Probability inheritance algorithm and its implementation

An algorithm for functional evaluation of the likelihood of paternal and maternal recombination fractions for pedigree data is proposed. The idea behind the algorithm is that the probability of affected status and certain marker genotypes of ancestors is inherited by their descendants along with the inheritance of certain haplotypes. In this algorithm, the likelihood is evaluated by a single recursive call for each terminal sibling set along with the inheritance flow. The advantage of the algorithm is not only the simplicity of its implementation, but also its functional form of evaluation. The likelihood is obtained as a polynomial of the recombination fractions, making it easier to validate the likelihood more carefully, resulting in a more accurate localization of the disease locus. We report an experimental implementation of this algorithm in R, together with several practical applications.     

Heuristic algorithms for finding inexpensive elimination schemes

The computational cost, in both storage requirements and calculation, of performing an elimination ordering is considered as a function of the order in which the vertices of a graph are eliminated. Several different heuristic and relaxed heuristic algorithms for finding low cost elimination orderings are described and compared. The new heuristic and relaxed heuristic algorithms proposed in this paper are shown to find computationally more efficient elimination orderings than previously proposed heuristic algorithms.     

A note on the estimates of multivariate Gaussian probability

ABSTRACT

Some lower and upper bounds of multivariate Gaussian probability are given based on the univariate Mills’ ratio. These bounds are sharper than known ones on the multivariate Mills’ ratio in many case.     

A note on the orthant probability of the elliptically contoured distribution

When a scale matrix satisfies certain conditions, the orthant probability of the elliptically contoured distribution with the scale matrix is expressed as the same probability of the equicorrelated normal distribution.     

Bayesian predictive power: choice of prior and some recommendations for its use as probability of success in drug development

Bayesian predictive power, the expectation of the power function with respect to a prior distribution for the true underlying effect size, is routinely used in drug development to quantify the probability of success of a clinical trial. Choosing the prior is crucial for the properties and interpretability of Bayesian predictive power. We review recommendations on the choice of prior for Bayesian predictive power and explore its features as a function of the prior. The density of power values induced by a given prior is derived analytically and its shape characterized. We find that for a typical clinical trial scenario, this density has a u‐shape very similar, but not equal, to a β‐distribution. Alternative priors are discussed, and practical recommendations to assess the sensitivity of Bayesian predictive power to its input parameters are provided. Copyright © 2016 John Wiley & Sons, Ltd.     

Decomposing Bayesian networks: triangulation of the moral graph with genetic algorithms

In this paper we consider the optimal decomposition of Bayesian networks. More concretely, we examine empirically the applicability of genetic algorithms to the problem of the triangulation of moral graphs. This problem constitutes the only difficult step in the evidence propagation algorithm of Lauritzen and Spiegelhalter (1988) and is known to be NP-hard (Wen, 1991). We carry out experiments with distinct crossover and mutation operators and with different population sizes, mutation rates and selection biasses. The results are analysed statistically. They turn out to improve the results obtained with most other known triangulation methods (Kjærulff, 1990) and are comparable to results obtained with simulated annealing (Kjærulff, 1990; Kjærulff, 1992).     

A note on the unimddality of discrete distributions

It is noted that the unimadility property is very inportant and necessary in many probabilistic-statistical models. In this paper, we consider the definition of discrete uniirodality such that the mode may be unique integer or a sequence of consecutive integers. It will be shown that the necessary and sufficient condition for the discrete distribution to be uninnodal about a can be given through some canonical representation of its characteristic function ‘ch.f.’ Further, characterization results for some well-known distributions are established.     

A further note on the bivariate normal distribution

It is indicated to what extent the conditional normality of the distribution of one comnonent of a bivariate random vector given the value of the other component together with a restricted type of conditional normality or the marginal normality for the other component is equivalent to the bivariate normality of this random vector.     

A note on the inequalities for tail probability of the multivariate normal distribution

In this paper inequalities given by Harkness Godambe (1976) for the rail probabilities of the multivariate normal distribution in the equicorrelated case are improved by using the properties of the characteristic roots of a matrix and of the convex function.     

A note on separated data and exact likelihood-ratio p-values in logistic regression

Exact conditional p-values based on the likelihood-ratio statistic in logistic regression require accurate computation of the supremum of the likelihood function, particularly for outcomes in the sample space that represent completely-separated or quasi-completely-separated data sets. Current software does not always handle these cases well. Three simple solutions are proposed.     

Functional data: local linear estimation of the conditional density and its application

In this paper, we introduce a new nonparametric estimation procedure of the conditional density of a scalar response variable given a random variable taking values in a semi-metric space. Under some general conditions, we establish both the pointwise and the uniform almost-complete consistencies with convergence rates of the conditional density estimator related to this estimation procedure. Moreover, we give some particular cases of our results which can also be considered as novel in the finite-dimensional setting. Notice also that the results of this paper are used to derive some asymptotic properties of the local linear estimator of the conditional mode.     

