Approximate Bayesian computation (ABC) is an approach for sampling from an approximate posterior distribution in the presence of a computationally intractable likelihood function. A common implementation is based on simulating model, parameter and dataset triples, (m,θ,y), from the prior, and then accepting as samples from the approximate posterior, those pairs (m,θ) for which y, or a summary of y, is "close" to the observed data. Closeness is typically determined though a distance measure and a kernel scale parameter, ε. Appropriate choice of εis important to producing a good quality approximation. This paper proposes diagnostic tools for the choice of εbased on assessing the coverage property, which asserts that credible intervals have the correct coverage levels. We provide theoretical results on coverage for both model and parameter inference, and adapt these into diagnostics for the ABC context. We re-analyse a study on human demographic history to determine whether the adopted posterior approximation was appropriate. R code implementing the proposed methodology is freely available in the package "abc."
Chain event graphs have been established as a practical Bayesian graphical tool. While bespoke diagnostics have been developed for Bayesian Networks, they have not yet been defined for the statistical class of Chain Event Graph models. Mirroring the methodology of prequential diagnostics for Bayesian Networks, in this paper we design a number of new Bayesian diagnostics for this new class. These can be used to check whether a selected model--presumably the best within the class--captures most of the salient features of the observed process. These are designed to check the continued validity of…
As a statistical tool to assist formative assessments in educational settings, diagnostic classification models (DCMs) have been increasingly used to provide diagnostic information regarding examinees' attributes. DCMs often adopt a dichotomous division such as the mastery and non-mastery of attributes to express the mastery states of attributes. However, many practical settings involve different levels of mastery states rather than a simple dichotomy in a single attribute. Although this practical demand can be addressed by polytomous-attribute DCMs, their computational cost in a Markov chain …
In many applications of Bayesian clustering, posterior sampling on the discrete state space of cluster allocations is achieved via Markov chain Monte Carlo (MCMC) techniques. As it is typically challenging to design transition kernels to explore this state space efficiently, MCMC convergence diagnostics for clustering applications is especially important. For general MCMC problems, state-of-the-art convergence diagnostics involve comparisons across multiple chains. However, single-chain alternatives can be appealing for computationally intensive and slowly-mixing MCMC, as is typically the case…
Bayesian hierarchical models are increasingly popular for realistic modelling and analysis of complex data. This trend is accompanied by the need for flexible, general, and computationally efficient methods for model criticism and conflict detection. Usually, a Bayesian hierarchical model incorporates a grouping of the individual data points, for example individuals in repeated measurement data. In such cases, the following question arises: Are any of the groups "outliers", or in conflict with the remaining groups? Existing general approaches aiming to answer such questions tend to be extremel…
Current literature on posterior approximation for Bayesian inference offers many alternative methods. Does our chosen approximation scheme work well on the observed data? The best existing generic diagnostic tools treating this kind of question by looking at performance averaged over data space, or otherwise lack diagnostic detail. However, if the approximation is bad for most data, but good at the observed data, then we may discard a useful approximation. We give graphical diagnostics for posterior approximation at the observed data. We estimate a "distortion map" that acts on univariate marg…
Fueled by the call for formative assessments, diagnostic classification models (DCMs) have recently gained popularity in psychometrics. Despite their potential for providing diagnostic information that aids in classroom instruction and students' learning, empirical applications of DCMs to classroom assessments have been highly limited. This is partly because how DCMs with different estimation methods perform in small sample contexts is not yet well-explored. Hence, this study aims to investigate the performance of respondent classification and item parameter estimation with a comprehensive sim…
There has been growing interest in the AI community for precise uncertainty quantification. Conditional density models f(y|x), where x represents potentially high-dimensional features, are an integral part of uncertainty quantification in prediction and Bayesian inference. However, it is challenging to assess conditional density estimates and gain insight into modes of failure. While existing diagnostic tools can determine whether an approximated conditional density is compatible overall with a data sample, they lack a principled framework for identifying, locating, and interpreting the nature…
