A stream of algorithmic advances has steadily increased the popularity of the Bayesian approach as an inference paradigm, both from the theoretical and applied perspective. Even with apparent successes in numerous application fields, a rising concern is the robustness of Bayesian inference in the presence of model misspecification, which may lead to undesirable extreme behavior of the posterior distributions for large sample sizes. Generalized belief updating with a loss function represents a central principle to making Bayesian inference more robust and less vulnerable to deviations from the assumed model. Here we consider such updates with $f$-divergences to quantify a discrepancy between the assumed statistical model and the probability distribution which generated the observed data. Since the latter is generally unknown, estimation of the divergence may be viewed as an intractable problem. We show that the divergence becomes accessible through the use of probabilistic classifiers that can leverage an estimate of the ratio of two probability distributions even when one or both of them is unknown. We demonstrate the behavior of generalized belief updates for various specific choices under the $f$-divergence family. We show that for specific divergence functions such an approach can even improve on methods evaluating the correct model likelihood function analytically.
Generalised Bayesian inference updates prior beliefs using a loss function, rather than a likelihood, and can therefore be used to confer robustness against possible mis-specification of the likelihood. Here we consider generalised Bayesian inference with a Stein discrepancy as a loss function, motivated by applications in which the likelihood contains an intractable normalisation constant. In this context, the Stein discrepancy circumvents evaluation of the normalisation constant and produces generalised posteriors that are either closed form or accessible using standard Markov chain Monte Ca…
We introduce a methodology for robust Bayesian estimation with robust divergence (e.g., density power divergence or γ-divergence), indexed by a single tuning parameter. It is well known that the posterior density induced by robust divergence gives highly robust estimators against outliers if the tuning parameter is appropriately and carefully chosen. In a Bayesian framework, one way to find the optimal tuning parameter would be using evidence (marginal likelihood). However, we numerically illustrate that evidence induced by the density power divergence does not work to select the optimal tunin…
The first investigation is made of designs for screening experiments where the response variable is approximated by a generalised linear model. A Bayesian information capacity criterion is defined for the selection of designs that are robust to the form of the linear predictor. For binomial data and logistic regression, the effectiveness of these designs for screening is assessed through simulation studies using all-subsets regression and model selection via maximum penalised likelihood and a generalised information criterion. For Poisson data and log-linear regression, similar assessments are…
In this paper, we discuss a method to define prior distributions for the threshold of a generalised Pareto distribution, in particular when its applications are directed to heavy-tailed data. We propose to assign prior probabilities to the order statistics of a given set of observations. In other words, we assume that the threshold coincides to one of the data points. We show two ways of defining a prior: by assigning equal mass to each order statistic, that is a uniform prior, and by considering the worth that every order statistic has in representing the true threshold. Both proposed priors …
The design of an experiment can be always be considered at least implicitly Bayesian, with prior knowledge used informally to aid decisions such as the variables to be studied and the choice of a plausible relationship between the explanatory variables and measured responses. Bayesian methods allow uncertainty in these decisions to be incorporated into design selection through prior distributions that encapsulate information available from scientific knowledge or previous experimentation. Further, a design may be explicitly tailored to the aim of the experiment through a decision-theoretic app…
In this work we propose a semiparametric bivariate copula whose density is defined by a piecewise constant function on disjoint squares. We obtain the maximum likelihood estimators of model parameters and prove that they reduce to the sample copula under specific conditions. We further propose to carry out a full Bayesian analysis of the model and introduce a spatial dependent prior distribution for the model parameters. This prior allows the parameters to borrow strength across neighbouring regions to produce smooth posterior estimates. To characterise the posterior distribution, via the full…
The Australian National University（澳大利亚国立大学）；NICTA（澳大利亚国家信息与通信技术研究院）；Microsoft Research（微软研究院）
Mixability of a loss is known to characterise when constant regret bounds are achievable in games of prediction with expert advice through the use of Vovk's aggregating algorithm. We provide a new interpretation of mixability via convex analysis that highlights the role of the Kullback-Leibler divergence in its definition. This naturally generalises to what we call $Φ$-mixability where the Bregman divergence $D_Φ$ replaces the KL divergence. We prove that losses that are $Φ$-mixable also enjoy constant regret bounds via a generalised aggregating algorithm that is similar to mirror descent.
