Approximate Bayesian Inference for Survival Models

Abstract. Bayesian analysis of time‐to‐event data, usually called survival analysis, has received increasing attention in the last years. In Cox‐type models it allows to use information from the full likelihood instead of from a partial likelihood, so that the baseline hazard function and the model parameters can be jointly estimated. In general, Bayesian methods permit a full and exact posterior inference for any parameter or predictive quantity of interest. On the other side, Bayesian inference often relies on Markov chain Monte Carlo (MCMC) techniques which, from the user point of view, may appear slow at delivering answers. In this article, we show how a new inferential tool named integrated nested Laplace approximations can be adapted and applied to many survival models making Bayesian analysis both fast and accurate without having to rely on MCMC‐based inference.     

Approximate Bayesian inference in semi-mechanistic models

Inference of interaction networks represented by systems of differential equations is a challenging problem in many scientific disciplines. In the present article, we follow a semi-mechanistic modelling approach based on gradient matching. We investigate the extent to which key factors, including the kinetic model, statistical formulation and numerical methods, impact upon performance at network reconstruction. We emphasize general lessons for computational statisticians when faced with the challenge of model selection, and we assess the accuracy of various alternative paradigms, including recent widely applicable information criteria and different numerical procedures for approximating Bayes factors. We conduct the comparative evaluation with a novel inferential pipeline that systematically disambiguates confounding factors via an ANOVA scheme.     

Approximate Bayesian computation with composite score functions

Both approximate Bayesian computation (ABC) and composite likelihood methods are useful for Bayesian and frequentist inference, respectively, when the likelihood function is intractable. We propose to use composite likelihood score functions as summary statistics in ABC in order to obtain accurate approximations to the posterior distribution. This is motivated by the use of the score function of the full likelihood, and extended to general unbiased estimating functions in complex models. Moreover, we show that if the composite score is suitably standardised, the resulting ABC procedure is invariant to reparameterisations and automatically adjusts the curvature of the composite likelihood, and of the corresponding posterior distribution. The method is illustrated through examples with simulated data, and an application to modelling of spatial extreme rainfall data is discussed.     

Non-linear regression models for Approximate Bayesian Computation

Approximate Bayesian inference on the basis of summary statistics is well-suited to complex problems for which the likelihood is either mathematically or computationally intractable. However the methods that use rejection suffer from the curse of dimensionality when the number of summary statistics is increased. Here we propose a machine-learning approach to the estimation of the posterior density by introducing two innovations. The new method fits a nonlinear conditional heteroscedastic regression of the parameter on the summary statistics, and then adaptively improves estimation using importance sampling. The new algorithm is compared to the state-of-the-art approximate Bayesian methods, and achieves considerable reduction of the computational burden in two examples of inference in statistical genetics and in a queueing model.     

偏度系数的近似线性贝叶斯估计

在概率统计中,偏度系数反映了随机变量的密度曲线的对称特征.由于偏度系数涉及到分布的前三阶矩,因此得到好的估计有一定的难度.文章建立贝叶斯模型,对偏度系数提出近似线性贝叶斯估计,并在多条数据结构下,对先验分布的超参数提出合适的估计,得到偏度系数的经验贝叶斯估计.     

Bayesian validation assessment of multivariate computational models

Multivariate model validation is a complex decision-making problem involving comparison of multiple correlated quantities, based upon the available information and prior knowledge. This paper presents a Bayesian risk-based decision method for validation assessment of multivariate predictive models under uncertainty. A generalized likelihood ratio is derived as a quantitative validation metric based on Bayes’ theorem and Gaussian distribution assumption of errors between validation data and model prediction. The multivariate model is then assessed based on the comparison of the likelihood ratio with a Bayesian decision threshold, a function of the decision costs and prior of each hypothesis. The probability density function of the likelihood ratio is constructed using the statistics of multiple response quantities and Monte Carlo simulation. The proposed methodology is implemented in the validation of a transient heat conduction model, using a multivariate data set from experiments. The Bayesian methodology provides a quantitative approach to facilitate rational decisions in multivariate model assessment under uncertainty.     

Approximate Bayesian Inference in Spatial Generalized Linear Mixed Models

Abstract.  In this paper we propose fast approximate methods for computing posterior marginals in spatial generalized linear mixed models. We consider the common geostatistical case with a high dimensional latent spatial variable and observations at known registration sites. The methods of inference are deterministic, using no simulation-based inference. The first proposed approximation is fast to compute and is 'practically sufficient', meaning that results do not show any bias or dispersion effects that might affect decision making. Our second approximation, an improvement of the first version, is 'practically exact', meaning that one would have to run MCMC simulations for very much longer than is typically done to detect any indication of error in the approximate results. For small-count data the approximations are slightly worse, but still very accurate. Our methods are limited to likelihood functions that give unimodal full conditionals for the latent variable. The methods help to expand the future scope of non-Gaussian geostatistical models as illustrated by applications of model choice, outlier detection and sampling design. The approximations take seconds or minutes of CPU time, in sharp contrast to overnight MCMC runs for solving such problems.     

