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 made using maximum likelihood and the Akaike information criterion for minimally-supported designs that are constructed analytically. The results show that effective screening, that is, high power with moderate type I error rate and false discovery rate, can be achieved through suitable choices for the number of design support points and experiment size. Logistic regression is shown to present a more challenging problem than log-linear regression. Some areas for future work are also indicated.
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…
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…
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…
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…
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…
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 …
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…
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.
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.
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 …
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 present a general framework for Bayesian estimation of incompletely observed multivariate diffusion processes. Observations are assumed to be discrete in time, noisy and incomplete. We assume the drift and diffusion coefficient depend on an unknown parameter. A data-augmentation algorithm for drawing from the posterior distribution is presented which is based on simulating diffusion bridges conditional on a noisy incomplete observation at an intermediate time. The dynamics of such filtered bridges are derived and it is shown how these can be simulated using a generalised version of the guid…
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 …
When can a posterior predictive check identify the learning rate? Exact degeneracy in Gaussian models and implications for Generalised Bayesian Inerence
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 …
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…
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…
Gaussian process latent variable models (GPLVM) are a flexible and non-linear approach to dimensionality reduction, extending classical Gaussian processes to an unsupervised learning context. The Bayesian incarnation of the GPLVM Titsias and Lawrence, 2010] uses a variational framework, where the posterior over latent variables is approximated by a well-behaved variational family, a factorized Gaussian yielding a tractable lower bound. However, the non-factories ability of the lower bound prevents truly scalable inference. In this work, we study the doubly stochastic formulation of the Bayesia…
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.…
Scalable Bayesian structure learning of directed acyclic graphs via Laplace approximation, with an application to breast cancer gene expression networks
Ferdowsi University of Mashhad（马什哈德 Ferdowsi 大学）；North Carolina Agricultural and Technical State University（北卡罗来纳农业与州立大学）
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.
Most estimates for penalised linear regression can be viewed as posterior modes for an appropriate choice of prior distribution. Bayesian shrinkage methods, particularly the horseshoe estimator, have recently attracted a great deal of attention in the problem of estimating sparse, high-dimensional linear models. This paper extends these ideas, and presents a Bayesian grouped model with continuous global-local shrinkage priors to handle complex group hierarchies that include overlapping and multilevel group structures. As the posterior mean is never a sparse estimate of the linear model coeffic…
Statisticians often face the choice between using probability models or a paradigm defined by minimising a loss function. Both approaches are useful and, if the loss can be re-cast into a proper probability model, there are many tools to decide which model or loss is more appropriate for the observed data, in the sense of explaining the data's nature. However, when the loss leads to an improper model, there are no principled ways to guide this choice. We address this task by combining the Hyvärinen score, which naturally targets infinitesimal relative probabilities, and general Bayesian updati…
Inria Lille - Nord Europe（法国国家信息与自动化研究所里尔-北欧中心）；University College London（伦敦大学学院）
Generalised Bayesian learning algorithms are increasingly popular in machine learning, due to their PAC generalisation properties and flexibility. The present paper aims at providing a self-contained survey on the resulting PAC-Bayes framework and some of its main theoretical and algorithmic developments.
Despite the popularity of sparse factor models, little attention has been given to formally address identifiability of these models beyond standard rotation-based identification such as the positive lower triangular constraint. To fill this gap, we provide a counting rule on the number of nonzero factor loadings that is sufficient for achieving uniqueness of the variance decomposition in the factor representation. Furthermore, we introduce the generalised lower triangular representation to resolve rotational invariance and show that within this model class the unknown number of common factors …
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）
Normalised generalised gamma processes are random probability measures that induce nonparametric prior distributions widely used in Bayesian statistics, particularly for mixture modelling. We construct a class of dependent normalised generalised gamma priors induced by a stationary population model of Moran type, which exploits a generalised Pólya urn scheme associated with the prior. We study the asymptotic scaling for the dynamics of the number of clusters in the sample, which in turn provides a dynamic measure of diversity in the underlying population. The limit is formalised to be a positi…
We explore the effect of finite population sampling in design problems with many variables cross-classified in many ways. In particular, we investigate designs where we wish to sample individuals belonging to different groups for which the underlying covariance matrices are separable between groups and variables. We exploit the generalised conditional independence structure of the model to show how the analysis of the full model can be reduced to an interpretable series of lower dimensional problems. The types of information we gain by sampling are identified with the orthogonal canonical dire…
In this paper, we introduce the notion of Gaussian processes indexed by probability density functions for extending the Matérn family of covariance functions. We use some tools from information geometry to improve the efficiency and the computational aspects of the Bayesian learning model. We particularly show how a Bayesian inference with a Gaussian process prior (covariance parameters estimation and prediction) can be put into action on the space of probability density functions. Our framework has the capacity of classifiying and infering on data observations that lie on nonlinear subspaces.…
University of Warwick（华威大学）；The Alan Turing Institute（艾伦·图灵研究所）
We introduce a framework for inference in general state-space hidden Markov models (HMMs) under likelihood misspecification. In particular, we leverage the loss-theoretic perspective of Generalized Bayesian Inference (GBI) to define generalised filtering recursions in HMMs, that can tackle the problem of inference under model misspecification. In doing so, we arrive at principled procedures for robust inference against observation contamination by utilising the $β$-divergence. Operationalising the proposed framework is made possible via sequential Monte Carlo methods (SMC), where most standard…