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 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…
Divergence-based Robust Generalised Bayesian Inference for Directional Data via von Mises-Fisher models
School of Data Science, Meisei University（明成大学数据科学学院）；RIKEN Center for Brain Science（理化学研究所脑科学中心）；School of Business Administration, Meiji University（明治大学商学院）；Department of Information Sciences, Tokyo University of Science（东京科学技术大学信息科学系）
The relationship between short-term exposure to air pollution and mortality or morbidity has been the subject of much recent research, in which the standard method of analysis uses Poisson linear or additive models. In this paper we use a Bayesian dynamic generalised linear model (DGLM) to estimate this relationship, which allows the standard linear or additive model to be extended in two ways: (i) the long-term trend and temporal correlation present in the health data can be modelled by an autoregressive process rather than a smooth function of calendar time; (ii) the effects of air pollution…
In this Note we introduce a new methodology for Bayesian inference through the use of $ϕ$-divergences and the duality technique. The asymptotic laws of the estimates are established.
Optimal sampling strategies are critical for surveys of deeper coral reef and shoal systems, due to the significant cost of accessing and field sampling these remote and poorly understood ecosystems. Additionally, well-established standard diver-based sampling techniques used in shallow reef systems cannot be deployed because of water depth. Here we develop a Bayesian design strategy to optimise sampling for a shoal deep reef system using three years of pilot data. Bayesian designs are generally found by maximising the expectation of a utility function with respect to the joint distribution of…
A two-stage normal hierarchical model called the Fay--Herriot model and the empirical Bayes estimator are widely used to provide indirect and model-based estimates of means in small areas. However, the performance of the empirical Bayes estimator might be poor when the assumed normal distribution is misspecified. In this article, we propose a simple modification by using density power divergence and suggest a new robust empirical Bayes small area estimator. The mean squared error and estimated mean squared error of the proposed estimator are derived based on the asymptotic properties of the ro…
Divergence is not only an important mathematical concept in information theory, but also applied to machine learning problems such as low-dimensional embedding, manifold learning, clustering, classification, and anomaly detection. We proposed a bayesian model---co-BPM---to characterize the discrepancy of two sample sets, i.e., to estimate the divergence of their underlying distributions. In order to avoid the pitfalls of plug-in methods that estimate each density independently, our bayesian model attempts to learn a coupled binary partition of the sample space that best captures the landscapes…
Generalised Robust Bayes for Joint Inference of Model and Contamination
The University of Osaka（大阪大学）；Lattice Lab.（晶格实验室）；Toyota Motor Corporation（丰田汽车公司）；RIKEN AIP（理化学研究所先进智能研究中心）；The University of Sydney Business School（悉尼大学商学院）
Traditional Item Response Theory models assume the distribution of the abilities of the population in study to be Gaussian. However, this may not always be a reasonable assumption, which motivates the development of more general models. This paper presents a generalised approach for the distribution of the abilities in dichotomous 3-parameter Item Response models. A mixture of normal distributions is considered, allowing for features like skewness, multimodality and heavy tails. A solution is proposed to deal with model identifiability issues without compromising the flexibility and practical …
In this paper we introduce objective proper prior distributions for hypothesis testing and model selection based on measures of divergence between the competing models; we call them divergence based (DB) priors. DB priors have simple forms and desirable properties, like information (finite sample) consistency; often, they are similar to other existing proposals like the intrinsic priors; moreover, in normal linear models scenarios, they exactly reproduce Jeffreys-Zellner-Siow priors. Most importantly, in challenging scenarios such as irregular models and mixture models, the DB priors are well …
In this paper, we compare the performance of two methods for estimating Bayesian networks from data containing exogenous variables and random effects. The first method is fully Bayesian in which a prior distribution is placed on the exogenous variables, whereas the second method, which we call the residual approach, accounts for the effects of exogenous variables by using the notion of restricted maximum likelihood. We review the two score-based metrics, then study their performance by measuring the Kullback Leibler divergence, or distance, between the two resulting posterior density functions…
The paper proposes a novel model assessment paradigm aiming to address shortcoming of posterior predictive $p-$values, which provide the default metric of fit for Bayesian structural equation modelling (BSEM). The model framework of the paper focuses on the approximate zero approach, according to which parameters that would before set to zero (e.g. factor loadings) are now formulated to be approximate zero via informative priors (Muthen and Asparouhov, 2012). The introduced model assessment procedure monitors the out-of-sample predictive performance of the fitted model, and together with a lis…
In this paper, a Bayesian accelerated life testing model is presented. The Weibull distribution is used as the life distribution and the generalised Eyring model as the time transformation function. This is a model that allows for the use of more than one stressor, whereas other commonly used acceleration models, such as the Arrhenius and power law models, allow for only one stressor. The generalised Eyring-Weibull model is used in an application, where MCMC methods are utilised to generate samples for posterior inference.
