The use of objective prior in Bayesian applications has become a common practice to analyze data without subjective information. Formal rules usually obtain these priors distributions, and the data provide the dominant information in the posterior distribution. However, these priors are typically improper and may lead to improper posterior. Here, we show, for a general family of distributions, that the obtained objective priors for the parameters either follow a power-law distribution or has an asymptotic power-law behavior. As a result, we observed that the exponents of the model are between 0.5 and 1. Understand these behaviors allow us to easily verify if such priors lead to proper or improper posteriors directly from the exponent of the power-law. The general family considered in our study includes essential models such as Exponential, Gamma, Weibull, Nakagami-m, Haf-Normal, Rayleigh, Erlang, and Maxwell Boltzmann distributions, to list a few. In summary, we show that comprehending the mechanisms describing the shapes of the priors provides essential information that can be used in situations where additional complexity is presented.
Study of the bivariate normal distribution raises the full range of issues involving objective Bayesian inference, including the different types of objective priors (e.g., Jeffreys, invariant, reference, matching), the different modes of inference (e.g., Bayesian, frequentist, fiducial) and the criteria involved in deciding on optimal objective priors (e.g., ease of computation, frequentist performance, marginalization paradoxes). Summary recommendations as to optimal objective priors are made for a variety of inferences involving the bivariate normal distribution. In the course of the investi…
In multi-parameter models, reference priors typically depend on the parameter or quantity of interest, and it is well known that this is necessary to produce objective posterior distributions with optimal properties. There are, however, many situations where one is simultaneously interested in all the parameters of the model or, more realistically, in functions of them that include aspects such as prediction, and it would then be useful to have a single objective prior that could safely be used to produce reasonable posterior inferences for all the quantities of interest. In this paper, we con…
The choice of the prior distribution is a key aspect of Bayesian analysis. For the spatial regression setting a subjective prior choice for the parameters may not be trivial, from this perspective, using the objective Bayesian analysis framework a reference is introduced for the spatial Student-t regression model with unknown degrees of freedom. The spatial Student-t regression model poses two main challenges when eliciting priors: one for the spatial dependence parameter and the other one for the degrees of freedom. It is well-known that the propriety of the posterior distribution over object…
Objective priors for sequential experiments are considered. Common priors, such as the Jeffreys prior and the reference prior, will typically depend on the stopping rule used for the sequential experiment. New expressions for reference priors are obtained in various contexts, and computational issues involving such priors are considered.
In objective Bayesian model selection, no single criterion has emerged as dominant in defining objective prior distributions. Indeed, many criteria have been separately proposed and utilized to propose differing prior choices. We first formalize the most general and compelling of the various criteria that have been suggested, together with a new criterion. We then illustrate the potential of these criteria in determining objective model selection priors by considering their application to the problem of variable selection in normal linear models. This results in a new model selection objective…
Autoregressive cokriging models have been widely used to emulate multiple computer models with different levels of fidelity. The dependence structures are modeled via Gaussian processes at each level of fidelity, where covariance structures are often parameterized up to a few parameters. The predictive distributions typically require intensive Monte Carlo approximations in previous works. This article derives new closed-form formulas to compute the means and variances of predictive distributions in autoregressive cokriging models that only depend on correlation parameters. For parameter estima…
This paper considers the problem of defining distributions over graphical structures. We propose an extension of the hyper Markov properties of Dawid and Lauritzen [Ann. Statist. 21 (1993) 1272-1317], which we term structural Markov properties, for both undirected decomposable and directed acyclic graphs, which requires that the structure of distinct components of the graph be conditionally independent given the existence of a separating component. This allows the analysis and comparison of multiple graphical structures, while being able to take advantage of the common conditional independence…
Informative Bayesian priors are often difficult to elicit, and when this is the case, modelers usually turn to noninformative or objective priors. However, objective priors such as the Jeffreys and reference priors are not tractable to derive for many models of interest. We address this issue by proposing techniques for learning reference prior approximations: we select a parametric family and optimize a black-box lower bound on the reference prior objective to find the member of the family that serves as a good approximation. We experimentally demonstrate the method's effectiveness by recover…
The construction of objective priors is, at best, challenging for multidimensional parameter spaces. A common practice is to assume independence and set up the joint prior as the product of marginal distributions obtained via "standard" objective methods, such as Jeffreys or reference priors. However, the assumption of independence a priori is not always reasonable, and whether it can be viewed as strictly objective is still open to discussion. In this paper, by extending a previously proposed objective approach based on scoring rules for the one dimensional case, we propose a novel objective …
For in vivo research experiments with small sample sizes and available historical data, we propose a sequential Bayesian method for the Behrens-Fisher problem. We consider it as a model choice question with two models in competition: one for which the two expectations are equal and one for which they are different. The choice between the two models is performed through a Bayesian analysis, based on a robust choice of combined objective and subjective priors, set on the parameters space and on the models space. Three steps are necessary to evaluate the posterior probability of each model using …
Objective prior distributions represent an important tool that allows one to have the advantages of using the Bayesian framework even when information about the parameters of a model is not available. The usual objective approaches work off the chosen statistical model and in the majority of cases the resulting prior is improper, which can pose limitations to a practical implementation, even when the complexity of the model is moderate. In this paper we propose to take a novel look at the construction of objective prior distributions, where the connection with a chosen sampling distribution mo…
Objective Bayesian inference procedures are derived for the parameters of the multivariate random effects model generalized to elliptically contoured distributions. The posterior for the overall mean vector and the between-study covariance matrix is deduced by assigning two noninformative priors to the model parameter, namely the Berger and Bernardo reference prior and the Jeffreys prior, whose analytical expressions are obtained under weak distributional assumptions. It is shown that the only condition needed for the posterior to be proper is that the sample size is larger than the dimension …
In this work we apply the methodology of integral priors to handle Bayesian model selection in binomial regression models with a general link function. These models are very often used to investigate associations and risks in epidemiological studies where one goal is to exhibit whether or not an exposure is a risk factor for developing a certain disease; the purpose of the current paper is to test the effect of specific exposure factors. We formulate the problem as a Bayesian model selection case and solve it using objective Bayes factors. To construct the reference prior distributions on the …
The Yule-Simon distribution is usually employed in the analysis of frequency data. As the Bayesian literature, so far, ignored this distribution, here we show the derivation of two objective priors for the parameter of the Yule-Simon distribution. In particular, we discuss the Jeffreys prior and a loss-based prior, which has recently appeared in the literature. We illustrate the performance of the derived priors through a simulation study and the analysis of real datasets.
