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We describe Bayes factors based on z, t, $\chi^2$, and F statistics when non-local moment prior distributions are used to define alternative hypotheses. The non-local alternative prior distributions are centered on standardized effects. The prior densities include a dispersion parameter that can be used to model prior precision and the variation of effect sizes across replicated experiments. We examine the convergence rates of Bayes factors under true null and true alternative hypotheses and show how these Bayes factors can be used to construct Bayes factor functions. An example illustrates the application of resulting Bayes factors to psychological experiments.

Bayes factors represent the ratio of probabilities assigned to data by competing scientific hypotheses. Drawbacks of Bayes factors are their dependence on prior specifications that define null and alternative hypotheses and difficulties encountered in their computation. To address these problems, we define Bayes factor functions (BFF) directly from common test statistics. BFFs depend on a single non-centrality parameter that can be expressed as a function of standardized effect sizes, and plots of BFFs versus effect size provide informative summaries of hypothesis tests that can be easily aggregated across studies. Such summaries eliminate the need for arbitrary P-value thresholds to define ``statistical significance.'' BFFs are available in closed form and can be computed easily from z, t, chi-squared, and F statistics.

For high dimensional data, where P features for N objects (P >> N) are represented in an NxP matrix X, we describe a clustering algorithm based on the normalized left Gram matrix, G = XX'/P. Under certain regularity conditions, the rows in G that correspond to objects in the same cluster converge to the same mean vector. By clustering on the row means, the algorithm does not require preprocessing by dimension reduction or feature selection techniques and does not require specification of tuning or hyperparameter values. Because it is based on the NxN matrix G, it has a lower computational cost than many methods based on clustering the feature matrix X. When compared to 14 other clustering algorithms applied to 32 benchmarked microarray datasets, the proposed algorithm provided the most accurate estimate of the underlying cluster configuration more than twice as often as its closest competitors.

We describe a modified sequential probability ratio test that can be used to reduce the average sample size required to perform statistical hypothesis tests at specified levels of significance and power. Examples are provided for $z$ tests, $t$ tests, and tests of binomial success probabilities. A description of a software package to implement the test designs is provided. We compare the sample sizes required in fixed design tests conducted at 5$\%$ significance levels to the average sample sizes required in sequential tests conducted at 0.5$\%$ significance levels, and we find that the two sample sizes are approximately equal., Comment: 50 pages, 22 figures

We propose an algorithm for clustering high dimensional data. If $P$ features for $N$ objects are represented in an $N\times P$ matrix ${\bf X}$, where $N\ll P$, the method is based on exploiting the cluster-dependent structure of the $N\times N$ matrix ${\bf XX}^T$. Computational burden thus depends primarily on $N$, the number of objects to be clustered, rather than $P$, the number of features that are measured. This makes the method particularly useful in high dimensional settings, where it is substantially faster than a number of other popular clustering algorithms. Aside from an upper bound on the number of potential clusters, the method is independent of tuning parameters. When compared to $16$ other clustering algorithms on $32$ genomic datasets with gold standards, we show that it provides the most accurate cluster configuration more than twice as often than its closest competitors. We illustrate the method on data taken from highly cited genomic studies.

Uniformly most powerful Bayesian tests (UMPBT's) are an objective class of Bayesian hypothesis tests that can be considered the Bayesian counterpart of classical uniformly most powerful tests. Because the rejection regions of UMPBT's can be matched to the rejection regions of classical uniformly most powerful tests (UMPTs), UMPBT's provide a mechanism for calibrating Bayesian evidence thresholds, Bayes factors, classical significance levels and p-values. The purpose of this article is to expand the application of UMPBT's outside the class of exponential family models. Specifically, we introduce sufficient conditions for the existence of UMPBT's and propose a unified approach for their derivation. An important application of our methodology is the extension of UMPBT's to testing whether the non-centrality parameter of a chi-squared distribution is zero. The resulting tests have broad applicability, providing default alternative hypotheses to compute Bayes factors in, for example, Pearson's chi-squared test for goodness-of-fit, tests of independence in contingency tables, and likelihood ratio, score and Wald tests., Comment: 16 pages, 3 figures

