We study a parametric estimation problem related to moment condition models. As an alternative to the generalized empirical likelihood (GEL) and the generalized method of moments (GMM), a Bayesian approach to the problem can be adopted, extending the MEM procedure to parametric moment conditions. We show in particular that a large number of GEL estimators can be interpreted as a maximum entropy solution. Moreover, we provide a more general field of applications by proving the method to be robust to approximate moment conditions.
Bayesian inference provides a flexible way of combining data with prior information. However, quantile regression is not equipped with a parametric likelihood, and therefore, Bayesian inference for quantile regression demands careful investigation. This paper considers the Bayesian empirical likelihood approach to quantile regression. Taking the empirical likelihood into a Bayesian framework, we show that the resultant posterior from any fixed prior is asymptotically normal; its mean shrinks toward the true parameter values, and its variance approaches that of the maximum empirical likelihood …
In this paper we are interested in empirical likelihood (EL) as a method of estimation, and we address the following two problems: (1) selecting among various empirical discrepancies in an EL framework and (2) demonstrating that EL has a well-defined probabilistic interpretation that would justify its use in a Bayesian context. Using the large deviations approach, a Bayesian law of large numbers is developed that implies that EL and the Bayesian maximum a posteriori probability (MAP) estimators are consistent under misspecification and that EL can be viewed as an asymptotic form of MAP. Estima…
The asymmetric Laplace density (ALD) is used as a working likelihood for Bayesian quantile regression. Sriram et al.(2013) derived posterior consistency for Bayesian linear quantile regression based on the misspecified ALD. While their paper also argued for $\sqrt{n}-$consistency, Sriram and Ramamoorthi (2017) highlighted that the argument was only valid for a rate less than $\sqrt{n}$. So, the question of $\sqrt{n}-$rate has remained unaddressed, but is necessary to carry out meaningful Bayesian inference based on the ALD. In this paper, we derive results to obtain posterior consistency rates…
Composite likelihoods are increasingly used in applications where the full likelihood is analytically unknown or computationally prohibitive. Although the maximum composite likelihood estimator has frequentist properties akin to those of the usual maximum likelihood estimator, Bayesian inference based on composite likelihoods has yet to be explored. In this paper we investigate the use of the Metropolis--Hastings algorithm to compute a pseudo-posterior distribution based on the composite likelihood. Two methodologies for adjusting the algorithm are presented and their performance on approximat…
Bayesian properties of the signed root likelihood ratio statistic are analysed. Conditions for first-order probability matching are derived by the examination of the Bayesian posterior and frequentist means of this statistic. Second-order matching conditions are shown to arise from matching of the Bayesian posterior and frequentist variances of a mean-adjusted version of the signed root statistic. Conditions for conditional probability matching in ancillary statistic models are derived and discussed.
