General Probability Measures Research Articles

Probabilistic and deterministic algorithms for space multidimensional irregular porous media equation

The object of this paper is a multi-dimensional generalized porous media equation (PDE) with not smooth and possibly discontinuous coefficient $\beta$, which is well-posed as an evolution problem in $L^1(\mathbb{R}^d)$. This work continues the study related to the one-dimensional case by the same authors. One expects that a solution of the mentioned PDE can be represented through the solution (in law) of a non-linear stochastic differential equation (NLSDE). A classical tool for doing this is a uniqueness argument for some Fokker-Planck type equations with measurable coefficients. When $\beta$ is possibly discontinuous, this is often possible in dimension $d = 1$. If $d > 1$, this problem is more complex than for $d = 1$. However, it is possible to exhibit natural candidates for the probabilistic representation and to use them for approximating the solution of (PDE) through a stochastic particle algorithm. We compare it with some numerical deterministic techniques that we have implemented adapting the method of a paper of Cavalli et al. whose convergence was established when $\beta$ is Lipschitz. Special emphasis is also devoted to the case when the initial condition is radially symmetric. On the other hand assuming that $\beta$ is continuous (even though not smooth), one provides existence results for a mollified version of the (NLSDE) and a related partial integro-differential equation, even if the initial condition is a general probability measure.

Necessary and sufficient condition for the existence of a Fréchet mean on the circle

Let ( \hbox{}S1,dS1 ) be the unit circle in ℝ2 endowed with the arclength distance. We give a sufficient and necessary condition for a general probability measure μ to admit a well defined Frechet mean on ( \hbox{}S1,dS1 ). We derive a new sufficient condition of existence P(α, ϕ) with no restriction on the support of the measure. Then, we study the convergence of the empirical Frechet mean to the Frechet mean and we give an algorithm to compute it.

The inclusion-exclusion principle for IF-events

The inclusion-exclusion principle is one of the basic theorems in combinatorics. In this paper the inclusion-exclusion principle for IF-sets on generalized probability measures is studied. The basic theorems are proved.

Stochastic Kolmogorov-type system with infinite delay

Abstract The paper considers a stochastic functional Kolmogorov-type population system with infinite delay under the general probability measures. Main aim is to show that the environment noise will not only suppress a potential population explosion but also make the solution be stochastically ultimately bounded and asymptotic stable. Moreover, two stochastic functional Lotka–Volterra equations as examples are provided to illustrate the main results.

Learning Theory and Approximation

Learning Theory and Approximation

Weighted Poincaré-type inequalities for Cauchy and other convex measures

Brascamp–Lieb-type, weighted Poincaré-type and related analytic inequalities are studied for multidimensional Cauchy distributions and more general κ-concave probability measures (in the hierarchy of convex measures). In analogy with the limiting (infinite-dimensional log-concave) Gaussian model, the weighted inequalities fully describe the measure concentration and large deviation properties of this family of measures. Cheeger-type isoperimetric inequalities are investigated similarly, giving rise to a common weight in the class of concave probability measures under consideration.

Bayesian Nonparametric Estimation of Continuous Monotone Functions with Applications to Dose–Response Analysis

In this article, we consider monotone nonparametric regression in a Bayesian framework. The monotone function is modeled as a mixture of shifted and scaled parametric probability distribution functions, and a general random probability measure is assumed as the prior for the mixing distribution. We investigate the choice of the underlying parametric distribution function and find that the two-sided power distribution function is well suited both from a computational and mathematical point of view. The model is motivated by traditional nonlinear models for dose-response analysis, and provides possibilities to elicitate informative prior distributions on different aspects of the curve. The method is compared with other recent approaches to monotone nonparametric regression in a simulation study and is illustrated on a data set from dose-response analysis.

Distributional properties of means of random probability measures

The present paper provides a review of the results concerning distributional properties of means of random probability measures. Our in- terest in this topic has originated from inferential problems in Bayesian Nonparametrics. Nonetheless, it is worth noting that these random quanti- ties play an important role in seemingly unrelated areas of research. In fact, there is a wealth of contributions both in the statistics and in the probabil- ity literature that we try to summarize in a unified framework. Particular attention is devoted to means of the Dirichlet process given the relevance of the Dirichlet process in Bayesian Nonparametrics. We then present a number of recent contributions concerning means of more general random probability measures and highlight connections with the moment problem, combinatorics, special functions,excursions of stochastic processes and sta- tistical physics.

