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Scalar products of elementary distributions
The field of real numbers being extended as a larger commutative field, we
investigate the possibility of defining a scalar product for the distributions
of finite discrete support. Then we focus on the most simple possible extension
(which is an ordered field), we provide explicit formulas for this scalar
product, and we exhibit a structure of positive definite inner-product space.
In a one-dimensional application to the Schroedinger equation, the
distributions supported by the origin are embedded into a bra-ket vector space,
where the "singular" potential describing point interaction is defined in a
natural way. A contact with the hyperreal numbers that arise in nonstandard
analysis is possible but not essential, our extensions of and
being obtained by a quite elementary method.Comment: 27 page
Group invariant inferred distributions via noncommutative probability
One may consider three types of statistical inference: Bayesian, frequentist,
and group invariance-based. The focus here is on the last method. We consider
the Poisson and binomial distributions in detail to illustrate a group
invariance method for constructing inferred distributions on parameter spaces
given observed results. These inferred distributions are obtained without using
Bayes' method and in particular without using a joint distribution of random
variable and parameter. In the Poisson and binomial cases, the final formulas
for inferred distributions coincide with the formulas for Bayes posteriors with
uniform priors.Comment: Published at http://dx.doi.org/10.1214/074921706000000563 in the IMS
Lecture Notes--Monograph Series
(http://www.imstat.org/publications/lecnotes.htm) by the Institute of
Mathematical Statistics (http://www.imstat.org
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