A note on the geometry of kullback-leibler information numbers

Csiszar (1975) has shown thdt kullhack-Leibler information numbers possess some geometrical properties much like those in Euclidean geometry. Thls paper extends these results by characterizing the shortest line between two ditributions as well as the midpoit of the line.     

A note on calibration weightings for stratified double sampling with equal probability

This study proposes a more efficient calibration estimator for estimating population mean in stratified double sampling using new calibration weights. The variance of the proposed calibration estimator has been derived under large sample approximation. Calibration asymptotic optimum estimator and its approximate variance estimator are derived for the proposed calibration estimator and existing calibration estimators in stratified double sampling. Analytical results showed that the proposed calibration estimator is more efficient than existing members of its class in stratified double sampling. Analysis and evaluation are presented.     

A note on exact calculation of the non central hypergeometric distribution

Direct calculation of the non central hypergeometric (NH) distribution and its moments can present computational issues in both efficiency and accuracy. In response, several methods, both approximate and exact, for calculating the NH mean and variance have appeared in the literature. We add to this body of work, a straight-forward, exact method that is easily programed, efficient, and computationally stable. Specifically, by considering the logs of the values of the NH probability mass function (pmf) and then shifting the exponents so that, prior to normalization, the mode acquires a value of 1, concerns for overflow are eliminated.     

A formulation of the concept of probability based on the use of experimental devices

This paper sets out to identify the abilities that a person needs to be able to successfully use an experimental device, such as a probability wheel or balls in an urn, for the elicitation of subjective probabilities. It is assumed that the successful use of the device requires that the person elicits unique probability values that obey the standard probability laws. This leads to a definition of probability based on the idea of the similarity between the likeliness of events and this concept is naturally extended to the idea that probabilities have strengths, which relates to information about the likeliness of an event that lies beyond a simple probability value. The latter notion is applied to the problem of explaining the Ellsberg paradox. To avoid the definition of probability being circular, probabilities are defined such that they depend on the choice of a reference set of events R which, in simple cases, corresponds to the raw outcomes produced by using an experimental device. However, it is shown that even when the events in R are considered as having an “equal chance” of occurring, the values and/or strengths of probabilities can still be affected by the choice of the set R.     

Naive method to test the convergence of simulation and its applications in the computation of bankruptcy probability

We analyse a naive method using sample mean and sample variance to test the convergence of simulation. We find this method is valid for identically, independently distributed samples, as well as correlated samples with correlation disappearing in long period. Our simulation results on the approximation to bankruptcy probability (BP) show the naive method compares well with the Half-Width, Geweke and CUSUM methods in terms of accuracy and time cost. There are clear evidences of variance reduction from tail-distribution sampling for all convergence test methods when the true BP is very low.     

A note on nonparametric estimation of circular conditional densities

ABSTRACT

The conditional density offers the most informative summary of the relationship between explanatory and response variables. We need to estimate it in place of the simple conditional mean when its shape is not well-behaved. A motivation for estimating conditional densities, specific to the circular setting, lies in the fact that a natural alternative of it, like quantile regression, could be considered problematic because circular quantiles are not rotationally equivariant. We treat conditional density estimation as a local polynomial fitting problem as proposed by Fan et al. [Estimation of conditional densities and sensitivity measures in nonlinear dynamical systems. Biometrika. 1996;83:189–206] in the Euclidean setting, and discuss a class of estimators in the cases when the conditioning variable is either circular or linear. Asymptotic properties for some members of the proposed class are derived. The effectiveness of the methods for finite sample sizes is illustrated by simulation experiments and an example using real data.     

A hybrid approach based on saddlepoint and importance sampling methods for bootstrap tail probability estimation

We propose a simple hybrid method which makes use of both saddlepoint and importance sampling techniques to approximate the bootstrap tail probability of an M-estimator. The method does not rely on explicit formula of the Lugannani-Rice type, and is computationally more efficient than both uniform bootstrap sampling and importance resampling suggested in earlier literature. The method is also applied to construct confidence intervals for smooth functions of M-estimands.     

A flexible procedure for formulating probability distributions on the unit interval with applications

Abstract

In this paper, we present a flexible mechanism for constructing probability distributions on a bounded intervals which is based on the composition of the baseline cumulative probability function and the quantile transformation from another cumulative probability distribution. In particular, we are interested in the (0, 1) intervals. The composite quantile family of probability distributions contains many models that have been proposed in the recent literature and new probability distributions are introduced on the unit interval. The proposed methodology is illustrated with two examples to analyze a poverty dataset in Peru from the Bayesian paradigm and Likelihood points of view.