on constrained black-box problems where the objective and constraints lack known special structure, and evaluations are expensive and even with noise. We develop a novel constrained Bayesian optimization approach based on the knowledge gradient method ($c-\rm{KG}$). A new acquisition function is propo…
Bayesian neural network (BNN) priors are defined in parameter space, making it hard to encode prior knowledge expressed in function space. We formulate a prior that incorporates functional constraints about what the output can or cannot be in regions of the input space. Output-Constrained BNNs (OC-BNN) represent an interpretable approach of enforcing a range of constraints, fully consistent with the Bayesian framework and amenable to black-box inference. We demonstrate how OC-BNNs improve model robustness and prevent the prediction of infeasible outputs in two real-world applications of health…
Motivated by the success of Bayesian optimisation algorithms in the Euclidean space, we propose a novel approach to construct Intrinsic Bayesian optimisation (In-BO) on manifolds with a primary focus on complex constrained domains or irregular-shaped spaces arising as submanifolds of R2, R3 and beyond. Data may be collected in a spatial domain but restricted to a complex or intricately structured region corresponding to a geographic feature, such as lakes. Traditional Bayesian Optimisation (Tra-BO) defined with a Radial basis function (RBF) kernel cannot accommodate these complex constrained c…
We propose a new distribution, called the soft tMVN distribution, which provides a smooth approximation to the truncated multivariate normal (tMVN) distribution with linear constraints. An efficient blocked Gibbs sampler is developed to sample from the soft tMVN distribution in high dimensions. We provide theoretical support to the approximation capability of the soft tMVN and provide further empirical evidence thereof. The soft tMVN distribution can be used to approximate simulations from a multivariate truncated normal distribution with linear constraints, or itself as a prior in shape-const…
The global optimization of a high-dimensional black-box function under black-box constraints is a pervasive task in machine learning, control, and engineering. These problems are challenging since the feasible set is typically non-convex and hard to find, in addition to the curses of dimensionality and the heterogeneity of the underlying functions. In particular, these characteristics dramatically impact the performance of Bayesian optimization methods, that otherwise have become the de facto standard for sample-efficient optimization in unconstrained settings, leaving practitioners with evolu…
ONERA（法国国家航空航天研究中心）；Université de Toulouse（图卢兹大学）；ISAE-SUPAERO（法国国立高等航空航天学院）
Bayesian optimization methods have been successfully applied to black box optimization problems that are expensive to evaluate. In this paper, we adapt the so-called super effcient global optimization algorithm to solve more accurately mixed constrained problems. The proposed approach handles constraints by means of upper trust bound, the latter encourages exploration of the feasible domain by combining the mean prediction and the associated uncertainty function given by the Gaussian processes. On top of that, a refinement procedure, based on a learning rate criterion, is introduced to enhance…
Bayesian optimization (BO) is a model-based approach to sequentially optimize expensive black-box functions, such as the validation error of a deep neural network with respect to its hyperparameters. In many real-world scenarios, the optimization is further subject to a priori unknown constraints. For example, training a deep network configuration may fail with an out-of-memory error when the model is too large. In this work, we focus on a general formulation of Gaussian process-based BO with continuous or binary constraints. We propose constrained Max-value Entropy Search (cMES), a novel info…