We reveal the relations between the conditional normalized maximum likelihood (CNML) distributions and Bayesian predictive densities based on the latent information priors (LIPs). In particular, CNML3, which is one type of CNML distributions, is investigated. The Bayes projection of a predictive density, which is an information projection of the predictive density on a set of Bayesian predictive densities, is considered. We prove that the sum of the Bayes projection divergence of CNML3 and the conditional mutual information is asymptotically constant. This result implies that the Bayes project…
Works, briefly surveyed here, are concerned with two basic methods: Maximum Probability and Bayesian Maximum Probability; as well as with their asymptotic instances: Relative Entropy Maximization and Maximum Non-parametric Likelihood. Parametric and empirical extensions of the latter methods - Empirical Maximum Maximum Entropy and Empirical Likelihood - are also mentioned. The methods are viewed as tools for solving certain ill-posed inverse problems, called Pi-problem, Phi-problem, respectively. Within the two classes of problems, probabilistic justification and interpretation of the respecti…
We consider the use of randomised forward models and log-likelihoods within the Bayesian approach to inverse problems. Such random approximations to the exact forward model or log-likelihood arise naturally when a computationally expensive model is approximated using a cheaper stochastic surrogate, as in Gaussian process emulation (kriging), or in the field of probabilistic numerical methods. We show that the Hellinger distance between the exact and approximate Bayesian posteriors is bounded by moments of the difference between the true and approximate log-likelihoods. Example applications of …
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…
In recent years, infinite-dimensional methods have been introduced for the Gaussian channels estimation. The aim of this paper is to study the application of similar methods to Poisson channels. In particular we compute the Bayesian estimator of a Poisson channel using the likelihood ratio and the discrete Malliavin gradient. This algorithm is suitable for numerical implementation via the Monte-Carlo scheme. As an application we provide an new proof of the formula obtained recently by Guo, Shamai and Verduú relating some derivatives of the input-output mutual information of a time-continuous P…
Bayesian network (BN) structure learning from complete data has been extensively studied in the literature. However, fewer theoretical results are available for incomplete data, and most are related to the Expectation-Maximisation (EM) algorithm. Balov (2013) proposed an alternative approach called Node-Average Likelihood (NAL) that is competitive with EM but computationally more efficient; and he proved its consistency and model identifiability for discrete BNs. In this paper, we give general sufficient conditions for the consistency of NAL; and we prove consistency and identifiability for co…
In this paper we consider the problem of inference in statistical models characterized by moment restrictions by casting the problem within the Exponentially Tilted Empirical Likelihood (ETEL) framework. Because the ETEL function has a well defined probabilistic interpretation and plays the role of a nonparametric likelihood, a fully Bayesian semiparametric framework can be developed. We establish a number of powerful results surrounding the Bayesian ETEL framework in such models. One major concern driving our work is the possibility of misspecification. To accommodate this possibility, we sho…
In this article, we investigate large sample properties of model selection procedures in a general Bayesian framework when a closed form expression of the marginal likelihood function is not available or a local asymptotic quadratic approximation of the log-likelihood function does not exist. Under appropriate identifiability assumptions on the true model, we provide sufficient conditions for a Bayesian model selection procedure to be consistent and obey the Occam's razor phenomenon, i.e., the probability of selecting the "smallest" model that contains the truth tends to one as the sample size…
We demonstrate how to calculate posteriors for general CRM-based priors and likelihoods for Bayesian nonparametric models. We further show how to represent Bayesian nonparametric priors as a sequence of finite draws using a size-biasing approach---and how to represent full Bayesian nonparametric models via finite marginals. Motivated by conjugate priors based on exponential family representations of likelihoods, we introduce a notion of exponential families for CRMs, which we call exponential CRMs. This construction allows us to specify automatic Bayesian nonparametric conjugate priors for exp…
We propose a modification of a maximum likelihood procedure for tuning parameter values in models, based upon the comparison of their output to field data. Our methodology, which uses polynomial approximations of the sample space to increase the computational efficiency, differs from similar Bayesian estimation frameworks in the use of an alternative likelihood distribution, is shown to better address problems in which covariance information is lacking, than its more conventional counterpart. Lack of covariance information is a frequent challenge in large-scale geophysical estimation. This is …
This article surveys computational methods for posterior inference with intractable likelihoods, that is where the likelihood function is unavailable in closed form, or where evaluation of the likelihood is infeasible. We review recent developments in pseudo-marginal methods, approximate Bayesian computation (ABC), the exchange algorithm, thermodynamic integration, and composite likelihood, paying particular attention to advancements in scalability for large datasets. We also mention R and MATLAB source code for implementations of these algorithms, where they are available.