An evaluation of Clenshaw–Curtis quadrature rule for integration w.r.t. singular measures

This work is devoted to the study of quadrature rules for integration with respect to (w.r.t.) general probability measures with known moments. Automatic calculation of the Clenshaw–Curtis rules is considered and analyzed. It is shown that it is possible to construct these rules in a stable manner for quadrature w.r.t. balanced measures. In order to make a comparison Gauss rules and their stable implementation for integration w.r.t. balanced measures are recalled. Convergence rates are tested in the case of binomial measures.

Extension of domains of states

We deal with categorical aspects of the extensions of generalized probability measures. In particular, we study various domains of fuzzy sets, describe the relationships between σ-fields of crisp sets and generated Łukasiewicz tribes of measurable functions, and mention some probabilistic aspects. D-posets and sequential continuity play an important role.

Dimensions of Points in Self-Similar Fractals

Self-similar fractals arise as the unique attractors of iterated function systems (IFSs) consisting of finitely many contracting similarities satisfying an open set condition. Each point $x$ in such a fractal $F$ arising from an IFS $S$ is naturally regarded as the “outcome” of an infinite coding sequence $T$ (which need not be unique) over the alphabet $\Sigma_k = \{0, \ldots, k-1\}$, where $k$ is the number of contracting similarities in $S$. A classical theorem of Moran (1946) and Falconer (1989) states that the Hausdorff and packing dimensions of a self-similar fractal coincide with its similarity dimension, which depends only on the contraction ratios of the similarities. The theory of computing has recently been used to provide a meaningful notion of the dimensions of individual points in Euclidean space. In this paper, we use (and extend) this theory to analyze the dimensions of individual points in fractals that are computably self-similar, meaning that they are unique attractors of IFSs that are computable and satisfy the open set condition. Our main theorem states that, if $F \subseteq \mathbb{R}^n$ is any computably self-similar fractal and $S$ is any IFS testifying to this fact, then the dimension identities $\operatorname{dim}(x) = \operatorname{sdim}(F) \operatorname{dim}^{\pi_S}(T)$ and $\operatorname{Dim}(x) = \operatorname{sdim}(F) \operatorname{Dim}^{\pi_S}(T)$ hold for all $x \in F$ and all coding sequences $T$ for $x$. In these equations, $\operatorname{sdim}(F)$ denotes the similarity dimension of the fractal $F$; $\operatorname{dim}(x)$ and $\operatorname{Dim}(x)$ denote the dimension and strong dimension, respectively, of the point $x$ in Euclidean space; and $\operatorname{dim}^{\pi_S}(T)$ and $\operatorname{Dim}^{\pi_S}(T)$ denote the dimension and strong dimension, respectively, of the coding sequence $T$ relative to a probability measure $\pi_S$ that the IFS $S$ induces on the alphabet $\Sigma_k$. The above-mentioned theorem of Moran and Falconer follows easily from our main theorem by relativization. Along the way to our main theorem, we develop the elements of the theory of constructive dimensions relative to general probability measures. The proof of our main theorem uses Kolmogorov complexity characterizations of these dimensions.

Some Bregman distances between financial diffusion processes

AbstractWe introduce Bregman distances between general probability measures, and discuss some link with Csiszar divergences. From this, we derive bounds on particular Bregman distances between non–lognormally distributed financial diffusion processes. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

An optimal Wegner estimate and its application to the global continuity of the integrated density of states for random Schrödinger operators

We prove that the integrated density of states (IDS) of random Schrödinger operators with Anderson-type potentials on L2(Rd) for d≥1 is locally Hölder continuous at all energies with the same Hölder exponent 0<α≤1 as the conditional probability measure for the single-site random variable. As a special case, we prove that if the probability distribution is absolutely continuous with respect to Lebesgue measure with a bounded density, then the IDS is Lipschitz continuous at all energies. The single-site potential u∈L0∞(Rd) must be nonnegative and compactly supported. The unperturbed Hamiltonian must be periodic and satisfy a unique continuation principle (UCP). We also prove analogous continuity results for the IDS of random Anderson-type perturbations of the Landau Hamiltonian in two dimensions. All of these results follow from a new Wegner estimate for local random Hamiltonians with rather general probability measures

Universal Algorithms for Learning Theory. Part II: Piecewise Polynomial Functions

This paper is concerned with estimating the regression function fρ in supervised learning by utilizing piecewise polynomial approximations on adaptively generated partitions. The main point of interest is algorithms that with high probability are optimal in terms of the least square error achieved for a given number m of observed data. In a previous paper [1], we have developed for each β > 0 an algorithm for piecewise constant approximation which is proven to provide such optimal order estimates with probability larger than 1- m-β. In this paper we consider the case of higher-degree polynomials. We show that for general probability measures ρ empirical least squares minimization will not provide optimal error estimates with high probability. We go further in identifying certain conditions on the probability measure ρ which will allow optimal estimates with high probability.