An Approximate Bayesian Marginal Likelihood Approach for Estimating Finite Mixtures

Estimation of finite mixture models when the mixing distribution support is unknown is an important problem. This article 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 approximation and simulated annealing procedure compares favorably with existing methods.     

Bayesian methods for mark-resighting surveys

We consider the problem of estimating the size of a closed population based on the results of a certain type of mark-resighting sampling design. The design is similar to the commonly used multiple capture-recapture design, yet in some cases economically more feasible and easy to use. Sampling is done by first tagging a number of randomly selected animals with visible markers and later randomly sighting them (for instance, for large animals by visually sampling from a helicopter) and counting the number of tagged animals. In this paper, we look at Bayesian methods for point and interval estimation of population size for this design. An example involving estimation of mountain sheep, a couple of simulated examples and simulation studies are given to demonstrate the advantages of the proposed procedure over the other available approximate procedures.     

Approximate likelihood methods for estimating local recombination rates

Summary. There is currently great interest in understanding the way in which recombination rates vary, over short scales, across the human genome. Aside from inherent interest, an understanding of this local variation is essential for the sensible design and analysis of many studies aimed at elucidating the genetic basis of common diseases or of human population histories. Standard pedigree-based approaches do not have the fine scale resolution that is needed to address this issue. In contrast, samples of deoxyribonucleic acid sequences from unrelated chromosomes in the population carry relevant information, but inference from such data is extremely challenging. Although there has been much recent interest in the development of full likelihood inference methods for estimating local recombination rates from such data, they are not currently practicable for data sets of the size being generated by modern experimental techniques. We introduce and study two approximate likelihood methods. The first, a marginal likelihood, ignores some of the data. A careful choice of what to ignore results in substantial computational savings with virtually no loss of relevant information. For larger sequences, we introduce a 'composite' likelihood, which approximates the model of interest by ignoring certain long-range dependences. An informal asymptotic analysis and a simulation study suggest that inference based on the composite likelihood is practicable and performs well. We combine both methods to reanalyse data from the lipoprotein lipase gene, and the results seriously question conclusions from some earlier studies of these data.     

Bayesian methods for analysing ringing data

A major recent development in statistics has been the use of fast computational methods of Markov chain Monte Carlo. These procedures allow Bayesian methods to be used in quite complex modelling situations. In this paper, we shall use a range of real data examples involving lapwings, shags, teal, dippers, and herring gulls, to illustrate the power and range of Bayesian techniques. The topics include: prior sensitivity; the use of reversible-jump MCMC for constructing model probabilities and comparing models, with particular reference to models with random effects; model-averaging; and the construction of Bayesian measures of goodness-of-fit. Throughout, there will be discussion of the practical aspects of the work - for instance explaining when and when not to use the BUGS package.     

Bayesian methods for analysing ringing data

A major recent development in statistics has been the use of fast computational methods of Markov chain Monte Carlo. These procedures allow Bayesian methods to be used in quite complex modelling situations. In this paper, we shall use a range of real data examples involving lapwings, shags, teal, dippers, and herring gulls, to illustrate the power and range of Bayesian techniques. The topics include: prior sensitivity; the use of reversible-jump MCMC for constructing model probabilities and comparing models, with particular reference to models with random effects; model-averaging; and the construction of Bayesian measures of goodness-of-fit. Throughout, there will be discussion of the practical aspects of the work - for instance explaining when and when not to use the BUGS package.     

Symmetry exploits for Bayesian cubature methods

Statistics and Computing - Bayesian cubature provides a flexible framework for numerical integration, in which a priori knowledge on the integrand can be encoded and exploited. This additional...     

Bayesian estimation methods for contingency tables

Summary A method of inputting prior opinion in contingency tables is described. The method can be used to incorporate beliefs of independence or symmetry but extensions are straightforward. Logistic normal distributions that express such beliefs are used as priors of the cell probabilities and posterior estimates are derived. Empirical Bayes methods are also discussed and approximate posterior variances are provided. The methods are illustrated by a numerical example.     

Bayesian parameter estimation via variational methods

We consider a logistic regression model with a Gaussian prior distribution over the parameters. We show that an accurate variational transformation can be used to obtain a closed form approximation to the posterior distribution of the parameters thereby yielding an approximate posterior predictive model. This approach is readily extended to binary graphical model with complete observations. For graphical models with incomplete observations we utilize an additional variational transformation and again obtain a closed form approximation to the posterior. Finally, we show that the dual of the regression problem gives a latent variable density model, the variational formulation of which leads to exactly solvable EM updates.     

Approximate Bayesian Computation for Exponential Random Graph Models for Large Social Networks

We consider the issue of sampling from the posterior distribution of exponential random graph (ERG) models and other statistical models with intractable normalizing constants. Existing methods based on exact sampling are either infeasible or require very long computing time. We study a class of approximate Markov chain Monte Carlo (MCMC) sampling schemes that deal with this issue. We also develop a new Metropolis–Hastings kernel to sample sparse large networks from ERG models. We illustrate the proposed methods on several examples.