Family Of Probability Distributions Research Articles

Multivariate Jacobi polynomials and the Selberg integral

This work is motivated by the problem of harmonic analysis on “big” groups and can be viewed as a continuation of the first author’s paper in Functional Anal. Appl. 37 (2003), no. 4, 281–301. Our main result is the proof of the existence of a family of probability distributions with infinite-dimensional support; these distributions are analogs of multidimensional Euler betadistributions that appear in the Selberg integral.

Line profiles from discrete kinematic data

We develop a method to extract the shape information of line profiles from discrete kinematic data. The Gauss-Hermite expansion, which is widely used to describe the line of sight velocity distributions extracted from absorption spectra of elliptical galaxies, is not readily applicable to samples of discrete stellar velocity measurements, accompanied by individual measurement errors and probabilities of membership. We introduce two parameter families of probability distributions describing symmetric and asymmetric distortions of the line profiles from Gaussianity. These are used as the basis of a maximum likelihood estimator to quantify the shape of the line profiles. Tests show that the method outperforms a Gauss-Hermite expansion for discrete data, with a lower limit for the relative gain of approx 2 for sample sizes N approx 800. To ensure that our methods can give reliable descriptions of the shape, we develop an efficient test to assess the statistical quality of the obtained fit. As an application, we turn our attention to the discrete velocity datasets of the dwarf spheroidals of the Milky Way. In Sculptor, Carina and Sextans the symmetric deviations are consistent with velocity distributions more peaked than Gaussian. In Fornax, instead, there is an evolution in the symmetric deviations of the line profile from a peakier to more flat-topped distribution on moving outwards. These results suggest a radially biased orbital structure for the outer parts of Sculptor, Carina and Sextans. On the other hand, tangential anisotropy is favoured in Fornax. This is all consistent with a picture in which Fornax may have had a different evolutionary history to Sculptor, Carina and Sextans.

Fuzzy Uncertainty Representations of Coseismic Displacement Measurements Issued From SAR Imagery

An emerging way to reduce the geodetic parameter uncertainty is to combine the large numbers of data provided by satellite synthetic aperture radar (SAR) images. However, the measurements by radar imagery are subject to both random and systematic uncertainties. Thus, mathematical theories that are adequate for each type of uncertainty representation and handling have to be selected. Probability theory is known as the adequate theory for uncertainties corresponding to random variables, but questionable for systematic uncertainties, arising from information incompleteness. Fuzzy theory, being a generalization of interval mathematics, is more adapted to such uncertainty. Moreover, it provides a bridge with probability theory by its ability to represent a family of probability distributions. Therefore, we consider here the conventional probability and the fuzzy approaches for handling the random and systematic uncertainties of displacement measurements derived from differential SAR interferometry (D-InSAR) and SAR amplitude image correlation. The applications are performed on the measurement of the displacement field due to the 2005 Kashmir earthquake. The fuzzy approach, being free from distribution and independence hypotheses, gives the most pessimistic uncertainty assessment, while the conventional probability approach gives the most optimistic uncertainty assessment. As confirmed by the Monte Carlo simulation applied to an earth deformation model, the actual uncertainty should be situated between the fuzzy and conventional uncertainties.

Geometry of deformed exponential families: Invariant, dually-flat and conformal geometries

An information-geometrical foundation is established for the deformed exponential families of probability distributions. Two different types of geometrical structures, an invariant geometry and a flat geometry, are given to a manifold of a deformed exponential family. The two different geometries provide respective quantities such as deformed free energies, entropies and divergences. The class belonging to both the invariant and flat geometries at the same time consists of exponential and mixture families. Theq-families are characterized from the viewpoint of the invariant and flat geometries. The q-exponential family is a unique class that has the invariant and flat geometries in the extended class of positive measures. Furthermore, it is the only class of which the Riemannian metric is conformally connected with the invariant Fisher metric.

