940,664 research outputs found
The interplay of classes of algorithmically random objects
We study algorithmically random closed subsets of , algorithmically
random continuous functions from to , and algorithmically
random Borel probability measures on , especially the interplay
between these three classes of objects. Our main tools are preservation of
randomness and its converse, the no randomness ex nihilo principle, which say
together that given an almost-everywhere defined computable map between an
effectively compact probability space and an effective Polish space, a real is
Martin-L\"of random for the pushforward measure if and only if its preimage is
random with respect to the measure on the domain. These tools allow us to prove
new facts, some of which answer previously open questions, and reprove some
known results more simply.
Our main results are the following. First we answer an open question of
Barmapalias, Brodhead, Cenzer, Remmel, and Weber by showing that
is a random closed set if and only if it is the
set of zeros of a random continuous function on . As a corollary we
obtain the result that the collection of random continuous functions on
is not closed under composition. Next, we construct a computable
measure on the space of measures on such that
is a random closed set if and only if
is the support of a -random measure. We also establish a
correspondence between random closed sets and the random measures studied by
Culver in previous work. Lastly, we study the ranges of random continuous
functions, showing that the Lebesgue measure of the range of a random
continuous function is always contained in
Maximum-entropy probability distributions under Lp-norm constraints
Continuous probability density functions and discrete probability mass functions are tabulated which maximize the differential entropy or absolute entropy, respectively, among all probability distributions with a given L sub p norm (i.e., a given pth absolute moment when p is a finite integer) and unconstrained or constrained value set. Expressions for the maximum entropy are evaluated as functions of the L sub p norm. The most interesting results are obtained and plotted for unconstrained (real valued) continuous random variables and for integer valued discrete random variables. The maximum entropy expressions are obtained in closed form for unconstrained continuous random variables, and in this case there is a simple straight line relationship between the maximum differential entropy and the logarithm of the L sub p norm. Corresponding expressions for arbitrary discrete and constrained continuous random variables are given parametrically; closed form expressions are available only for special cases. However, simpler alternative bounds on the maximum entropy of integer valued discrete random variables are obtained by applying the differential entropy results to continuous random variables which approximate the integer valued random variables in a natural manner. All the results are presented in an integrated framework that includes continuous and discrete random variables, constraints on the permissible value set, and all possible values of p. Understanding such as this is useful in evaluating the performance of data compression schemes
A mean value theorem for systems of integrals
More than a century ago, G. Kowalewski stated that for each n continuous
functions on a compact interval [a,b], there exists an n-point quadrature rule
(with respect to Lebesgue measure on [a,b]), which is exact for given
functions. Here we generalize this result to continuous functions with an
arbitrary positive and finite measure on an arbitrary interval. The proof
relies on a version of Caratheodory's convex hull theorem for a continuous
curve, that we also prove in the paper. As applications, we give a
representation of the covariance for two continuous functions of a random
variable, and a most general version of Gruess' inequality.Comment: 7 page
Complementary Lipschitz continuity results for the distribution of intersections or unions of independent random sets in finite discrete spaces
We prove that intersections and unions of independent random sets in finite
spaces achieve a form of Lipschitz continuity. More precisely, given the
distribution of a random set , the function mapping any random set
distribution to the distribution of its intersection (under independence
assumption) with is Lipschitz continuous with unit Lipschitz constant if
the space of random set distributions is endowed with a metric defined as the
norm distance between inclusion functionals also known as commonalities.
Moreover, the function mapping any random set distribution to the distribution
of its union (under independence assumption) with is Lipschitz continuous
with unit Lipschitz constant if the space of random set distributions is
endowed with a metric defined as the norm distance between hitting
functionals also known as plausibilities.
Using the epistemic random set interpretation of belief functions, we also
discuss the ability of these distances to yield conflict measures. All the
proofs in this paper are derived in the framework of Dempster-Shafer belief
functions. Let alone the discussion on conflict measures, it is straightforward
to transcribe the proofs into the general (non necessarily epistemic) random
set terminology
Random samples generation with Stata from continuous and discrete distributions
Simulations are nowadays a very important way of analyzing new improvements in different areas before the physical implementation, which may require hard resources which could only be affronted in case of a high probability of success. The use of random samples from different distributions are a must in simulations.
In this talk we introduce new Stata functions for generating random samples from continuous and discrete distributions that are not considered in the defined Stata random-number generation functions. In addition, we will also introduce new Stata functions for generating random samples as an alternative of the build-in Stata functions.
The goodness of the generated samples will be checked using the mean squared error (MSE) of the differences between the frequencies of the sample and the theoretical expected ones. We will also provide bar charts which will allow the user to compare graphically the sample with the exact distribution function of the random distribution which is being sampled.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tec
Efficient computation of updated lower expectations for imprecise continuous-time hidden Markov chains
We consider the problem of performing inference with imprecise
continuous-time hidden Markov chains, that is, imprecise continuous-time Markov
chains that are augmented with random output variables whose distribution
depends on the hidden state of the chain. The prefix `imprecise' refers to the
fact that we do not consider a classical continuous-time Markov chain, but
replace it with a robust extension that allows us to represent various types of
model uncertainty, using the theory of imprecise probabilities. The inference
problem amounts to computing lower expectations of functions on the state-space
of the chain, given observations of the output variables. We develop and
investigate this problem with very few assumptions on the output variables; in
particular, they can be chosen to be either discrete or continuous random
variables. Our main result is a polynomial runtime algorithm to compute the
lower expectation of functions on the state-space at any given time-point,
given a collection of observations of the output variables
Heterogeneity and the nonparametric analysis of consumer choice: conditions for invertibility
This paper considers structural nonparametric random utility models for continuous
choice variables. It provides sufficient conditions on random preferences to yield reduced-
form systems of nonparametric stochastic demand functions that allow global invertibility
between demands and random utility components. Invertibility is essential for global
identifcation of structural consumer demand models, for the existence of well-specified
probability models of choice and for the nonparametric analysis of revealed stochastic
preference
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