In this paper we revisit the weighted likelihood bootstrap, a method that generates samples from an approximate Bayesian posterior of a parametric model. We show that the same method can be derived, without approximation, under a Bayesian nonparametric model with the parameter of interest defined as minimising an expected negative log-likelihood under an unknown sampling distribution. This interpretation enables us to extend the weighted likelihood bootstrap to posterior sampling for parameters minimizing an expected loss. We call this method the loss-likelihood bootstrap. We make a connection…
Implementing Bayesian inference is often computationally challenging in applications involving complex models, and sometimes calculating the likelihood itself is difficult. Synthetic likelihood is one approach for carrying out inference when the likelihood is intractable, but it is straightforward to simulate from the model. The method constructs an approximate likelihood by taking a vector summary statistic as being multivariate normal, with the unknown mean and covariance matrix estimated by simulation for any given parameter value. Our article makes three contributions. The first shows that…
Approximate Bayesian computing is a powerful likelihood-free method that has grown increasingly popular since early applications in population genetics. However, complications arise in the theoretical justification for Bayesian inference conducted from this method with a non-sufficient summary statistic. In this paper, we seek to re-frame approximate Bayesian computing within a frequentist context and justify its performance by standards set on the frequency coverage rate. In doing so, we develop a new computational technique called approximate confidence distribution computing, yielding theor…
Two data-dependent information metrics are developed to quantify the information of the prior and likelihood functions within a parametric Bayesian model, one of which is closely related to the reference priors from Berger, Bernardo, and Sun, and information measure introduced by Lindley. A combination of theoretical, empirical, and computational support provides evidence that these information-theoretic metrics may be useful diagnostic tools when performing a Bayesian analysis.
Gibbs random fields play an important role in statistics, for example the autologistic model is commonly used to model the spatial distribution of binary variables defined on a lattice. However they are complicated to work with due to an intractability of the likelihood function. It is therefore natural to consider tractable approximations to the likelihood function. Composite likelihoods offer a principled approach to constructing such approximation. The contribution of this paper is to examine the performance of a collection of composite likelihood approximations in the context of Bayesian i…
Bayesian synthetic likelihood (BSL) is a popular method for performing approximate Bayesian inference when the likelihood function is intractable. In synthetic likelihood methods, the likelihood function is approximated parametrically via model simulations, and then standard likelihood-based techniques are used to perform inference. The Gaussian synthetic likelihood estimator has become ubiquitous in BSL literature, primarily for its simplicity and ease of implementation. However, it is often too restrictive and may lead to poor posterior approximations. Recently, a more flexible semi-parametr…
We propose a Bayesian elastic net that uses empirical likelihood and develop an efficient tuning of Hamiltonian Monte Carlo for posterior sampling. The proposed model relaxes the assumptions on the identity of the error distribution, performs well when the variables are highly correlated, and enables more straightforward inference by providing posterior distributions of the regression coefficients. The Hamiltonian Monte Carlo method implemented in Bayesian empirical likelihood overcomes the challenges that the posterior distribution lacks a closed analytic form and its domain is nonconvex. We …
It is common practice to use Laplace approximations to compute marginal likelihoods in Bayesian versions of generalised linear models (GLM). Marginal likelihoods combined with model priors are then used in different search algorithms to compute the posterior marginal probabilities of models and individual covariates. This allows performing Bayesian model selection and model averaging. For large sample sizes, even the Laplace approximation becomes computationally challenging because the optimisation routine involved needs to evaluate the likelihood on the full set of data in multiple iterations…
This article presents a Bayesian inferential method where the likelihood for a model is unknown but where data can easily be simulated from the model. We discretize simulated (continuous) data to estimate the implicit likelihood in a Bayesian analysis employing a Markov chain Monte Carlo algorithm. Three examples are presented as well as a small study on some of the method's properties.