Commutative algebra of statistical ranking

A model for statistical ranking is a family of probability distributions whose states are orderings of a fixed finite set of items. We represent the orderings as maximal chains in a graded poset. The most widely used ranking models are parameterized by rational function in the model parameters, so they define algebraic varieties. We study these varieties from the perspective of combinatorial commutative algebra. One of our models, the Plackett–Luce model, is non-toric. Five others are toric: the Birkhoff model, the ascending model, the Csiszár model, the inversion model, and the Bradley–Terry model. For these models we examine the toric algebra, its lattice polytope, and its Markov basis.

Jensen divergence based on Fisher’s information

The measure of Jensen–Fisher divergence between probability distributions is introduced and its theoretical grounds set up. This quantity, in contrast to the remaining Jensen divergences, grasps the fluctuations of the probability distributions because it is controlled by the (local) Fisher information, which is a gradient functional of the distribution. So it is appropriate and informative when studying the similarity of distributions, mainly for those having oscillatory character. The new Jensen–Fisher divergence shares with the Jensen–Shannon divergence the following properties: non-negativity, additivity when applied to an arbitrary number of probability densities, symmetry under exchange of these densities, vanishing under certain conditions and definiteness even when these densities present non-common zeros. Moreover, the Jensen–Fisher divergence is shown to be expressed in terms of the relative Fisher information as the Jensen–Shannon divergence does in terms of the Kullback–Leibler or relative Shannon entropy. Finally, the Jensen–Shannon and Jensen–Fisher divergences are compared for the following three large, non-trivial and qualitatively different families of probability distributions: the sinusoidal, generalized gamma-like and Rakhmanov–Hermite distributions, which are closely related to the quantum-mechanical probability densities of numerous physical systems.

A Moment Preserving Finitization Across the Power Series Family of Probability Distributions

Finitization transforms a discrete distribution into a distribution with smaller support of specified size. In special cases finitization preserves moments (moments of the order n finitization coincide with those of the parent distribution). We create a moment preserving finitization method for power series distributions by introducing an alternative representation and showing how to finitize members of this new class in a manner that preserves moments of the parent distribution. We provide results on convolutions and a reproductive property for power series distributions that have been finitized in this manner, and show how these finitized distributions accelerate variate generation in simulation.

Comparison of Quantum Statistical Models: Equivalent Conditions for Sufficiency

A family of probability distributions (i.e. a statistical model) is said to be sufficient for another, if there exists a transition matrix transforming the probability distributions in the former to the probability distributions in the latter. The Blackwell-Sherman-Stein (BSS) theorem provides necessary and sufficient conditions for one statistical model to be sufficient for another, by comparing their information values in statistical decision problems. In this paper we extend the BSS theorem to quantum statistical decision theory, where statistical models are replaced by families of density matrices defined on finite-dimensional Hilbert spaces, and transition matrices are replaced by completely positive, trace-preserving maps (i.e. coarse-grainings). The framework we propose is suitable for unifying results that previously were independent, like the BSS theorem for classical statistical models and its analogue for pairs of bipartite quantum states, recently proved by Shmaya. An important role in this paper is played by statistical morphisms, namely, affine maps whose definition generalizes that of coarse-grainings given by Petz and induces a corresponding criterion for statistical sufficiency that is weaker, and hence easier to be characterized, than Petz's.

On φ-Families of Probability Distributions

We generalize the exponential family of probability distributions. In our approach, the exponential function is replaced by a φ-function, resulting in a φ-family of probability distributions. We show how φ-families are constructed. In a φ-family, the analogue of the cumulant-generating function is a normalizing function. We define the φ-divergence as the Bregman divergence associated to the normalizing function, providing a generalization of the Kullback–Leibler divergence. A formula for the φ-divergence where the φ-function is the Kaniadakis κ-exponential function is derived.

