Computing Bayesian predictive distributions: The K-square and K-prime distributions

The computation of two Bayesian predictive distributions which are discrete mixtures of incomplete beta functions is considered. The number of iterations can easily become large for these distributions and thus, the accuracy of the result can be questionable. Therefore, existing algorithms for that class of mixtures are improved by introducing round-off error calculation into the stopping rule. A further simple modification is proposed to deal with possible underflows that may prevent recurrence to work properly

A quadratic Poisson Gel'fand-Kirillov problem in prime characteristic

The quadratic Poisson Gel’fand-Kirillov problem asks whether the field of fractions of a Poisson algebra is Poisson birationally equivalent to a Poisson affine space, i.e. to a polyno-mial algebra K[X1,..., Xn] with Poisson bracket defined by {Xi, Xj} = λijXiXj for some skew-symmetric matrix (λij) ∈Mn(K). This problem was studied in [9] over a field of charac-teristic 0 by using a Poisson version of the deleting derivation homomorphism of Cauchon. In this paper, we study the quadratic Poisson Gel’fand-Kirillov problem over a field of arbitrary characteristic. In particular, we prove that the quadratic Poisson Gel’fand-Kirillov problem is satisfied for a large class of Poisson algebras arising as semiclassical limits of quantised co-ordinate rings. For, we introduce the concept of higher Poisson derivation which allows us to extend the Poisson version of the deleting derivation homomorphism from the characteristic 0 case to the case of arbitrary characteristic. When a torus is acting rationally by Poisson automorphisms on a Poisson polynomial algebra arising as the semiclassical limit of a quantised coordinate ring, we prove (under some technical assumptions) that quotients by Poisson prime torus-invariant ideals also satisfy the quadratic Poisson Gel’fand-Kirillov problem. In particular, we show that coordinate rings of determinantal varieties satisfy the quadratic Poisson Gel’fand-Kirillov problem

Killeen's Probability of Replication and Predictive Probabilities: How to Compute, Use, and Interpret Them

International audienceP. R. Killeen's (2005a) [Killeen, P. R. - An alternative to null-hypothesis significance tests, Psychological Science, 2005, 16, 345–353] probability of replication (prep) of an experimental result is the fiducial Bayesian predictive probability of finding a same-sign effect in a replication of an experiment. prep is now routinely reported in Psychological Science and has also begun to appear in other journals. However, there is little concrete, practical guidance for use of prep and the procedure has not received the scrutiny that it deserves. Furthermore, only a solution that assumes a known variance has been implemented. A practical problem with prep is identified: in many articles prep appears to be incorrectly computed, due to the confusion between 1-tailed and 2-tailed p-values. Experimental findings reveal the risk of misinterpreting prep as the predictive probability of finding a same-sign and significant effect in a replication (psrep). Conceptual and practical guidelines are given to avoid these pitfalls. They include the extension to the case of unknown variance. Moreover, other uses of fiducial Bayesian predictive probabilities, for analyzing, designing ("how many subjects?") and monitoring ("when to stop?") experiments, are presented. Concluding remarks emphasize the role of predictive procedures in statistical methodology

A Simple Model to Generate Hard Satisfiable Instances

In this paper, we try to further demonstrate that the models of random CSP instances proposed by [Xu and Li, 2000; 2003] are of theoretical and practical interest. Indeed, these models, called RB and RD, present several nice features. First, it is quite easy to generate random instances of any arity since no particular structure has to be integrated, or property enforced, in such instances. Then, the existence of an asymptotic phase transition can be guaranteed while applying a limited restriction on domain size and on constraint tightness. In that case, a threshold point can be precisely located and all instances have the guarantee to be hard at the threshold, i.e., to have an exponential tree-resolution complexity. Next, a formal analysis shows that it is possible to generate forced satisfiable instances whose hardness is similar to unforced satisfiable ones. This analysis is supported by some representative results taken from an intensive experimentation that we have carried out, using complete and incomplete search methods.Comment: Proc. of 19th IJCAI, pp.337-342, Edinburgh, Scotland, 2005. For more information, please click http://www.nlsde.buaa.edu.cn/~kexu/papers/ijcai05-abstract.ht

STR2: Optimized Simple Tabular Reduction for Table Constraints

International audienceTable constraints play an important role within constraint programming. Recently, many schemes or algorithms have been proposed to propagate table constraints and/or to compress their representation. In this paper, we describe an optimization of simple tabular reduction (STR), a technique proposed by J. Ullmann to dynamically maintain the tables of supports when generalized arc consistency (GAC) is enforced/maintained. STR2, the new refined GAC algorithm we propose, allows us to limit the number of operations related to validity checking and search of supports. Interestingly enough, this optimization makes simple tabular reduction potentially r times faster where r is the arity of the constraint(s). The results of an extensive experimentation that we have conducted with respect to random and structured instances indicate that STR2 is usually around twice as fast as the original STR, two or three times faster than the approach based on the hidden variable encoding, and can be up to one order of magnitude faster than previously state-of-the art (generic) GAC algorithms on some series of instances. When comparing STR2 with the more recently developed algorithm based on multi-valued decision diagrams (MDDs), we show that both approaches are rather complementary

Another Look at the Confidence Intervals for the Noncentral T Distribution

An alternative approach to the computation of confidence intervals for the noncentrality parameter of the Noncentral t distribution is proposed. It involves the percent points of a statistical distribution. This conceptual improvement renders the technical process for deriving the limits more comprehensible. Accurate approximations can be derived and easily used

Polynomial Poisson algebras: Gel'fand-Kirillov problem and Poisson spectra

We study the fields of fractions and the Poisson spectra of polynomial Poisson algebras. First we investigate a Poisson birational equivalence problem for polynomial Poisson algebras over a field of arbitrary characteristic. Namely, the quadratic Poisson Gel'fand-Kirillov problem asks whether the field of fractions of a Poisson algebra is isomorphic to the field of fractions of a Poisson affine space, i.e. a polynomial algebra such that the Poisson bracket of two generators is equal to their product (up to a scalar). We answer positively the quadratic Poisson Gel'fand-Kirillov problem for a large class of Poisson algebras arising as semiclassical limits of quantised coordinate rings, as well as for their quotients by Poisson prime ideals that are invariant under the action of a torus. In particular, we show that coordinate rings of determinantal Poisson varieties satisfy the quadratic Poisson Gel'fand-Kirillov problem. Our proof relies on the so-called characteristic-free Poisson deleting derivation homomorphism. Essentially this homomorphism allows us to simplify Poisson brackets of a given polynomial Poisson algebra by localising at a generator. Next we develop a method, the characteristic-free Poisson deleting derivations algorithm, to study the Poisson spectrum of a polynomial Poisson algebra. It is a Poisson version of the deleting derivations algorithm introduced by Cauchon [8] in order to study spectra of some noncommutative noetherian algebras. This algorithm allows us to define a partition of the Poisson spectrum of certain polynomial Poisson algebras, and to prove the Poisson Dixmier-Moeglin equivalence for those Poisson algebras when the base field is of characteristic zero. Finally, using both Cauchon's and our algorithm, we compare combinatorially spectra and Poisson spectra in the framework of (algebraic) deformation theory. In particular we compare spectra of quantum matrices with Poisson spectra of matrix Poisson varieties