Estimates of national immunization coverage are crucial for guiding policy and decision-making in national immunization programs and setting the global immunization agenda. WHO and UNICEF estimates of national immunization coverage (WUENIC) are produced annually for various vaccine-dose combinations and all WHO Member States using information from multiple data sources and a deterministic computational logic approach. This approach, however, is incapable of characterizing the uncertainties inherent in coverage measurement and estimation. It also provides no statistically principled way of expl…
In this paper, we study frequentist coverage errors of Bayesian credible sets for an approximately linear regression model with (moderately) high dimensional regressors, where the dimension of the regressors may increase with but is smaller than the sample size. Specifically, we consider quasi-Bayesian inference on the slope vector under the quasi-likelihood with Gaussian error distribution. Under this setup, we derive finite sample bounds on frequentist coverage errors of Bayesian credible rectangles. Derivation of those bounds builds on a novel Berry--Esseen type bound on quasi-posterior dis…
The development of a new diagnostic test ideally follows a sequence of stages which, amongst other aims, evaluate technical performance. This includes an analytical validity study, a diagnostic accuracy study and an interventional clinical utility study. Current approaches to the design and analysis of the diagnostic accuracy study can suffer from prohibitively large sample sizes and interval estimates with undesirable properties. In this paper, we propose a novel Bayesian approach which takes advantage of information available from the analytical validity stage. We utilise assurance to calcul…
Complex stochastic models represented by directed acyclic graphs (DAGs) are increasingly employed to synthesise multiple, imperfect and disparate sources of evidence, to estimate quantities that are difficult to measure directly. The various data sources are dependent on shared parameters and hence have the potential to conflict with each other, as well as with the model. In a Bayesian framework, the model consists of three components: the prior distribution, the assumed form of the likelihood and structural assumptions. Any of these components may be incompatible with the observed data. The d…
Multivariate nonnegative orthant data are real vectors bounded to the left by the null vector, and they can be continuous, discrete or mixed. We first review the recent relative variability indexes for multivariate nonnegative continuous and count distributions. As a prelude, the classification of two comparable distributions having the same mean vector is done through under-, equi- and over-variability with respect to the reference distribution. Multivariate associated kernel estimators are then reviewed with new proposals that can accommodate any nonnegative orthant dataset. We focus on band…
Frequentist coverage of $(1-α)$-highest posterior density (HPD) credible sets is studied in a signal plus noise model under a large class of noise distributions. We consider a specific class of spike-and-slab prior distributions. Different regimes are identified and we derive closed form expressions for the $(1-α)$-HPD on each of these regimes. Similar to the earlier work by Marchand and Strawderman, it is shown that under suitable conditions, the frequentist coverage can drop to $1-3α/2.$
In a bivariate meta-analysis the number of diagnostic studies involved is often very low so that frequentist methods may result in problems. Bayesian inference is attractive as informative priors that add small amount of information can stabilise the analysis without overwhelming the data. However, Bayesian analysis is often computationally demanding and the selection of the prior for the covariance matrix of the bivariate structure is crucial with little data. The integrated nested Laplace approximations (INLA) method provides an efficient solution to the computational issues by avoiding any …
This paper introduces the \proglang{R} package \pkg{meta4diag} for implementing Bayesian bivariate meta-analyses of diagnostic test studies. Our package \pkg{meta4diag} is a purpose-built front end of the \proglang{R} package \pkg{INLA}. While \pkg{INLA} offers full Bayesian inference for the large set of latent Gaussian models using integrated nested Laplace approximations, \pkg{meta4diag} extracts the features needed for bivariate meta-analysis and presents them in an intuitive way. It allows the user a straightforward model-specification and offers user-specific prior distributions. Further…
University of Tübingen（蒂宾根大学）；Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）