Göteborgs Universitet（哥德堡大学）；Chalmers University of Technology（查尔姆斯理工大学）
Bayesian methods suffer from the problem of how to specify prior beliefs. One interesting idea is to consider worst-case priors. This requires solving a stochastic zero-sum game. In this paper, we extend well-known results from bandit theory in order to discover minimax-Bayes policies and discuss when they are practical.
Formulating a statistical inverse problem as one of inference in a Bayesian model has great appeal, notably for what this brings in terms of coherence, the interpretability of regularisation penalties, the integration of all uncertainties, and the principled way in which the set-up can be elaborated to encompass broader features of the context, such as measurement error, indirect observation, etc. The Bayesian formulation comes close to the way that most scientists intuitively regard the inferential task, and in principle allows the free use of subject knowledge in probabilistic model building…
We investigate Bayesian shrinkage methods for constructing predictive distributions. We consider the multivariate Normal model with a known covariance matrix and show that the Bayesian predictive density with respect to Stein's harmonic prior dominates the best invariant Bayesian predictive density, when the dimension is greater than three. Alpha-divergence from the true distribution to a predictive distribution is adopted as a loss function.
We consider the problem of parameter estimation in a Bayesian setting and propose a general lower-bound that includes part of the family of $f$-Divergences. The results are then applied to specific settings of interest and compared to other notable results in the literature. In particular, we show that the known bounds using Mutual Information can be improved by using, for example, Maximal Leakage, Hellinger divergence, or generalizations of the Hockey-Stick divergence.
Filtering and smoothing with a generalised representation of uncertainty is considered. Here, uncertainty is represented using a class of outer measures. It is shown how this representation of uncertainty can be propagated using outer-measure-type versions of Markov kernels and generalised Bayesian-like update equations. This leads to a system of generalised smoothing and filtering equations where integrals are replaced by supremums and probability density functions are replaced by positive functions with supremum equal to one. Interestingly, these equations retain most of the structure found …
Divergence time estimation is crucial to provide temporal signals for dating biologically important events, from species divergence to viral transmissions in space and time. With the advent of high-throughput sequencing, recent Bayesian phylogenetic studies have analyzed hundreds to thousands of sequences. Such large-scale analyses challenge divergence time reconstruction by requiring inference on highly-correlated internal node heights that often become computationally infeasible. To overcome this limitation, we explore a ratio transformation that maps the original N - 1 internal node heights…
This paper deals with measuring the Bayesian robustness of classes of contaminated priors. Two different classes of priors in the neighborhood of the elicited prior are considered. The first one is the well-known $ε$-contaminated class, while the second one is the geometric mixing class. The proposed measure of robustness is based on computing the curvature of Rényi divergence between posterior distributions. Examples are used to illustrate the results by using simulated and real data sets.
