Noise modelling and evaluating learning from examples

A rigorous lower bound is obtained for the average resolution of any estimate of a shift parameter, such as an optical phase shift or a spatial translation. The bound has the asymptotic form kI/h2|G|i where G is the generator of the shift (with an arbitrary discrete or continuous spectrum), and hence establishes a universally applicable bound of the same form as the usual Heisenberg limit. The scaling constant kI depends on prior information about the shift parameter. For example, in phase sensing regimes, where the phase shift is confined to some small interval of length L, the relative resolution ˆ

https://iopscience.iop.org/article/10.1088/1367-2630/14/3/033040/pdf

Heisenberg-style bounds for arbitrary estimates of shift parameters including prior information

The Kumaraswamy distribution is very similar to the Beta distribution, but has the important advantage of an invertible closed-form cumulative distribution function. The parameterization of the distribution in terms of shape parameters and the lack of simple expressions for its mean and variance hinder, however, its utilization with modeling purposes. The paper presents two median-dispersion reparameterizations of the Kumaraswamy distribution aimed at facilitating its use in regression models in which both the location and the dispersion parameters are functions of their own distinct sets of covariates, and in latent-variable and other models estimated through simulation-based methods. In both re-parameterizations the dispersion parameter establishes a quantile-spread order among Kumaraswamy distributions with the same median and support. The paper also describes the behavior of the re-parameterized distributions, determines some of their limiting distributions, and discusses the potential comparative advantages of using them in the context of regression modeling and simulation based estimation

The Kumaraswamy Distribution: Median-Dispersion Re-Parameterizations for Regression Modeling and Simulation-Based Estimation

The Kumaraswamy distribution is very similar to the Beta distribution, but has the important advantage of an invertible closed form cumulative distribution function. The parameterization of the distribution in terms of shape parameters and the lack of simple expressions for its mean and variance hinder, however, its utilization with modeling purposes. The paper presents two median-dispersion re-parameterizations of the Kumaraswamy distribution aimed at facilitating its use in regression models in which both the location and the dispersion parameters are functions of their own distinct sets of covariates, and in latent-variable and other models estimated through simulation-based methods. In both re-parameterizations the dispersion parameter establishes a quantile-spread order among Kumaraswamy distributions with the same median and support. The study also describes the behavior of the re-parameterized distributions, determines some of their limiting distributions, and discusses the potential comparative advantages of using them in the context of regression modeling and simulation-based estimation.

The Kumaraswamy distribution: median-dispersion re-parameterizations for regression modeling and simulation-based estimation

LetX andY be two random variables with finite expectationsE
 X andE
 Y, respectively. ThenX is said to be smaller thanY in the dilation order ifE[ϕ(X-E
 X)]≤E[ϕ(Y-E
 Y)] for any convex functionϕ for which the expectations exist. In this paper we obtain a new characterization of the dilation order. This characterization enables us to give new interpretations to the dilation order, and using them we identify conditions which imply the dilation order. A sample of applications of the new characterization is given.

A Characterization of the Dilation Order and its Applications

On compare l'approche de Lewis-Thompson (1981) a l'ordre de dispersion obtenu par majorisation

Concepts of dispersion in distributions: a comparative note

Majorisation is used to compare discrete distributions in terms of randomness or, equivalently, informativeness. Some properties of measures of randomness are discussed. The effect on degree of randomness of a change in parameter value is investigated for some distributions including the binomial and Poisson.

http://www.jstor.org/stable/pdfplus/3213602.pdf

Majorisation, randomness and some discrete distributions

Majorisation is used to compare continuous distributions in terms of randomness. General results on randomness in the continuous case are given and these are used to investigate the connection between randomness and parameter values in some well-known families of distributions including the normal and gamma.

Continuous majorisation and randomness

The properties of a measure of randomness on the class of discrete probability distributions are discussed. A result concerning finiteness is obtained for a family of measures which includes the entropy function.

Raymond J. Hickey

Papers

Concepts of dispersion in distributions: a comparative note

Majorisation, randomness and some discrete distributions

Continuous majorisation and randomness

A note on the measurement of randomness