Related papers: Test Martingales, Bayes Factors and $p$-Values
Certain countably and finitely additive measures can be associated to a given nonnegative supermartingale. Under weak assumptions on the underlying probability space, existence and (non)uniqueness results for such measures are proven.
Ratios of universal enumerable semimeasures corresponding to hypotheses are investigated as a solution for statistical composite hypotheses testing if an unbounded amount of computation time can be assumed. Influence testing for discrete…
We develop a new method for frequentist multiple testing with Bayesian prior information. Our procedure finds a new set of optimal p-value weights called the Bayes weights. Prior information is relevant to many multiple testing problems.…
Significance tests are probably the most extended form of inference in empirical research, and significance is often interpreted as providing greater informational content than non-significance. In this article we show, however, that…
A Bayesian approach is developed to determine quantum mechanical potentials from empirical data. Bayesian methods, combining empirical measurements and "a priori" information, provide flexible tools for such empirical learning problems. The…
We give tight concentration bounds for mixtures of martingales that are simultaneously uniform over (a) mixture distributions, in a PAC-Bayes sense; and (b) all finite times. These bounds are proved in terms of the martingale variance,…
There have been many attempts to solve the P versus NP problem. However, with a new proof method, P not equal NP can be proved. A time limit is set for an arbitrary Turing machine and an input word is rejected on a timeout. The time limit…
This work provides a novel convergence analysis for stochastic optimization in terms of stopping times, addressing the practical reality that algorithms are often terminated adaptively based on observed progress. Unlike prior approaches,…
There is a third way of implementing probability models and practicing. This is to answer questions put in terms of observables. This eliminates frequentist hypothesis testing and Bayes factors and it also eliminates parameter estimation.…
Priors in which a large number of parameters are specified to be independent are dangerous; they make it hard to learn from data. I present a couple of examples from the literature and work through a bit of large sample theory to show what…
We study the asymptotic behavior of the weighted least squares estimators of the unknown parameters of bifurcating integer-valued autoregressive processes. Under suitable assumptions on the immigration, we establish the almost sure…
We study set-valued classification for a Bayesian model where data originates from one of a finite number $N$ of possible hypotheses. Thus we consider the scenario where the size of the classified set of categories ranges from 0 to $N$.…
This paper considers the problem of making statistical inferences about a parameter when a narrow interval centred at a given value of the parameter is considered special, which is interpreted as meaning that there is a substantial degree…
In physics the value of a theory is measured by its agreement with experimental data. But how should the physics community gauge the value of an emerging theory that has not been tested experimentally as of yet? With no reality check, a…
It is widely acknowledged that the biomedical literature suffer from a surfeit of false positive results. Part of the reason for this is the persistence of the myth that observation of a p value less than 0.05 is sufficient justification to…
In statistics, there are a variety of methods for performing model selection that all stem from slightly different paradigms of statistical inference. The reasons for choosing one particular method over another seem to be based entirely on…
Using martingale methods, we provide bounds for the entropy of a probability measure on $\mathbb {R}^d$ with the right-hand side given in a certain integral form. As a corollary, in the one-dimensional case, we obtain a weighted log-Sobolev…
A new empirical Bayes approach to variable selection in the context of generalized linear models is developed. The proposed algorithm scales to situations in which the number of putative explanatory variables is very large, possibly much…
Factorization of numbers with the help of Gauss sums relies on an intimate relationship between the maxima of these functions and the factors. Indeed, when we restrict ourselves to integer arguments of the Gauss sum we profit from a…
We consider the common setting where one observes probability estimates for a large number of events, such as default risks for numerous bonds. Unfortunately, even with unbiased estimates, selecting events corresponding to the most extreme…