When dealing with Bayesian inference the choice of the prior often remains a debatable question. Empirical Bayes methods offer a data-driven solution to this problem by estimating the prior itself from an ensemble of data. In the nonparametric case, the maximum likelihood estimate is known to overfit the data, an issue that is commonly tackled by regularization. However, the majority of regularizations are ad hoc choices which lack invariance under reparametrization of the model and result in inconsistent estimates for equivalent models. We introduce a non-parametric, transformation invariant …
Bayesian methods are increasingly applied in these days in the theory and practice of statistics. Any Bayesian inference depends on a likelihood and a prior. Ideally one would like to elicit a prior from related sources of information or past data. However, in its absence, Bayesian methods need to rely on some "objective" or "default" priors, and the resulting posterior inference can still be quite valuable. Not surprisingly, over the years, the catalog of objective priors also has become prohibitively large, and one has to set some specific criteria for the selection of such priors. Our aim i…
We consider the Jeffreys-Lindley paradox from an objective Bayesian perspective by attempting to find priors representing complete indifference to sample size in the problem. This means that we ensure that the prior for the unknown mean and the prior predictive for the $t$-statistic are independent of the sample size. If successful, this would lead to Bayesian model comparison that was independent of sample size and ameliorate the paradox. Unfortunately, it leads to an improper scale-invariant prior for the unknown mean. We show, however, that a truncated scale-invariant prior delays the depen…
An objective Bayesian approach to estimate the number of degrees of freedom $(ν)$ for the multivariate $t$ distribution and for the $t$-copula, when the parameter is considered discrete, is proposed. Inference on this parameter has been problematic for the multivariate $t$ and, for the absence of any method, for the $t$-copula. An objective criterion based on loss functions which allows to overcome the issue of defining objective probabilities directly is employed. The support of the prior for $ν$ is truncated, which derives from the property of both the multivariate $t$ and the $t$-copula of …
In this paper we propose to make Bayesian inferences for the parameters of the Lomax distribution using non-informative priors, namely the Jeffreys prior and the reference prior. We assess Bayesian estimation through a Monte Carlo study with 500 simulated data sets. To evaluate the possible impact of prior specification on estimation, two criteria were considered: the bias and square root of the mean square error. The developed procedures are illustrated on a real data set.
In this paper we present a loss-based approach to change point analysis. In particular, we look at the problem from two perspectives. The first focuses on the definition of a prior when the number of change points is known a priori. The second contribution aims to estimate the number of change points by using a loss-based approach recently introduced in the literature. The latter considers change point estimation as a model selection exercise. We show the performance of the proposed approach on simulated data and real data sets.
London School of Economics（伦敦政治经济学院）；Royal Holloway College（皇家霍洛威学院）；University of London（伦敦大学）
In this paper the problem of learning appropriate bias for an environment of related tasks is examined from a Bayesian perspective. The environment of related tasks is shown to be naturally modelled by the concept of an {\em objective} prior distribution. Sampling from the objective prior corresponds to sampling different learning tasks from the environment. It is argued that for many common machine learning problems, although we don't know the true (objective) prior for the problem, we do have some idea of a set of possible priors to which the true prior belongs. It is shown that under these …
Deakin University（迪肯大学）；Office of National Statistics（国家统计局）；University of Western Sydney（西悉尼大学）；National University of Singapore（新加坡国立大学）
Bayesian optimization (BO) is a widely-used method for optimizing expensive (to evaluate) problems. At the core of most BO methods is the modeling of the objective function using a Gaussian Process (GP) whose covariance is selected from a set of standard covariance functions. From a weight-space view, this models the objective as a linear function in a feature space implied by the given covariance K, with an arbitrary Gaussian weight prior ${\bf w} \sim \mathcal{N} ({\bf 0}, {\bf I})$. In many practical applications there is data available that has a similar (covariance) structure to the objec…
Student-$t$ processes have recently been proposed as an appealing alternative non-parameteric function prior. They feature enhanced flexibility and predictive variance. In this work the use of Student-$t$ processes are explored for multi-objective Bayesian optimization. In particular, an analytical expression for the hypervolume-based probability of improvement is developed for independent Student-$t$ process priors of the objectives. Its effectiveness is shown on a multi-objective optimization problem which is known to be difficult with traditional Gaussian processes.
