The two-parameter Poisson-Dirichlet distribution, denoted pd(ff; `), is a distribution on the set of decreasing positive sequences with sum 1. The usual Poisson-Dirichlet distribution with a single parameter `, introduced by Kingman, is pd(0; `). Known properties of pd(0; `), including the Markov chain description due to Vershik-Shmidt-Ignatov, are generalized to the two-parameter case. The size-biased random permutation of pd(ff; `) is a simple residual allocation model proposed by Engen in the context of species diversity, and rediscovered by Perman and the authors in the study of excursions of Brownian motion and Bessel processes. For 0 ! ff ! 1, pd(ff; 0) is the asymptotic distribution of ranked lengths of excursions of a Markov chain away from a state whose recurrence time distribution is in the domain of attraction of a stable law of index ff. Formulae in this case trace back to work of Darling, Lamperti and Wendel in the 1950's and 60's. The distribution of ranked lengths of e...

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Abstract — In this paper, we briefly review recent advances in blind source separation (BSS) for nonlinear mixing models. After a general introduction to the nonlinear BSS and ICA (independent Component Analysis) problems, we discuss in more detail uniqueness issues, presenting some new results. A fundamental difficulty in the nonlinear BSS problem and even more so in the nonlinear ICA problem is that they are nonunique without extra constraints, which are often implemented by using a suitable regularization. Post-nonlinear mixtures are an important special case, where a nonlinearity is applied to linear mixtures. For such mixtures, the ambiguities are essentially the same as for the linear ICA or BSS problems. In the later part of this paper, various separation techniques proposed for post-nonlinear mixtures and general nonlinear mixtures are reviewed. I. THE NONLINEAR ICA AND BSS PROBLEMS Consider Æ samples of the observed data vector Ü, modeled by

The first part of this paper is concerned by the history of source separation. It include our comments and those of a few other researchers on the development of this new research field. The second part is focused on recent developments of the separation in nonlinear mixtures.

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We study fundamental properties of the gamma process and their relation to various topics such as Poisson–Dirichlet measures and stable processes. We prove the quasi-invariance of the gamma process with respect to a large group of linear transformations. We also show that it is a renormalized limit of the stable processes and has an equivalent sigma-finite measure (quasi-Lebesgue) with important invariance properties. New properties of the gamma process can be applied to the Poisson—Dirichlet measures. We also emphasize the deep similarity between the gamma process and the Brownian motion. The connection of the above topics makes more transparent some old and new facts about stable and gamma processes, and the Poisson-Dirichlet measures. Keywords. Lévy processes, gamma process, stable processes, Poisson–Dirichlet measures, multiplicative quasi-invariance, quasi-Lebesgue measure, Markov–Krein identity.

We investigate a multiple hypothesis test designed for detecting signals embedded in noisy observations of a sensor array. The global level of the multiple test is controlled by the false discovery rate (FDR) criterion recently suggested by Benjamini and Hochberg instead of the classical familywise error rate (FWE) criterion. In the previous study [3], the suggested procedure has shown promising results on simulated data. Here we carefully examine the independence condition required by the Benjamini Hochberg procedure to ensure the control of FDR. Based on the properties of beta distribution, we proved the applicability of the Benjamini Hochberg procedure to the underlying test. Further simulation results show that the false alarm rate is less than 0.02 for a choice of FDR as high as 0.1, implying the reliability of the test has not been affected by the increase in power. 1.

The principles underlying this paper can be applied not only to C2I systems, but also to many other complex structures, such as those involving medical, fault or configuration diagnoses and analyses. In short, a new relatively simple look-up table of formulas is presented which can be used as a practical aid in C2 decision-making nodes for deducing conditional or unconditional conclusions from (conditional or unconditional) premises in probability form. This results from a recent breakthrough yielding a new Cognitive Probability Logic-- or Logic of Averages. This logic is actually a natural weighting modification of Adams ’ well-known High Probability Logic. Consequently, a number of long-standing conflicts between ordinary probability logic and “commonsense ” reasoning are resolved for the first time, including the well-known transitivitysyllogism problem. These results are based upon completely rigorous universal second order probability principles, together with use of the newly emerging field of product space conditional event algebra. Surprisingly, both disciplines are actually technically entirely within the purview of classical logic and basic probability theory. Applications to linguistic-based information can also be obtained by use of these techniques, together with one-point random set coverage representations of fuzzy logic and an extension of product space conditional event algebra, dubbed relational event algebra.

Abstract: An important line of research is the investigation of the laws of random variables known as Dirichlet means as discussed in Cifarelli and Regazzini (7). However there is not much information on inter-relationships between different Dirichlet means. Here we introduce two distributional operations, which consist of multiplying a mean functional by an independent beta random variable and an operation involving an exponential change of measure. These operations identify relationships between different means and their densities. This allows one to use the often considerable analytic work to obtain results for one Dirichlet mean to obtain results for an entire family of otherwise seemingly unrelated Dirichlet means. Additionally, it allows one to obtain explicit densities for the related class of random variables that have generalized gamma convolution distributions, and the finite-dimensional distribution of their associated Lévy processes. This has implications in, for instance, the explicit description of Bayesian nonparametric prior and posterior models, and more generally in a variety of applications in probability and statistics involving Lévy processes.

We consider the problem of simulating X conditional on the value of X +Y, when X and Y are independent positive random variables. We propose approximate methods for sampling (X|X +Y) by approximating the fraction (X/z|X + Y = z) with a beta random variable. We discuss applications to Lévy processes and infinitely divisible distributions, and we report numerical tests for Poisson processes, tempered stable processes, and the Heston stochastic volatility model. 1