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Quantification of prior impact in terms of effective current sample size
Bayesian methods allow borrowing of historical information through prior distributions. The concept of prior effective sample size (prior ESS) facilitates quantification and communication of such prior information by equating it to a sample size. Prior information can arise from historical observations; thus, the traditional approach identifies the ESS with such a historical sample size. However, this measure is independent of newly observed data, and thus would not capture an actual “loss of information” induced by the prior in case of priordata conflict. We build on a recent work to relate prior impact to the number of (virtual) samples from the current data model and introduce the effective current sample size (ECSS) of a prior, tailored to the application in Bayesian clinical trial designs. Special emphasis is put on robust mixture, power, and commensurate priors. We apply the approach to an adaptive design in which the number of recruited patients is adjusted depending on the effective sample size at an interim analysis. We argue that the ECSS is the appropriate measure in this case, as the aim is to save current (as opposed to historical) patients from recruitment. Furthermore, the ECSS can help overcome lack of consensus in the ESS assessment of mixture priors and can, more broadly, provide further insights into the impact of priors. An R package accompanies the paper.
Journal Article
Adaptive Prior Weighting in Generalized Regression
The prior distribution is a key ingredient in Bayesian inference. Prior information on regression coefficients may come from different sources and may or may not be in conflict with the observed data. Various methods have been proposed to quantify a potential prior-data conflict, such as Box's p-value. However, there are no clear recommendations how to react to possible prior-data conflict in generalized regression models. To address this deficiency, we propose to adaptively weight a prespecified multivariate normal prior distribution on the regression coefficients. To this end, we relate empirical Bayes estimates of prior weight to Box's p-value and propose alternative fully Bayesian approaches. Prior weighting can be done for the joint prior distribution of the regression coefficients or—under prior independence—separately for prespecified blocks of regression coefficients. We outline how the proposed methodology can be implemented using integrated nested Laplace approximations (INLA) and illustrate the applicability with a Bayesian logistic regression model for data from a cross-sectional study. We also provide a simulation study that shows excellent performance of our approach in the case of prior misspecification in terms of root mean squared error and coverage. Supplementary Materials give details on software implementation and code and another application to binary longitudinal data from a randomized clinical trial using a Bayesian generalized linear mixed model.
Journal Article
Penalising Model Component Complexity: A Principled, Practical Approach to Constructing Priors
In this paper, we introduce a new concept for constructing prior distributions. We exploit the natural nested structure inherent to many model components, which defines the model component to be a flexible extension of a base model. Proper priors are defined to penalise the complexity induced by deviating from the simpler base model and are formulated after the input of a user-defined scaling parameter for that model component, both in the univariate and the multivariate case. These priors are invariant to reparameterisations, have a natural connection to Jeffreys' priors, are designed to support Occam's razor and seem to have excellent robustness properties, all which are highly desirable and allow us to use this approach to define default prior distributions. Through examples and theoretical results, we demonstrate the appropriateness of this approach and how it can be applied in various situations.
Journal Article
Predictively consistent prior effective sample sizes
Determining the sample size of an experiment can be challenging, even more so when incorporating external information via a prior distribution. Such information is increasingly used to reduce the size of the control group in randomized clinical trials. Knowing the amount of prior information, expressed as an equivalent prior effective samplesize (ESS), clearly facilitates trial designs. Various methods to obtain a prior’s ESS have been proposed recently. They have been justified by the fact that they give the standard ESS for one-parameter exponential families. However, despite being based on similar information-based metrics, they may lead to surprisingly different ESS for nonconjugate settings, which complicates many designs with prior information. We show that current methods fail a basic predictive consistency criterion, which requires the expected posterior-predictive ESS for a sample of size N to be the sum of the prior ESS and N. The expected local-information-ratio ESS is introduced and shown to be predictively consistent. It corrects the ESS of current methods, as shown for normally distributed data with a heavy-tailed Student-t prior and exponential data with a generalized Gamma prior. Finally, two applications are discussed: the prior ESS for the control group derived from historical data and the posterior ESS for hierarchical subgroup analyses.
Journal Article
The Prior Can Often Only Be Understood in the Context of the Likelihood
A key sticking point of Bayesian analysis is the choice of prior distribution, and there is a vast literature on potential defaults including uniform priors, Jeffreys’ priors, reference priors, maximum entropy priors, and weakly informative priors. These methods, however, often manifest a key conceptual tension in prior modeling: a model encoding true prior information should be chosen without reference to the model of the measurement process, but almost all common prior modeling techniques are implicitly motivated by a reference likelihood. In this paper we resolve this apparent paradox by placing the choice of prior into the context of the entire Bayesian analysis, from inference to prediction to model evaluation.
Journal Article
The Formal Definition of Reference Priors
Reference analysis produces objective Bayesian inference, in the sense that inferential statements depend only on the assumed model and the available data, and the prior distribution used to make an inference is least informative in a certain information-theoretic sense. Reference priors have been rigorously defined in specific contexts and heuristically defined in general, but a rigorous general definition has been lacking. We produce a rigorous general definition here and then show how an explicit expression for the reference prior can be obtained under very weak regularity conditions. The explicit expression can be used to derive new reference priors both analytically and numerically.
Journal Article
Bayesian Model Selection in High-Dimensional Settings
Standard assumptions incorporated into Bayesian model selection procedures result in procedures that are not competitive with commonly used penalized likelihood methods. We propose modifications of these methods by imposing nonlocal prior densities on model parameters. We show that the resulting model selection procedures are consistent in linear model settings when the number of possible covariates p is bounded by the number of observations n, a property that has not been extended to other model selection procedures. In addition to consistently identifying the true model, the proposed procedures provide accurate estimates of the posterior probability that each identified model is correct. Through simulation studies, we demonstrate that these model selection procedures perform as well or better than commonly used penalized likelihood methods in a range of simulation settings. Proofs of the primary theorems are provided in the Supplementary Material that is available online.
Journal Article
Toward a New Framework to Evaluate Process‐Based Model Configurations and Quantify Data Worth Prior to Calibration
Model criticism, discrimination, and selection methods often rely on calibrated model outputs. Because calibration can be computationally expensive, model criticism can first be undertaken by assessing model outputs obtained from limited prior parameter ensembles. However, such prior‐based methods are often heuristic and do not formalize the notion of balancing model consistency with data and model complexity (i.e., model adequacy). We present a new framework to discriminate among candidate models prior to calibration that formalizes prior‐to‐calibration model adequacy into a metric to implicitly balance prior model output data coverage with model complexity represented by prior output (co)variance. The prior model adequacy metric “Mahalanobis distance deviation” quantifies the deviation of (a) the set of squared Mahalanobis distances of data from a prior model output distribution from (b) the set of squared Mahalanobis distances of data from their own distribution. A new data worth metric “discernment value” is also presented which quantifies the value of data for screening less‐adequate models prior to calibration. Discernment value is calculated from the change in variance of a weighted average of prior model outputs from all candidate models due to less‐adequate model outputs receiving lower weight. The framework is demonstrated using a one‐dimensional groundwater flow model with eight possible configurations. A synthetic data network is used to test the framework. Results show the framework identifies the candidate models most similar to the true model used to create the synthetic data. Discernment values show variation in the value of different data types and locations for screening less‐adequate models.
Journal Article