Efficient variable selection in high-dimensional cancer genomic studies is critical for discovering genes associated with specific cancer types and for predicting response to treatment. Censored survival data is prevalent in such studies. In this article we introduce a Bayesian variable selection procedure that uses a mixture prior composed of a point mass at zero and an inverse moment prior in conjunction with the partial likelihood defined by the Cox proportional hazard model. The procedure is implemented in the R package BVSNLP, which supports parallel computing and uses a stochastic search method to explore the model space. Bayesian model averaging is used for prediction. The proposed algorithm provides better performance than other variable selection procedures in simulation studies, and appears to provide more consistent variable selection when applied to actual genomic datasets., Comment: 33 pages, 11 figures

We introduce a new shrinkage prior on function spaces, called the functional horseshoe prior (fHS), that encourages shrinkage towards parametric classes of functions. Unlike other shrinkage priors for parametric models, the fHS shrinkage acts on the shape of the function rather than inducing sparsity on model parameters. We study the efficacy of the proposed approach by showing an adaptive posterior concentration property on the function. We also demonstrate consistency of the model selection procedure that thresholds the shrinkage parameter of the functional horseshoe prior. We apply the fHS prior to nonparametric additive models and compare its performance with procedures based on the standard horseshoe prior and several penalized likelihood approaches. We find that the new procedure achieves smaller estimation error and more accurate model selection than other procedures in several simulated and real examples. The supplementary material for this article, which contains additional simulated and real data examples, MCMC diagnostics, and proofs of the theoretical results, is available online.

Bayesian model selection procedures based on nonlocal alternative prior densities are extended to ultrahigh dimensional settings and compared to other variable selection procedures using precision-recall curves. Variable selection procedures included in these comparisons include methods based on $g$-priors, reciprocal lasso, adaptive lasso, scad, and minimax concave penalty criteria. The use of precision-recall curves eliminates the sensitivity of our conclusions to the choice of tuning parameters. We find that Bayesian variable selection procedures based on nonlocal priors are competitive to all other procedures in a range of simulation scenarios, and we subsequently explain this favorable performance through a theoretical examination of their consistency properties. When certain regularity conditions apply, we demonstrate that the nonlocal procedures are consistent for linear models even when the number of covariates $p$ increases sub-exponentially with the sample size $n$. A model selection procedure based on Zellner's $g$-prior is also found to be competitive with penalized likelihood methods in identifying the true model, but the posterior distribution on the model space induced by this method is much more dispersed than the posterior distribution induced on the model space by the nonlocal prior methods. We investigate the asymptotic form of the marginal likelihood based on the nonlocal priors and show that it attains a unique term that cannot be derived from the other Bayesian model selection procedures. We also propose a scalable and efficient algorithm called Simplified Shotgun Stochastic Search with Screening (S5) to explore the enormous model space, and we show that S5 dramatically reduces the computing time without losing the capacity to search the interesting region in the model space. The S5 algorithm is available in an \verb R ~package {\it BayesS5} on \texttt{CRAN}.

Uniformly most powerful tests are statistical hypothesis tests that provide the greatest power against a fixed null hypothesis among all tests of a given size. In this article, the notion of uniformly most powerful tests is extended to the Bayesian setting by defining uniformly most powerful Bayesian tests to be tests that maximize the probability that the Bayes factor, in favor of the alternative hypothesis, exceeds a specified threshold. Like their classical counterpart, uniformly most powerful Bayesian tests are most easily defined in one-parameter exponential family models, although extensions outside of this class are possible. The connection between uniformly most powerful tests and uniformly most powerful Bayesian tests can be used to provide an approximate calibration between p-values and Bayes factors. Finally, issues regarding the strong dependence of resulting Bayes factors and p-values on sample size are discussed., Comment: Published in at http://dx.doi.org/10.1214/13-AOS1123 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Comment: Bayesian Checking of the Second Levels of Hierarchical Models [arXiv:0802.0743], Comment: Published in at http://dx.doi.org/10.1214/07-STS235D the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org)

This article describes an extension of classical \chi^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 \chi^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 interpretation, computational convenience and favorable power properties. The proposed diagnostics can be used to assess the adequacy of a broad class of Bayesian models, essentially requiring only a finite-dimensional parameter vector and conditionally independent observations., Comment: Published at http://dx.doi.org/10.1214/009053604000000616 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)