Almost the Best of Three Worlds: Risk, Consistency and Optional Stopping for the Switch Criterion in Nested Model Selection

We study the switch distribution, introduced by Van Erven et al. (2012), applied to model selection and subsequent estimation. While switching was known to be strongly consistent, here we show that it achieves minimax optimal parametric risk rates up to a $\log\log n$ factor when comparing two nested exponential families, partially confirming a conjecture by Lauritzen (2012) and Cavanaugh (2012) that switching behaves asymptotically like the Hannan-Quinn criterion. Moreover, like Bayes factor model selection but unlike standard significance testing, when one of the models represents a simple hypothesis, the switch criterion defines a robust null hypothesis test, meaning that its Type-I error probability can be bounded irrespective of the stopping rule. Hence, switching is consistent, insensitive to optional stopping and almost minimax risk optimal, showing that, Yang's (2005) impossibility result notwithstanding, it is possible to `almost' combine the strengths of AIC and Bayes factor model selection.Comment: To appear in Statistica Sinic

Adaptive posterior contraction rates for the horseshoe

We investigate the frequentist properties of Bayesian procedures for estimation based on the horseshoe prior in the sparse multivariate normal means model. Previous theoretical results assumed that the sparsity level, that is, the number of signals, was known. We drop this assumption and characterize the behavior of the maximum marginal likelihood estimator (MMLE) of a key parameter of the horseshoe prior. We prove that the MMLE is an effective estimator of the sparsity level, in the sense that it leads to (near) minimax optimal estimation of the underlying mean vector generating the data. Besides this empirical Bayes procedure, we consider the hierarchical Bayes method of putting a prior on the unknown sparsity level as well. We show that both Bayesian techniques lead to rate-adaptive optimal posterior contraction, which implies that the horseshoe posterior is a good candidate for generating rate-adaptive credible sets.Comment: arXiv admin note: substantial text overlap with arXiv:1607.0189

Conditions for Posterior Contraction in the Sparse Normal Means Problem

The first Bayesian results for the sparse normal means problem were proven for spike-and-slab priors. However, these priors are less convenient from a computational point of view. In the meanwhile, a large number of continuous shrinkage priors has been proposed. Many of these shrinkage priors can be written as a scale mixture of normals, which makes them particularly easy to implement. We propose general conditions on the prior on the local variance in scale mixtures of normals, such that posterior contraction at the minimax rate is assured. The conditions require tails at least as heavy as Laplace, but not too heavy, and a large amount of mass around zero relative to the tails, more so as the sparsity increases. These conditions give some general guidelines for choosing a shrinkage prior for estimation under a nearly black sparsity assumption. We verify these conditions for the class of priors considered by Ghosh and Chakrabarti (2015), which includes the horseshoe and the normal-exponential gamma priors, and for the horseshoe+, the inverse-Gaussian prior, the normal-gamma prior, and the spike-and-slab Lasso, and thus extend the number of shrinkage priors which are known to lead to posterior contraction at the minimax estimation rate

Almost the best of three worlds: Risk, consistency and optional stopping for the switch criterion in nested model selection

We study the switch distribution, introduced by van Erven, Grünwald and De Rooij (2012), applied to model selection and subsequent estimation. While switching was known to be strongly consistent, here we show that it achieves minimax optimal parametric risk rates up to a log log n factor when comparing two nested exponential families, partially confirming a conjecture by Lauritzen (2012) and Cavanaugh (2012) that switching behaves asymptotically like the Hannan-Quinn criterion. Moreover, like Bayes factor model selection, but unlike standard significance testing, when one of the models represents a simple hypothesis, the switch criterion defines a robust null hypothesis test, meaning that its Type-I error probability can be bounded irrespective of the stopping rule. Hence, switching is consistent, insensitive to optional stopping and almost minimax risk optimal, showing that, Yang's (2005) impossibility result notwithstanding, it is possible to `almost' combine the strengths of AIC and Bayes factor model selection

The Prognostic Value of Troponin-T in Out-of-Hospital Cardiac Arrest Without ST-Segment Elevation: A COACT Substudy

Background: In out-of-hospital cardiac arrest (OHCA) without ST-elevation, predictive markers that can identify those with a high risk of acute coronary syndrome are lacking. Methods: In this post hoc analysis of the Coronary Angiography after Cardiac Arrest (COACT) trial, the baseline, median, peak, and time-concentration curves of troponin-T (cTnT) (T-AUC) in OHCA patients without ST-elevation were studied. cTnT values were obtained at predefined time points at 0, 3, 6, 12, 24, 36, 28, and 72 hours after admission. All patients who died within the measurement period were not included. The primary outcome was the association between cTnT and 90-day survival. Secondary outcomes included the association of cTnT and acute thrombotic occlusions, acute unstable lesions, and left ventricular function. Results: In total, 352 patients were included in the analysis. The mean age was 64 ± 13 years (80.4% men). All cTnT measures were independent prognostic factors for mortality after adjustment for potential confounders age, sex, history of coronary artery disease, witnessed arrest, time to BLS, and time to return of spontaneous circulation (eg, for T-AUC: hazard ratio, 1.44; 95% CI, 1.06-1.94; P = .02; P value for all variables ≤ .02). Median cTnT (odds ratio [OR], 1.58; 95% CI, 1.18-2.12; P = .002) and T-AUC (OR, 2.03; 95% CI, 1.25-3.29; P = .004) were independent predictors for acute unstable lesions. Median cTnT (OR, 1.62; 95% CI, 1.17-2.23; P = .003) and T-AUC (OR, 2.16; 95% CI, 1.27-3.68; P = .004) were independent predictors for acute thrombotic occlusions. CTnT values were not associated with the left ventricular function (eg, for T-AUC: OR, 2.01; 95% CI, 0.65-6.19; P = .22; P value for all variables ≥ .14) Conclusion: In OHCA patients without ST-segment elevation, cTnT release during the first 72 hours after return of spontaneous circulation was associated with clinical outcomes

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