Revealing a temporal symmetry/asymmetry dichotomy in a Markovian setting, and a parameterization based on fractional low-order joint moments

2020 ◽  
Author(s):  
Alin Andrei Carsteanu ◽  
Andreas Langousis
Keyword(s):  
Stochastic Process ◽  
Probability Distributions ◽  
Arrow Of Time ◽  
Marginal Probability ◽  
Joint Moments ◽  
Statistical Tool ◽  
Joint Distributions ◽  
Striking Result ◽  
Temporal Symmetry ◽  
Low Order

<p>We show that "an arrow of time", which is reflected by the joint distributions of successive variables in a stochastic process, may exist (or not) solely on grounds of marginal probability distributions, without affecting stationarity or involving the structural dependencies within the process. The temporal symmetry/asymmetry dichotomy thus revealed, is exemplified for the simplest case of stably-distributed Markovian recursions, where the lack of Gaussianity, even when the increments of the process are independent and identically distributed (i.i.d.) with symmetric marginal, is generating a break of temporal symmetry. We devise a statistical tool to evidence this striking result, based on fractional low-order joint moments, whose existence is guaranteed even for the case of "fat-tailed" strictly-stable distributions, and is thereby suited for parameterizing structural dependencies within such a process.</p>

scholarly journals Linear Response Algorithms for Approximate Inference in Graphical Models

2004 ◽  
Vol 16 (1) ◽  
pp. 197-221 ◽  
Author(s):  
Max Welling ◽  
Yee Whye Teh
Keyword(s):  
Graphical Models ◽  
Random Fields ◽  
Linear Response ◽  
Belief Propagation ◽  
Probability Distributions ◽  
Gaussian Random Fields ◽  
Marginal Probability ◽  
Joint Distributions ◽  
Propagation Algorithm ◽  
New Algorithms

Belief propagation (BP) on cyclic graphs is an efficient algorithm for computing approximate marginal probability distributions over single nodes and neighboring nodes in the graph. However, it does not prescribe a way to compute joint distributions over pairs of distant nodes in the graph. In this article, we propose two new algorithms for approximating these pairwise probabilities, based on the linear response theorem. The first is a propagation algorithm that is shown to converge if BP converges to a stable fixed point. The second algorithm is based on matrix inversion. Applying these ideas to gaussian random fields, we derive a propagation algorithm for computing the inverse of a matrix.

scholarly journals Is Einsteinian no-signalling violated in Bell tests?

Open Physics ◽  
2017 ◽  
Vol 15 (1) ◽  
pp. 739-753 ◽  
Author(s):  
Marian Kupczynski
Keyword(s):  
Quantum Electrodynamics ◽  
Probability Distributions ◽  
Selection Procedure ◽  
Quantum Correlations ◽  
Spin Projection ◽  
Marginal Probability ◽  
Strong Correlations ◽  
Relativistic Invariance ◽  
Long Range Correlations ◽  
Apparent Violation

AbstractRelativistic invariance is a physical law verified in several domains of physics. The impossibility of faster than light influences is not questioned by quantum theory. In quantum electrodynamics, in quantum field theory and in the standard model relativistic invariance is incorporated by construction. Quantum mechanics predicts strong long range correlations between outcomes of spin projection measurements performed in distant laboratories. In spite of these strong correlations marginal probability distributions should not depend on what was measured in the other laboratory what is called shortly: non-signalling. In several experiments, performed to test various Bell-type inequalities, some unexplained dependence of empirical marginal probability distributions on distant settings was observed. In this paper we demonstrate how a particular identification and selection procedure of paired distant outcomes is the most probable cause for this apparent violation of no-signalling principle. Thus this unexpected setting dependence does not prove the existence of superluminal influences and Einsteinian no-signalling principle has to be tested differently in dedicated experiments. We propose a detailed protocol telling how such experiments should be designed in order to be conclusive. We also explain how magical quantum correlations may be explained in a locally causal way.

