7,709 research outputs found
On the half-Cauchy prior for a global scale parameter
This paper argues that the half-Cauchy distribution should replace the
inverse-Gamma distribution as a default prior for a top-level scale parameter
in Bayesian hierarchical models, at least for cases where a proper prior is
necessary. Our arguments involve a blend of Bayesian and frequentist reasoning,
and are intended to complement the original case made by Gelman (2006) in
support of the folded-t family of priors. First, we generalize the half-Cauchy
prior to the wider class of hypergeometric inverted-beta priors. We derive
expressions for posterior moments and marginal densities when these priors are
used for a top-level normal variance in a Bayesian hierarchical model. We go on
to prove a proposition that, together with the results for moments and
marginals, allows us to characterize the frequentist risk of the Bayes
estimators under all global-shrinkage priors in the class. These theoretical
results, in turn, allow us to study the frequentist properties of the
half-Cauchy prior versus a wide class of alternatives. The half-Cauchy occupies
a sensible 'middle ground' within this class: it performs very well near the
origin, but does not lead to drastic compromises in other parts of the
parameter space. This provides an alternative, classical justification for the
repeated, routine use of this prior. We also consider situations where the
underlying mean vector is sparse, where we argue that the usual conjugate
choice of an inverse-gamma prior is particularly inappropriate, and can lead to
highly distorted posterior inferences. Finally, we briefly summarize some open
issues in the specification of default priors for scale terms in hierarchical
models
Good, great, or lucky? Screening for firms with sustained superior performance using heavy-tailed priors
This paper examines historical patterns of ROA (return on assets) for a
cohort of 53,038 publicly traded firms across 93 countries, measured over the
past 45 years. Our goal is to screen for firms whose ROA trajectories suggest
that they have systematically outperformed their peer groups over time. Such a
project faces at least three statistical difficulties: adjustment for relevant
covariates, massive multiplicity, and longitudinal dependence. We conclude
that, once these difficulties are taken into account, demonstrably superior
performance appears to be quite rare. We compare our findings with other recent
management studies on the same subject, and with the popular literature on
corporate success. Our methodological contribution is to propose a new class of
priors for use in large-scale simultaneous testing. These priors are based on
the hypergeometric inverted-beta family, and have two main attractive features:
heavy tails and computational tractability. The family is a four-parameter
generalization of the normal/inverted-beta prior, and is the natural conjugate
prior for shrinkage coefficients in a hierarchical normal model. Our results
emphasize the usefulness of these heavy-tailed priors in large multiple-testing
problems, as they have a mild rate of tail decay in the marginal likelihood
---a property long recognized to be important in testing.Comment: Published in at http://dx.doi.org/10.1214/11-AOAS512 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Time-Frequency Analysis Reveals Pairwise Interactions in Insect Swarms
The macroscopic emergent behavior of social animal groups is a classic example of dynamical self-organization, and is thought to arise from the local interactions between individuals. Determining these interactions from empirical data sets of real animal groups, however, is challenging. Using multicamera imaging and tracking, we studied the motion of individual flying midges in laboratory mating swarms. By performing a time-frequency analysis of the midge trajectories, we show that the midge behavior can be segmented into two distinct modes: one that is independent and composed of low-frequency maneuvers, and one that consists of higher-frequency nearly harmonic oscillations conducted in synchrony with another midge. We characterize these pairwise interactions, and make a hypothesis as to their biological function
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