Showing papers by "James O. Berger published in 2020"

Bayesian analysis of the covariance matrix of a multivariate normal distribution with a new class of priors

Abstract: Bayesian analysis for the covariance matrix of a multivariate normal distribution has received a lot of attention in the last two decades. In this paper, we propose a new class of priors for the covariance matrix, including both inverse Wishart and reference priors as special cases. The main motivation for the new class is to have available priors—both subjective and objective—that do not “force eigenvalues apart,” which is a criticism of inverse Wishart and Jeffreys priors. Extensive comparison of these “shrinkage priors” with inverse Wishart and Jeffreys priors is undertaken, with the new priors seeming to have considerably better performance. A number of curious facts about the new priors are also observed, such as that the posterior distribution will be proper with just three vector observations from the multivariate normal distribution—regardless of the dimension of the covariance matrix—and that useful inference about features of the covariance matrix can be possible. Finally, a new MCMC algorithm is developed for this class of priors and is shown to be computationally effective for matrices of up to 100 dimensions.

Volcanic Hazard Assessment for an Eruption Hiatus, or Post-eruption Unrest Context: Modeling Continued Dome Collapse Hazards for Soufrière Hills Volcano

Abstract: Effective volcanic hazard management in regions where populations live in close proximity to persistent volcanic activity involves understanding the dynamic nature of hazards, and associated risk. Emphasis until now has been placed on identification and forecasting of the escalation of activity, in order to provide adequate warning. However, understanding eruption hiatus and post-eruption unrest hazard, or how to quantify hazard after the end of an eruption, is equally important and often key to post-eruption recovery. Unfortunately, in many cases when the level of activity lessens, the hazards, although reduced, do not necessarily cease. This is due to both the imprecise nature of determination of the ``end'' of an eruptive phase as well as to the possibility that post-eruption hazardous processes may continue to occur. An example of this is continued dome collapse hazard from lava domes which have ceased to grow, or sector collapse of a volcano. We present a new probabilistic model for forecasting pyroclastic density current (PDC) activity that takes into account the heavy-tailed distribution of the lengths of eruptive phases, the periods of relative quiescence, and the desired forecast window of interest. Further we combine this model with physics-based simulations of PDCs at Soufriere Hills to produce a series of probabilistic hazard maps that offer evidence-based guidance for dome collapse hazards that can be used to inform decision-making around access and reoccupation in areas around the volcano.

Frequentist Properties of Bayesian Multiplicity Control for Multiple Testing of Normal Means

Abstract: We consider the standard problem of multiple testing of normal means, obtaining Bayesian multiplicity control by assuming that the prior inclusion probability (the assumed equal prior probability that each mean is nonzero) is unknown and assigned a prior distribution. The asymptotic frequentist behavior of the Bayesian procedure is studied, as the number of tests grows. Studied quantities include the false positive probability, which is shown to go to zero asymptotically. The asymptotics of a Bayesian decision-theoretic approach are also presented.

Restricted Type II Maximum Likelihood Priors on Regression Coefficients

Abstract: In Bayesian hypothesis testing and model selection, prior distributions must be chosen carefully. For example, setting arbitrarily large prior scales for location parameters, which is common practice in estimation problems, can lead to undesirable behavior in testing (see Lindley’s paradox; Lindley (1957)). We study the properties of some restricted type II maximum likelihood (type II ML) priors on regression coefficients. In type II ML, hyperparameters are “estimated” by maximizing the marginal likelihood of a model. In this article, we define priors by estimating their variances or covariance matrices, adding restrictions which ensure that the resulting priors are at least as vague as conventional proper priors for model uncertainty. We find that these type II ML priors typically yield results that are close to answers obtained with the Bayesian Information Criterion (BIC; Schwarz (1978)).