We develop simple methods for constructing likelihoods and parameter priors for learning about the parameters and structure of a Bayesian network. In particular, we introduce several assumptions that permit the construction of likelihoods and parameter priors for a large number of Bayesian-network structures from a small set of assessments. The most notable assumption is that of likelihood equivalence, which says that data can not help to discriminate network structures that encode the same assertions of conditional independence. We describe the constructions that follow from these assumptions…
This paper proposes a new approach for Bayesian and maximum likelihood parameter estimation for stationary Gaussian processes observed on a large lattice with missing values. We propose an MCMC approach for Bayesian inference, and a Monte Carlo EM algorithm for maximum likelihood inference. Our approach uses data augmentation and circulant embedding of the covariance matrix, and provides exact inference for the parameters and the missing data. Using simulated data and an application to satellite sea surface temperatures in the Pacific Ocean, we show that our method provides accurate inference …
Expectation-Propagation for Bayesian Empirical Likelihood Inference
College of Computing and Data Science, Nanyang Technological University（南洋理工大学计算与数据科学学院）；School of Mathematics and Statistics, The University of Melbourne（墨尔本大学数学与统计学院）
Bayesian approaches to data analysis and machine learning are widespread and popular as they provide intuitive yet rigorous axioms for learning from data; see Bernardo and Smith (2004) and Bishop (2006). However, this rigour comes with a caveat that the Bayesian model is a precise reflection of Nature. There has been a recent trend to address potential model misspecification by raising the likelihood function to a power, primarily for robustness reasons, though not exclusively. In this paper we provide a coherent specification of the power parameter once the Bayesian model has been specified i…
We consider Bayesian inference problems with computationally intensive likelihood functions. We propose a Gaussian process (GP) based method to approximate the joint distribution of the unknown parameters and the data. In particular, we write the joint density approximately as a product of an approximate posterior density and an exponentiated GP surrogate. We then provide an adaptive algorithm to construct such an approximation, where an active learning method is used to choose the design points. With numerical examples, we illustrate that the proposed method has competitive performance agains…
Statistical analysis of max-stable processes used to model spatial extremes has been limited by the difficulty in calculating the joint likelihood function. This precludes all standard likelihood-based approaches, including Bayesian approaches. In this paper we present a Bayesian approach through the use of approximate Bayesian computing. This circumvents the need for a joint likelihood function by instead relying on simulations from the (unavailable) likelihood. This method is compared with an alternative approach based on the composite likelihood. We demonstrate that approximate Bayesian com…
A widely applicable Bayesian information criterion (Watanabe, 2013) is applicable for both regular and singular models in the model selection problem. This criterion tends to overestimate the log marginal likelihood. We identify an overestimating term of a widely applicable Bayesian information criterion. Adjustment of the term gives an asymptotically unbiased estimator of the leading two terms of asymptotic expansion of the log marginal likelihood. In numerical experiments on regular and singular models, the adjustment resulted in smaller bias than the original criterion.
Helsinki Institute for Information Technology HIIT（赫尔辛基信息技术研究所）；Aalto University（阿尔托大学）；University of Edinburgh（爱丁堡大学）
We consider Bayesian inference when only a limited number of noisy log-likelihood evaluations can be obtained. This occurs for example when complex simulator-based statistical models are fitted to data, and synthetic likelihood (SL) method is used to form the noisy log-likelihood estimates using computationally costly forward simulations. We frame the inference task as a sequential Bayesian experimental design problem, where the log-likelihood function is modelled with a hierarchical Gaussian process (GP) surrogate model, which is used to efficiently select additional log-likelihood evaluation…
After an autoencoder (AE) has learnt to reconstruct one dataset, it might be expected that the likelihood on an out-of-distribution (OOD) input would be low. This has been studied as an approach to detect OOD inputs. Recent work showed this intuitive approach can fail for the dataset pairs FashionMNIST vs MNIST. This paper suggests this is due to the use of Bernoulli likelihood and analyses why this is the case, proposing two fixes: 1) Compute the uncertainty of likelihood estimate by using a Bayesian version of the AE. 2) Use alternative distributions to model the likelihood.