scholarly journals The Sum and Difference of Two Lognormal Random Variables

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
C. F. Lo
Keyword(s):  
Stochastic Process ◽  
Probability Distributions ◽  
Operator Splitting ◽  
Splitting Method ◽  
Unified Approach ◽  
New Approach ◽  
Numerical Examples ◽  
Operator Splitting Method ◽  
Approximate Probability ◽  
Analytical Series

We have presented a new unified approach to model the dynamics of both the sum and difference of two correlated lognormal stochastic variables. By the Lie-Trotter operator splitting method, both the sum and difference are shown to follow a shifted lognormal stochastic process, and approximate probability distributions are determined in closed form. Illustrative numerical examples are presented to demonstrate the validity and accuracy of these approximate distributions. In terms of the approximate probability distributions, we have also obtained an analytical series expansion of the exact solutions, which can allow us to improve the approximation in a systematic manner. Moreover, we believe that this new approach can be extended to study both (1) the algebraic sum ofNlognormals, and (2) the sum and difference of other correlated stochastic processes, for example, two correlated CEV processes, two correlated CIR processes, and two correlated lognormal processes with mean-reversion.

The Discrete Optimal Assignment Problem

2016 ◽  
Author(s):  
Alfred Galichon
Keyword(s):  
Linear Programming ◽  
Assignment Problem ◽  
Linear Programming Problem ◽  
Matrix Algebra ◽  
Probability Distributions ◽  
Marginal Probability ◽  
Finite Support ◽  
Complementary Slackness ◽  
Finite Dimensional ◽  
Finite Dimensional Case

This chapter considers the finite-dimensional case, which is the case when the marginal probability distributions are discrete with finite support. In this case, the Monge–Kantorovich problem becomes a finite-dimensional linear programming problem; the primal and the dual solutions are related by complementary slackness, which is interpreted in terms of stability. The solutions can be conveniently computed by linear programming solvers, and the chapter shows how this is done using some matrix algebra and Gurobi.

Parameterization of multifractal cascade models based on their breakdown coefficients

2020 ◽  
Author(s):  
César Aguilar Flores ◽  
Alin Andrei Carsteanu
Keyword(s):  
Time Series ◽  
Probability Distribution ◽  
Rainfall Intensity ◽  
Probability Distributions ◽  
Asymptotic Properties ◽  
Distribution Functions ◽  
Marginal Probability ◽  
Probability Distribution Functions ◽  
Marginal Probability Distribution ◽  
Cascade Models

<p>Breakdown coefficients of multifractal cascades have been shown, in various contexts, to be ergodic in their (marginal) probability distribution functions, however the necessary connection between the cascading process (or a tracer thereof, such as rainfall) and the breakdown coefficients of the measure generated by the cascade, was missing. This work presents a method of parameterization of certain types of multiplicative cascades, using the breakdown coefficients of the measures they generate. The method is based on asymptotic properties of the probability distributions of the breakdown coefficients in “dressed” cascades, as compared with the respective distributions of the cascading weights. An application to rainfall intensity time series is presented.</p>

Generic Identication of Binary-Valued Hidden Markov Processes

2014 ◽  
Vol 5 (1) ◽  
Author(s):  
Alexander Schönhuth
Keyword(s):  
Stochastic Process ◽  
Markov Process ◽  
General Solution ◽  
Parameter Space ◽  
Markov Processes ◽  
Hidden Markov ◽  
Probability Distributions ◽  
Algebraic Statistics ◽  
Hidden Markov Processes ◽  
Lower Dimensional

The generic identication problem is to decide whether a stochastic process (Xt) is ahidden Markov process and if yes to infer its parameters for all but a subset of parametrizationsthat form a lower-dimensional subvariety in parameter space. Partial answers so far availabledepend on extra assumptions on the processes, which are usually centered around stationarity.Here we present a general solution for binary-valued hidden Markov processes. Our approach isrooted in algebraic statistics hence it is geometric in nature. We nd that the algebraic varietiesassociated with the probability distributions of binary-valued hidden Markov processes are zerosets of determinantal equations which draws a connection to well-studied objects from algebra. Asa consequence, our solution allows for algorithmic implementation based on elementary (linear)algebraic routines.