Regularity-informed data assimilation: A hierarchical Bayesian approach to ensemble Kalman filtering for hyperbolic conservation laws
Division of Applied Mathematics, Linköping University（林雪平大学应用数学系）；Center for Computational Science and Engineering, Massachusetts Institute of Technology（麻省理工学院计算科学与工程中心）
The beta-Bernoulli process provides a Bayesian nonparametric prior for models involving collections of binary-valued features. A draw from the beta process yields an infinite collection of probabilities in the unit interval, and a draw from the Bernoulli process turns these into binary-valued features. Recent work has provided stick-breaking representations for the beta process analogous to the well-known stick-breaking representation for the Dirichlet process. We derive one such stick-breaking representation directly from the characterization of the beta process as a completely random measure…
When two nested models are compared, using a Bayes factor, from an objective standpoint, two seemingly conflicting issues emerge at the time of choosing parameter priors under the two models. On the one hand, for moderate sample sizes, the evidence in favor of the smaller model can be inflated by diffuseness of the prior under the larger model. On the other hand, asymptotically, the evidence in favor of the smaller model typically accumulates at a slower rate. With reference to finitely discrete data models, we show that these two issues can be dealt with jointly, by combining intrinsic priors…
We present an objective Bayes method for covariance selection in Gaussian multivariate regression models whose error term has a covariance structure which is Markov with respect to a Directed Acyclic Graph (DAG). The scope is covariate-adjusted sparse graphical model selection, a topic of growing importance especially in the area of genetical genomics (eQTL analysis). Specifically, we provide a closed-form expression for the marginal likelihood of any DAG (with small parent sets) whose computation virtually requires no subjective elicitation by the user and involves only conjugate matrix norma…
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…
An initial screening experiment may lead to ambiguous conclusions regarding the factors which are active in explaining the variation of an outcome variable: thus adding follow-up runs becomes necessary. We propose a fully Bayes objective approach to follow-up designs, using prior distributions suitably tailored to model selection. We adopt a model criterion based on a weighted average of Kullback-Leibler divergences between predictive distributions for all possible pairs of models. When applied to real data, our method produces results which compare favorably to previous analyses based on subj…
One of the main approaches used to construct prior distributions for objective Bayes methods is the concept of random imaginary observations. Under this setup, the expected-posterior prior (EPP) offers several advantages, among which it has a nice and simple interpretation and provides an effective way to establish compatibility of priors among models. In this paper, we study the power-expected posterior prior as a generalization to the EPP in objective Bayesian model selection under normal linear models. We prove that it can be represented as a mixture of $g$-prior, like a wide range of prior…
This paper presents objective priors for robust Bayesian estimation against outliers based on divergences. The minimum $γ$-divergence estimator is well-known to work well estimation against heavy contamination. The robust Bayesian methods by using quasi-posterior distributions based on divergences have been also proposed in recent years. In objective Bayesian framework, the selection of default prior distributions under such quasi-posterior distributions is an important problem. In this study, we provide some properties of reference and moment matching priors under the quasi-posterior distribu…
Variational Autoencoder is a scalable method for learning latent variable models of complex data. It employs a clear objective that can be easily optimized. However, it does not explicitly measure the quality of learned representations. We propose a Variational Mutual Information Maximization Framework for VAE to address this issue. It provides an objective that maximizes the mutual information between latent codes and observations. The objective acts as a regularizer that forces VAE to not ignore the latent code and allows one to select particular components of it to be most informative with …
The purpose of this paper is to present a mathematical theory that can be used as a foundation for statistics that include improper priors. This theory includes improper laws in the initial axioms and has in particular Bayes theorem as a consequence. Another consequence is that some of the usual calculation rules are modified. This is important in relation to common statistical practice which usually include improper priors, but tends to use unaltered calculation rules. In some cases the results are valid, but in other cases inconsistencies may appear. The famous marginalization paradoxes exem…
Let $X_1,...,X_n$ be iid random vectors and $f\ge 0$ be a non-negative function. Let also $k(n) = {\rm Argmax}_{i=1,...,n} f(X_i)$. We are interested in the distribution of $X_{k(n)}$ and their limit theorems. In other words, what is the distribution the random vector where a function of its components is extreme? This question is motivated by a kind of inverse problem where one wants to determine the extremal behavior of $X$ when only explicitly observing $f(X)$. We shall refer to such types of results as to implicit extremes. It turns out that, as in the usual case of explicit extremes, all …