Full Bayesian posteriors are rarely analytically tractable, which is why real-world Bayesian inference heavily relies on approximate techniques. Approximations generally differ from the true posterior and require diagnostic tools to assess whether the inference can still be trusted. We investigate a new approach to diagnosing approximate inference: the approximation mismatch is attributed to a change in the inductive bias by treating the approximations as exact and reverse-engineering the corresponding prior. We show that the problem is more complicated than it appears to be at first glance, b…
For estimating a lower bounded parametric function in the framework of Marchand and Strawderman (2006), we provide through a unified approach a class of Bayesian confidence intervals with credibility $1-α$ and frequentist coverage probability bounded below by $\frac{1-α}{1+α}$. In cases where the underlying pivotal distribution is symmetric, the findings represent extensions with respect to the specification of the credible set achieved through the choice of a {\it spending function}, and include Marchand and Strawderman's HPD procedure result. For non-symmetric cases, the determination of a s…
Model misspecification constitutes a major obstacle to reliable inference in many inverse problems. Inverse problems in seismology, for example, are particularly affected by misspecification of wave propagation velocities. In this paper, we focus on a specific seismic inverse problem - full-waveform moment tensor inversion - and develop a Bayesian framework that seeks robustness to velocity misspecification. A novel element of our framework is the use of transport-Lagrangian (TL) distances between observed and model predicted waveforms to specify a loss function, and the use of this loss to de…
We consider the nonparametric multivariate isotonic regression problem, where the regression function is assumed to be nondecreasing with respect to each predictor. Our goal is to construct a Bayesian credible interval for the function value at a given interior point with assured limiting frequentist coverage. We put a prior on unrestricted step-functions, but make inference using the induced posterior measure by an "immersion map" from the space of unrestricted functions to that of multivariate monotone functions. This allows maintaining the natural conjugacy for posterior sampling. A natural…
The prediction interval has been increasingly used in meta-analyses as a useful measure for assessing the magnitude of treatment effect and between-studies heterogeneity. In calculations of the prediction interval, although the Higgins-Thompson-Spiegelhalter method is used most often in practice, it might not have adequate coverage probability for the true treatment effect of a future study under realistic situations. An effective alternative candidate is the Bayesian prediction interval, which has also been widely used in general prediction problems. However, these prediction intervals are co…
We investigate the frequentist coverage of Bayesian credible sets in a nonparametric setting. We consider a scale of priors of varying regularity and choose the regularity by an empirical Bayes method. Next we consider a central set of prescribed posterior probability in the posterior distribution of the chosen regularity. We show that such an adaptive Bayes credible set gives correct uncertainty quantification of "polished tail" parameters, in the sense of high probability of coverage of such parameters. On the negative side, we show by theory and example that adaptation of the prior necessar…
We investigate the frequentist coverage properties of Bayesian credible sets in a general, adaptive, nonparametric framework. It is well known that the construction of adaptive and honest confidence sets is not possible in general. To overcome this problem we introduce an extra assumption on the functional parameters, the so called "general polished tail" condition. We then show that under standard assumptions both the hierarchical and empirical Bayes methods results in honest confidence sets for sieve type of priors in general settings and we characterize their size. We apply the derived abst…
For estimating a lower bounded location or mean parameter for a symmetric and logconcave density, we investigate the frequentist performance of the $100(1-α)%$ Bayesian HPD credible set associated with priors which are truncations of flat priors onto the restricted parameter space. Various new properties are obtained. Namely, we identify precisely where the minimum coverage is obtained and we show that this minimum coverage is bounded between $1-\frac{3α}{2}$ and $1-\frac{3α}{2}+\frac{α^2}{1+α}$; with the lower bound $1-\frac{3α}{2}$ improving (for $α\leq 1/3$) on the previously established ([…
Many countries conduct a full census survey to report official population statistics. As no census survey ever achieves 100 per cent response rate, a post-enumeration survey (PES) is usually conducted and analysed to assess census coverage and produce official population estimates by geographic area and demographic attributes. Considering the usually small size of PES, direct estimation at the desired level of disaggregation is not feasible. Design-based estimation with sampling weight adjustment is a commonly used method but is difficult to implement when survey non-response patterns cannot b…