HRF Estimation in fMRI Data With an Unknown Drift Matrix by Iterative Minimization of the Kullback–Leibler Divergence

Hemodynamic response function (HRF) estimation in noisy functional magnetic resonance imaging (fMRI) plays an important role when investigating the temporal dynamic of a brain region response during activations. Nonparametric methods which allow more flexibility in the estimation by inferring the HRF at each time sample have provided improved performance in comparison to the parametric methods. In this paper, the mixed-effects model is used to derive a new algorithm for nonparametric maximum likelihood HRF estimation. In this model, the random effect is used to better account for the variability of the drift. Contrary to the usual approaches, the proposed algorithm has the benefit of considering an unknown and therefore flexible drift matrix. This allows the effective representation of a broader class of drift signals and therefore the reduction of the error in approximating the drift component. Estimates of the HRF and the hyperparameters are derived by iterative minimization of the Kullback-Leibler divergence between a model family of probability distributions defined using the mixed-effects model and a desired family of probability distributions constrained to be concentrated on the observed data. The performance of proposed method is demonstrated on simulated and real fMRI data, the latter originating from both event-related and block design fMRI experiments.

Geometry of q-Exponential Family of Probability Distributions

The Gibbs distribution of statistical physics is an exponential family of probability distributions, which has a mathematical basis of duality in the form of the Legendre transformation. Recent studies of complex systems have found lots of distributions obeying the power law rather than the standard Gibbs type distributions. The Tsallis q-entropy is a typical example capturing such phenomena. We treat the q-Gibbs distribution or the q-exponential family by generalizing the exponential function to the q-family of power functions, which is useful for studying various complex or non-standard physical phenomena. We give a new mathematical structure to the q-exponential family different from those previously given. It has a dually flat geometrical structure derived from the Legendre transformation and the conformal geometry is useful for understanding it. The q-version of the maximum entropy theorem is naturally induced from the q-Pythagorean theorem. We also show that the maximizer of the q-escort distribution is a Bayesian MAP (Maximum A posteriori Probability) estimator.

The centred parameterization and related quantities of the skew-[formula omitted] distribution

The skew-t family, in its univariate and multivariate versions, is a parametric family of probability distributions which is currently under intense investigation because of several appealing properties. The present paper addresses the question of the choice of its parameterization, and more generally of the selection of quantities of interest associated to this distribution.

Stochastic approximation learning for mixtures of multivariate elliptical distributions

Most of the current approaches to mixture modeling consider mixture components from a few families of probability distributions, in particular from the Gaussian family. The reasons of these preferences can be traced to their training algorithms, typically versions of the Expectation-Maximization (EM) method. The re-estimation equations needed by this method become very complex as the mixture components depart from the simplest cases. Here we propose to use a stochastic approximation method for probabilistic mixture learning. Under this method it is straightforward to train mixtures composed by a wide range of mixture components from different families. Hence, it is a flexible alternative for mixture learning. Experimental results are presented to show the probability density and missing value estimation capabilities of our proposal.

A family of empirical likelihood functions and estimators for the binary response model

The minimum discrimination information principle is used to identify an appropriate parametric family of probability distributions and the corresponding maximum likelihood estimators for binary response models. Estimators in the family subsume the conventional logit model and form the basis for a set of parametric estimation alternatives with the usual asymptotic properties. Sampling experiments are used to assess finite sample performance.

Optimal assignment of resources to strengthen the weakest link in an uncertain environment

The weakest link principle applies to many real-life situations where a system is as productive as its bottleneck. The purpose of this paper is to show how the efficiency of a manufacturing or service process (i.e., the strength of a chain) may be maximized by optimal allocation of resources to improve the performance of the bottleneck (i.e., strengthen the weakest link) in an uncertain environment. Specifically, we consider two different versions of the stochastic bottleneck assignment problem (SBAP), which is a variation of the classic assignment problem (AP), with respective goals of minimizing the expected longest processing time and maximizing the expected lowest production rate. It is proven that each of the two intrinsically difficult SBAPs is reducible to an efficiently solvable AP provided that the processing times or the production rates are independent random variables following some families of probability distributions.

