Julio Michael Stern

Can a significance test be genuinely Bayesian

The Full Bayesian Signicance Test, FBST, is extensively reviewed. Its test statistic, a genuine Bayesian measure of evidence, is discussed in detail. Its behavior in some problems of statistical inference like testing for independence in contingency tables is discussed.

Bayesian evidence test for precise hypotheses

Abstract The full Bayesian significance test (FBST) for precise hypotheses is presented, with some illustrative applications. In the FBST we compute the evidence against the precise hypothesis. We discuss some of the theoretical properties of the FBST, and provide an invariant formulation for coordinate transformations, provided a reference density has been established. This evidence is the probability of the highest relative surprise set, “tangential” to the sub-manifold (of the parameter space) that defines the null hypothesis.

The Rules of Logic Composition for the Bayesian Epistemic e-Values

In this paper, the relationship between the e-value of a complex hypothesis, H, and those of its constituent elementary hypotheses, H j , j = 1… k, is analyzed, in the independent setup. The e-value of a hypothesis H, ev(H), is a Bayesian epistemic, credibility or truth value defined under the Full Bayesian Significance Testing (FBST) mathematical apparatus. The questions addressed concern the important issue of how the truth value of H, and the truth function of the corresponding FBST structure M, relate to the truth values of its elementary constituents, H j , and to the truth functions of their corresponding FBST structures M j , respectively.

A straightforward multiallelic significance test for the Hardy-Weinberg equilibrium law

Much forensic inference based upon DNA evidence is made assuming Hardy-Weinberg Equilibrium (HWE) for the genetic loci being used. Several statistical tests to detect and measure deviation from HWE have been devised, and their limitations become more obvious when testing for deviation within multiallelic DNA loci. The most popular methods-Chi-square and Likelihood-ratio tests-are based on asymptotic results and cannot guarantee a good performance in the presence of low frequency genotypes. Since the parameter space dimension increases at a quadratic rate on the number of alleles, some authors suggest applying sequential methods, where the multiallelic case is reformulated as a sequence of “biallelic” tests. However, in this approach it is not obvious how to assess the general evidence of the original hypothesis; nor is it clear how to establish the significance level for its acceptance/rejection. In this work, we introduce a straightforward method for the multiallelic HWE test, which overcomes the aforementioned issues of sequential methods. The core theory for the proposed method is given by the Full Bayesian Significance Test (FBST), an intuitive Bayesian approach which does not assign positive probabilities to zero measure sets when testing sharp hypotheses. We compare FBST performance to Chi-square, Likelihood-ratio and Markov chain tests, in three numerical experiments. The results suggest that FBST is a robust and high performance method for the HWE test, even in the presence of several alleles and small sample sizes.

Testing the independence of Poisson variates under the Holgate bivariate distribution: the power of a new evidence test

A new Evidence Test is applied to the problem of testing whether two Poisson random variables are dependent. The dependence structure is that of Holgates bivariate distribution. These bivariate distribution depends on three parameters, 0

Simulated Annealing with a Temperature Dependent Penalty Function

We formulate the problem of permuting a matrix to block angular form as the combinatorial minimization of an objective function. We motivate the use of simulated annealing (SA) as an optimization tool. We then introduce a heuristic temperature dependent penalty function in the simulated annealing cost function, to be used instead of the real objective function being minimized. Finally we show that this temperature dependent penalty function version of simulated annealing consistently outperforms the standard simulated annealing approach, producing, with smaller running times, better solutions. We believe that the use of a temperature dependent penalty function may be useful in developing SA algorithms for other combinatorial problems. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

Bayesian epistemic values: focus on surprise, measure probability!

The e-value or epistemic value, ev(H ), measures the statistical significance of H , a hypothesis about the parameter θ of a Bayesian model. The e-value is obtained by a probability-possibility transformation of the model’s posterior measure, p(θ ), and can, in turn, be used to define the FBST or Full Bayesian Significance Test. This article investigates the relation of this novel approach to more standard probability-possibility transformations. In particular, we show how and why the e-value focus on or conforms with s(θ ) = p(θ )/r(θ ), the model’s surprise function relative to the reference density r(θ ), while it keeps itself consistent with the model’s posterior probability measure. In addition, we investigate traditional objections raised in decision theoretic Bayesian statistics against measures of significance engendered by probability-possibility transformations.

Paraconsistent Sensitivity Analysis for Bayesian Significance Tests

In this paper, the notion of degree of inconsistency is introduced as a tool to evaluate the sensitivity of the Full Bayesian Significance Test (FBST) value of evidence with respect to changes in the prior or reference density. For that, both the definition of the FBST, a possibilistic approach to hypothesis testing based on Bayesian probability procedures, and the use of bilattice structures, as introduced by Ginsberg and Fitting, in paraconsistent logics, are reviewed. The computational and theoretical advantages of using the proposed degree of inconsistency based sensitivity evaluation as an alternative to traditional statistical power analysis is also discussed.

Nested dissection for sparse nullspace bases

We propose a nested dissection approach to finding a fundamental cycle basis in a planar graph. the cycle basis corresponds to a fundamental nullspace basis of the adjacency matrix. This problem is meant to model sparse null basis computations occurring in a variety of settings. We achieve an O(n**3/2) bound on the nullspace basis size and an O(nlogn) bound on the size in the special case of grid graphs.

Symmetry, Invariance and Ontology in Physics and Statistics

This paper has three main objectives: (a) Discuss the formal analogy between some important symmetry-invariance arguments used in physics, probability and statistics. Specifically, we will focus on Noether’s theorem in physics, the maximum entropy principle in probability theory, and de Finetti-type theorems in Bayesian statistics; (b) Discuss the epistemological and ontological implications of these theorems, as they are interpreted in physics and statistics. Specifically, we will focus on the positivist (in physics) or subjective (in statistics) interpretations vs. objective interpretations that are suggested by symmetry and invariance arguments; (c) Introduce the cognitive constructivism epistemological framework as a solution that overcomes the realism-subjectivism dilemma and its pitfalls. The work of the physicist and philosopher Max Born will be particularly important in our discussion.