In this chapter, we will first present the most standard computational challenges met in Bayesian Statistics, focussing primarily on mixture estimation and on model choice issues, and then relate these problems with computational solutions. Of course, this chapter is only a terse introduction to the problems and solutions related to Bayesian computations. For more complete references, see Robert and Casella (2004, 2009), or Marin and Robert (2007), among others. We also restrain from providing an introduction to Bayesian Statistics per se and for comprehensive coverage, address the reader to R…
Modeling non-linear temporal trajectories is of fundamental interest in many application areas, such as in longitudinal microbiome analysis. Many existing methods focus on estimating mean trajectories, but it is also often of value to assess temporal patterns of individual subjects. Sparse principal components analysis (SFPCA) serves as a useful tool for assessing individual variation in non-linear trajectories; however its application to real data often requires careful model selection criteria and diagnostic tools. Here, we propose a Bayesian approach to SFPCA, which allows users to use the …
This paper provides two different novel approaches of slice sampling to estimate the parameters of absolute continuous Marshall-Olkin bivariate Pareto distribution with location and scale parameters. We carry out the bayesian analysis taking gamma prior for shape and scale parameters and truncated normal for location parameters. Credible intervals and coverage probabilities are also provided for all methods. A real-life data analysis is shown for illustrative purpose.
Ideally, any statistical inference should be robust to local influences. Although there are simple ways to check about leverage points in independent and linear problems, more complex models require more sophisticated methods. Kullback-Leiber and Bregman divergences were already applied in Bayesian inference to measure the isolated impact of each observation in a model. We extend these ideas to models for dependent data and with non-normal probability distributions such as time series, spatial models and generalized linear models. We also propose a strategy to rescale the functional Bregman di…
Accurate diagnosis of disease is of fundamental importance in clinical practice and medical research. Before a medical diagnostic test is routinely used in practice, its ability to distinguish between diseased and nondiseased states must be rigorously assessed through statistical analysis. The receiver operating characteristic (ROC) curve is the most popular used tool for evaluating the discriminatory ability of continuous-outcome diagnostic tests. It has been acknowledged that several factors (e.g., subject-specific characteristics, such as age and/or gender) can affect the test's accuracy be…
We develop a fully Bayesian hierarchical model for trend filtering, itself a new development in nonparametric, univariate regression. The framework more broadly applies to the generalized lasso, but focus is on Bayesian trend filtering. We compare two shrinkage priors, double exponential and generalized double Pareto. A simulation study, comparing Bayesian trend filtering to the original formulation and a number of other popular methods shows our method to improve estimation error while maintaining if not improving coverage probability. Two time series data sets demonstrate Bayesian trend filt…
This article describes an extension of classical χ^2 goodness-of-fit tests to Bayesian model assessment. The extension, which essentially involves evaluating Pearson's goodness-of-fit statistic at a parameter value drawn from its posterior distribution, has the important property that it is asymptotically distributed as a χ^2 random variable on K-1 degrees of freedom, independently of the dimension of the underlying parameter vector. By examining the posterior distribution of this statistic, global goodness-of-fit diagnostics are obtained. Advantages of these diagnostics include ease of interp…
The current standard Bayesian approach to model calibration, which assigns a Gaussian process prior to the discrepancy term, often suffers from issues of unidentifiability and computational complexity and instability. When the goal is to quantify uncertainty in physical parameters for extrapolative prediction, then there is no need to perform inference on the discrepancy term. With this in mind, we introduce Gibbs posteriors as an alternative Bayesian method for model calibration, which updates the prior with a loss function connecting the data to the parameter. The target of inference is the …
In this paper we describe the main featuress of the Bergm package for the open-source R software which provides a comprehensive framework for Bayesian analysis for exponential random graph models: tools for parameter estimation, model selection and goodness-of-fit diagnostics. We illustrate the capabilities of this package describing the algorithms through a tutorial analysis of two well-known network datasets.