An iterative procedure for general probability measures to obtain I-projections onto intersections of convex sets

The iterative proportional fitting procedure (IPFP) was introduced formally by Deming and Stephan in 1940. For bivariate densities, this procedure has been investigated by Kullback and Rüschendorf. It is well known that the IPFP is a sequence of successive I-projections onto sets of probability measures with fixed marginals. However, when finding the I-projection onto the intersection of arbitrary closed, convex sets (e.g., marginal stochastic orders), a sequence of successive I-projections onto these sets may not lead to the actual solution. Addressing this situation, we present a new iterative I-projection algorithm. Under reasonable assumptions and using tools from Fenchel duality, convergence of this algorithm to the true solution is shown. The cases of infinite dimensional IPFP and marginal stochastic orders are worked out in this context.

On the projections of measures invariant under the geodesic flow

We consider general probability measures μ on SM, the unit tangent bundle of a Riemannian surface M, invariant under the geodesic flow, and study their projection to M. We show that if dim μ ≤ dim M(= 2) the projected measure has the same dimension as μ, and if dim μ > dim M the projection is regular with respect to the Lebesgue measure on M.

A general duality approach to I-projections

I-projection problems arise in a myriad of situations and settings. In this paper, it is shown that under reasonable assumptions, a Fenchel type dual optimization problem exists which is equivalent to the stated I-projection problem for very general probability measures. This dual problem is often much more tractable than the original I-projection problem. I-projection problems which are equivalent to least square problems are also identified; primarily through the dual formulation. Several examples are examined to illustrate the duality structure and theorems.

On a Geometrical Representation of Probability Distributions and its use in Statistical Inference.

ABSTRACT: This paper has been divided into six sections. The first three sections are confined to the consideration of the geometry of a sa.mple space with finite number of points : the first is concerned with the representation of a single probability distribution by a point in a finite dimensional Euclidean space and the geometrical interpretation of the laws of probability theory; the second deals with the relationship between two probability distributions defined on the same space and tbe third gives some results of differential geometry applied to such a space and geometrical representation of Fisher's information matrix. The fourth section indicates bow the concepts can be extended to general probability measures defined on a general measurable space. The fifth section gives an inadequate account of the nse of the concepts developed earlier to problems of statistical inference. The last section is not directly concerned with distance, but contains a numder of allied concepts.

Partitioning General Probability Measures

Suppose $\mu_1,\ldots,\mu_n$ are probability measures on the same measurable space $(\Omega, \mathscr{F})$. Then if all atoms of each $\mu_i$ have mass $\alpha$ or less, there is a measurable partition $A_1,\ldots, A_n$ of $\Omega$ so that $\mu_i(A_i) \geq V_n(\alpha)$ for all $i = 1,\ldots, n$, where $V_n(\cdot)$ is an explicitly given piecewise linear nonincreasing continuous function on [0, 1]. Moreover, the bound $V_n(\alpha)$ is attained for all $n$ and all $\alpha$. Applications are given to $L_1$ spaces, to statistical decision theory, and to the classical nonatomic case.

On the Asymptotic Formula for the Probability of a Type I Error of Mixture Type Power One Tests

Let $X_1,X_2,\cdots$ be iid with density $f_y$ with respect to a sigma finite measure $\mu$ where ${f_y}_(y\in\omega$, $\omega\subseteqR$ is an exponential family. Let F be a probability measure on $\omega$ and let $\theta_0\in\omega$. Define $T(B,F)=\min \big\{n \left| \int_omega \frac{f_y(X_1) \1dots f_y(X_n)} {f_{\theta_0}(X_1)\dots f_{\theta_0 (X_1) \1dots f_{\theta_0} (X_n)} dF(y)\geq B \big\}$, $T (B,F) =\infty$ if no such n exists. Previous studies have found that if F has a positive and continuous density with respect to Lebesgue measure on $\omega$, then $BP_\theta\0(t(B,F)<\infty)\rigtharrow_{B\rigtharrow\infty}\int_\omega\ int^\infty_0\exp\{-x\}dH_\theta(x)dF(\theta)$, where $H_\theta$ are certain measures arising in a renewal-theoretic context. Here we show that in a nonlattice context, this convergence holds for general probability measures F. We also show that the convergence is uniform for all probability measures F whose support is contained in an arbitrary interval [a,b] interior to $\omega$, if the distribution of $X_1$ is strongly nonlattice for all $y\in\omega$.