Likelihood-free Bayesian inference algorithms are popular methods for calibrating the parameters of complex, stochastic models, required when the likelihood of the observed data is intractable. These algorithms characteristically rely heavily on repeated model simulations. However, whenever the computational cost of simulation is even moderately expensive, the significant burden incurred by likelihood-free algorithms leaves them unviable in many practical applications. The multifidelity approach has been introduced (originally in the context of approximate Bayesian computation) to reduce the s…
CREST（Crest研究中心）；ENSAE（法国国立统计与经济管理学院）；Institut Polytechnique de Paris（巴黎理工学院）；RIKEN Center for AI Project（理化学研究所人工智能项目中心）
In some misspecified settings, the posterior distribution in Bayesian statistics may lead to inconsistent estimates. To fix this issue, it has been suggested to replace the likelihood by a pseudo-likelihood, that is the exponential of a loss function enjoying suitable robustness properties. In this paper, we build a pseudo-likelihood based on the Maximum Mean Discrepancy, defined via an embedding of probability distributions into a reproducing kernel Hilbert space. We show that this MMD-Bayes posterior is consistent and robust to model misspecification. As the posterior obtained in this way mi…
Estimation of finite mixture models when the mixing distribution support is unknown is an important problem. This paper gives a new approach based on a marginal likelihood for the unknown support. Motivated by a Bayesian Dirichlet prior model, a computationally efficient stochastic approximation version of the marginal likelihood is proposed and large-sample theory is presented. By restricting the support to a finite grid, a simulated annealing method is employed to maximize the marginal likelihood and estimate the support. Real and simulated data examples show that this novel stochastic appro…
For many decades, statisticians have made attempts to prepare the Bayesian omelette without breaking the Bayesian eggs; that is, to obtain probabilistic likelihood-based inferences without relying on informative prior distributions. A recent example is Murray Aitkin's recent book, {\em Statistical Inference}, which presents an approach to statistical hypothesis testing based on comparisons of posterior distributions of likelihoods under competing models. Aitkin develops and illustrates his method using some simple examples of inference from iid data and two-way tests of independence. We analyz…
Pohang University of Science and Technology（浦项科技大学）；BARO AI
We propose a practical Bayesian optimization method using Gaussian process regression, of which the marginal likelihood is maximized where the number of model selection steps is guided by a pre-defined threshold. Since Bayesian optimization consumes a large portion of its execution time in finding the optimal free parameters for Gaussian process regression, our simple, but straightforward method is able to mitigate the time complexity and speed up the overall Bayesian optimization procedure. Finally, the experimental results show that our method is effective to reduce the execution time in mos…
A composite likelihood is a non-genuine likelihood function that allows to make inference on limited aspects of a model, such as marginal or conditional distributions. Composite likelihoods are not proper likelihoods and need therefore calibration for their use in inference, from both a frequentist and a Bayesian perspective. The maximizer to the composite likelihood can serve as an estimator and its variance is assessed by means of a suitably defined sandwich matrix. In the Bayesian setting, the composite likelihood can be adjusted by means of magnitude and curvature methods. Magnitude method…
A spectral approach to Bayesian inference is presented. It pursues the emulation of the posterior probability density. The starting point is a series expansion of the likelihood function in terms of orthogonal polynomials. From this spectral likelihood expansion all statistical quantities of interest can be calculated semi-analytically. The posterior is formally represented as the product of a reference density and a linear combination of polynomial basis functions. Both the model evidence and the posterior moments are related to the expansion coefficients. This formulation avoids Markov chain…
In this paper, we study the effects of different prior and likelihood choices for Bayesian matrix factorisation, focusing on small datasets. These choices can greatly influence the predictive performance of the methods. We identify four groups of approaches: Gaussian-likelihood with real-valued priors, nonnegative priors, semi-nonnegative models, and finally Poisson-likelihood approaches. For each group we review several models from the literature, considering sixteen in total, and discuss the relations between different priors and matrix norms. We extensively compare these methods on eight re…