Probabilistic Framework for Uncertainty Propagation With Both Probabilistic and Interval Variables

This paper develops and illustrates a probabilistic approach for uncertainty representation and propagation in system analysis, when the information on the uncertain input variables and/or their distribution parameters may be available as either probability distributions or simply intervals (single or multiple). A unique aggregation technique is used to combine multiple interval data and to compute rigorous bounds on the system response cumulative distribution function. The uncertainty described by interval data is represented through a flexible family of probability distributions. Conversion of interval data to a probabilistic format enables the use of computationally efficient methods for probabilistic uncertainty propagation. Two methods are explored for the implementation of the proposed approach, based on (1) sampling and (2) optimization. The sampling-based strategy is more expensive and tends to underestimate the output bounds. The optimization-based methodology improves both aspects. The proposed methods are used to develop new solutions to challenge problems posed by the Sandia epistemic uncertainty workshop (Oberkampf et al., 2004, “Challenge Problems: Uncertainty in System Response Given Uncertain Parameters,” Reliab. Eng. Syst. Saf., 85, pp. 11–19). Results for the challenge problems are compared with earlier solutions.

A simple derivation and classification of common probability distributions based on information symmetry and measurement scale

Commonly observed patterns typically follow a few distinct families of probability distributions. Over one hundred years ago, Karl Pearson provided a systematic derivation and classification of the common continuous distributions. His approach was phenomenological: a differential equation that generated common distributions without any underlying conceptual basis for why common distributions have particular forms and what explains the familial relations. Pearson's system and its descendants remain the most popular systematic classification of probability distributions. Here, we unify the disparate forms of common distributions into a single system based on two meaningful and justifiable propositions. First, distributions follow maximum entropy subject to constraints, where maximum entropy is equivalent to minimum information. Second, different problems associate magnitude to information in different ways, an association we describe in terms of the relation between information invariance and measurement scale. Our framework relates the different continuous probability distributions through the variations in measurement scale that change each family of maximum entropy distributions into a distinct family. From our framework, future work in biology can consider the genesis of common patterns in a new and more general way. Particular biological processes set the relation between the information in observations and magnitude, the basis for information invariance, symmetry and measurement scale. The measurement scale, in turn, determines the most likely probability distributions and observed patterns associated with particular processes. This view presents a fundamentally derived alternative to the largely unproductive debates about neutrality in ecology and evolution.

A Kullback–Leibler Divergence Approach to Blind Image Restoration

A new algorithm for maximum-likelihood blind image restoration is presented in this paper. It is obtained by modeling the original image and the additive noise as multivariate Gaussian processes with unknown covariance matrices. The blurring process is specified by its point spread function, which is also unknown. Estimations of the original image and the blur are derived by alternating minimization of the Kullback-Leibler divergence between a model family of probability distributions defined using the linear image degradation model and a desired family of probability distributions constrained to be concentrated on the observed data. The algorithm presents the advantage to provide closed form expressions for the parameters to be updated and to converge only after few iterations. A simulation example that illustrates the effectiveness of the proposed algorithm is presented.

A Kullback-Leibler Methodology for Unconditional ML DOA Estimation in Unknown Nonuniform Noise

Maximum likelihood (ML) direction-of arrival (DOA) estimation of multiple narrowband sources in unknown nonuniform white noise is considered. A new iterative algorithm for stochastic ML DOA estimation is presented. The stepwise concentration of the log-likelihood (LL) function with respect to the signal and noise nuisance parameters is derived by alternating minimization of the Kullback-Leibler divergence between a model family of probability distributions defined on the unconditional model and a desired family of probability distributions constrained to be concentrated on the observed data. The new algorithm presents the advantage to provide closed-form expressions for the signal and noise nuisance parameter estimates which results in a substantial reduction of the parameter space required for numerical optimization. The proposed algorithm converges only after a few iterations and its effectiveness is confirmed in a simulation example.

Uwagi o testach Jarque’a-Bera

The paper involves a description of simulation experiments aimed at determining the size of Jargue-Bera tests for skewness and kurtosis, as well as selected modifications of those tests. All experiments refer to four different families of probability distributions and consist of one thousand runs each.