A Method Based on Taking the Average of Probabilities to Compute the Flow Duration Curve

2000 ◽  
Vol 31 (3) ◽  
pp. 187-206 ◽  
Author(s):  
Hikmet Kerem Cigizoglu
Keyword(s):  
Probability Distributions ◽  
Maximum Flow ◽  
Exceedance Probability ◽  
Flow Duration Curve ◽  
Marginal Probability ◽  
Lognormal Distributions ◽  
Type Iii ◽  
Flow Duration ◽  
Pearson Type ◽  
Flow Duration Curves

In this study a method based on taking the average of the probabilities is presented to obtain flow duration curve. In this method the exceedance probability for each flow value is computed repeatedly for all time periods within a year. The final representing exceedance is just simply the average of all these probabilities. The applicability of the method to daily mean flows is tested assuming various marginal probability distributions like normal, Pearson type III, log-Pearson type III, 2-parameter lognormal and 3-parameter lognormal distributions. It is seen that the observed flow duration curves were quite well approximated by the 2-parameter lognormal average of probabilities curves. In that case the method requires the computation of the daily mean and standard deviation values of the observed flow data. The method curve enables extrapolation of the available data providing the exceedance probabilities for the flows higher than the observed maximum flow. The method is applied to the missing data and ungauged site problems and the results are quite satisfactory.

Bounds on Previsions and Conditional Probabilities on Joint Finite Spaces Under the Assumption of Independence in Imprecise Probability

2009 ◽  
Author(s):  
Fulvio Tonon ◽  
Xiaomin You ◽  
Alberto Bernardini
Keyword(s):  
Optimization Problems ◽  
Probability Distributions ◽  
Imprecise Probabilities ◽  
Imprecise Probability ◽  
Marginal Distributions ◽  
Joint Distributions ◽  
Probability Bounds ◽  
Finite Spaces ◽  
Conditional Probability Bounds

The primary difference between precise and imprecise probability theories lies in the allowance for imprecision, or a gap between upper and lower expectations (also called previsions) of bounded real functions. This gap generates a set of probability distributions or measures. As a result, in imprecise probabilities, the notion of independence on joint spaces is not unique; for example, notions of unknown interaction, epistemic irrelevance/independence and strong independence have been proposed in the literature. After introducing the three concepts of independence, various algorithms are proposed to calculate, through the different definitions of independence, both prevision and conditional probability bounds generated by marginal distributions over finite joint spaces. All algorithms are designed to accommodate two different types of constraints that define the sets of marginal distributions: previsions bounds or extreme distributions. Algorithms are applied to simple examples that show the role of the different quantities introduced and the equivalence of the two types of constraints. It is shown that, in epistemic irrelevance/independence, re-writing algorithms in terms of joint distributions turn quadratic optimization problems into linear ones.

scholarly journals Density estimation with Gaussian processes for gravitational wave posteriors

2021 ◽  
Vol 508 (2) ◽  
pp. 2090-2097
Author(s):  
V D’Emilio ◽  
R Green ◽  
V Raymond
Keyword(s):  
Gaussian Processes ◽  
Posterior Probability ◽  
Probability Function ◽  
Probability Distributions ◽  
Source Parameters ◽  
Kernel Density ◽  
Marginal Probability ◽  
Stochastic Sampling ◽  
Interpolation Accuracy ◽  
Non Gaussian

ABSTRACT The properties of black hole and neutron-star binaries are extracted from gravitational waves (GW) signals using Bayesian inference. This involves evaluating a multidimensional posterior probability function with stochastic sampling. The marginal probability distributions of the samples are sometimes interpolated with methods such as kernel density estimators. Since most post-processing analysis within the field is based on these parameter estimation products, interpolation accuracy of the marginals is essential. In this work, we propose a new method combining histograms and Gaussian processes (GPs) as an alternative technique to fit arbitrary combinations of samples from the source parameters. This method comes with several advantages such as flexible interpolation of non-Gaussian correlations, Bayesian estimate of uncertainty, and efficient resampling with Hamiltonian Monte Carlo.

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