We propose an objective Bayesian approach to the selection of covariates and their penalised splines transformations in generalised additive models. Specification of a reasonable default prior for the model parameters and combination with a multiplicity-correction prior for the models themselves is crucial for this task. Here we use well-studied and well-behaved continuous mixtures of g-priors as default priors. We introduce the methodology in the normal model and extend it to non-normal exponential families. A simulation study and an application from the literature illustrate the proposed app…
When can a posterior predictive check identify the learning rate? Exact degeneracy in Gaussian models and implications for Generalised Bayesian Inerence
When it is acknowledged that all candidate parameterised statistical models are misspecified relative to the data generating process, the decision maker (DM) must currently concern themselves with inference for the parameter value minimising the KL-divergence between the model and the process (Walker, 2013). However, it has long been known that minimising the KL-divergence places a large weight on correctly capturing the tails of the sample distribution. As a result the DM is required to worry about the robustness of their model to tail misspecifications if they want to conduct principled infe…
Bayesian nonparametric statistics is an area of considerable research interest. While recently there has been an extensive concentration in developing Bayesian nonparametric procedures for model checking, the use of the Dirichlet process, in its simplest form, along with the Kullback-Leibler divergence is still an open problem. This is mainly attributed to the discreteness property of the Dirichlet process and that the Kullback-Leibler divergence between any discrete distribution and any continuous distribution is infinity. The approach proposed in this paper, which is based on incorporating 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…
The Alan Turing Institute（艾伦·图灵研究所）；University of Warwick（华威大学）
We present the very first robust Bayesian Online Changepoint Detection algorithm through General Bayesian Inference (GBI) with $β$-divergences. The resulting inference procedure is doubly robust for both the parameter and the changepoint (CP) posterior, with linear time and constant space complexity. We provide a construction for exponential models and demonstrate it on the Bayesian Linear Regression model. In so doing, we make two additional contributions: Firstly, we make GBI scalable using Structural Variational approximations that are exact as $β\to 0$. Secondly, we give a principled way o…
This paper considers estimation of the predictive density for a normal linear model with unknown variance under alpha-divergence loss for -1 <= alpha <= 1. We first give a general canonical form for the problem, and then give general expressions for the generalized Bayes solution under the above loss for each alpha. For a particular class of hierarchical generalized priors studied in Maruyama and Strawderman (2005, 2006) for the problems of estimating the mean vector and the variance respectively, we give the generalized Bayes predictive density. Additionally, we show that, for a subclass of t…
Forecasts of mortality provide vital information about future populations, with implications for pension and health-care policy as well as for decisions made by private companies about life insurance and annuity pricing. Stochastic mortality forecasts allow the uncertainty in mortality predictions to be taken into consideration when making policy decisions and setting product prices. Longer lifespans imply that forecasts of mortality at ages 90 and above will become more important in such calculations. This paper presents a Bayesian approach to the forecasting of mortality that jointly estimat…
Romanian Institute of Science and Technology（罗马尼亚科学技术研究院）
Bayesian Neural Networks (BNNs) often result uncalibrated after training, usually tending towards overconfidence. Devising effective calibration methods with low impact in terms of computational complexity is thus of central interest. In this paper we present calibration methods for BNNs based on the alpha divergences from Information Geometry. We compare the use of alpha divergence in training and in calibration, and we show how the use in calibration provides better calibrated uncertainty estimates for specific choices of alpha and is more efficient especially for complex network architectur…
University of Oxford（牛津大学）；Alan Turing Institute（艾伦·图灵研究所）
The Asymptotic Randomised Control (ARC) algorithm provides a rigorous approximation to the optimal strategy for a wide class of Bayesian bandits, while retaining low computational complexity. In particular, the ARC approach provides nearly optimal choices even when the payoffs are correlated or more than the reward is observed. The algorithm is guaranteed to asymptotically optimise the expected discounted payoff, with error depending on the initial uncertainty of the bandit. In this paper, we extend the ARC framework to consider a batched bandit problem where observations arrive from a general…
Period estimation is one of the central topics in astronomical time series analysis, where data is often unevenly sampled. Especially challenging are studies of stellar magnetic cycles, as there the periods looked for are of the order of the same length than the datasets themselves. The datasets often contain trends, the origin of which is either a real long-term cycle or an instrumental effect, but these effects cannot be reliably separated, while they can lead to erroneous period determinations if not properly handled. In this study we aim at developing a method that can handle the trends pr…
We develop a Bayesian approach for selecting the model which is the most supported by the data within a class of marginal models for categorical variables formulated through equality and/or inequality constraints on generalised logits (local, global, continuation or reverse continuation), generalised log-odds ratios and similar higher-order interactions. For each constrained model, the prior distribution of the model parameters is formulated following the encompassing prior approach. Then, model selection is performed by using Bayes factors which are estimated by an importance sampling method.…
We investigate optimal posteriors for recently introduced \cite{begin2016pac} chi-squared divergence based PAC-Bayesian bounds in terms of nature of their distribution, scalability of computations, and test set performance. For a finite classifier set, we deduce bounds for three distance functions: KL-divergence, linear and squared distances. Optimal posterior weights are proportional to deviations of empirical risks, usually with subset support. For uniform prior, it is sufficient to search among posteriors on classifier subsets ordered by these risks. We show the bound minimization for linea…
