1,267 research outputs found
Optimal sequential procedures with Bayes decision rules
In this article, a general problem of sequential statistical inference for
general discrete-time stochastic processes is considered. The problem is to
minimize an average sample number given that Bayesian risk due to incorrect
decision does not exceed some given bound. We characterize the form of optimal
sequential stopping rules in this problem. In particular, we have a
characterization of the form of optimal sequential decision procedures when the
Bayesian risk includes both the loss due to incorrect decision and the cost of
observations.Comment: Shortened version for print publication, 17 page
An entropy for groups of intermediate growth
One of the few accepted dynamical foundations of non-additive
"non-extensive") statistical mechanics is that the choice of the appropriate
entropy functional describing a system with many degrees of freedom should
reflect the rate of growth of its configuration or phase space volume. We
present an example of a group, as a metric space, that may be used as the phase
space of a system whose ergodic behavior is statistically described by the
recently proposed -entropy. This entropy is a one-parameter variation
of the Boltzmann/Gibbs/Shannon functional and is quite different, in form, from
the power-law entropies that have been recently studied. We use the first
Grigorchuk group for our purposes. We comment on the connections of the above
construction with the conjectured evolution of the underlying system in phase
space.Comment: 19 pages, No figures, LaTeX2e. Version 2: change of affiliation,
addition of acknowledgement. Accepted for publication to "Advances in
Mathematical Physics
The Legendre Transform in Non-additive Thermodynamics and Complexity
We present an argument which purports to show that the use of the standard
Legendre transform in non-additive Statistical Mechanics is not appropriate.
For concreteness, we use as paradigm, the case of systems which are
conjecturally described by the (non-additive) Tsallis entropy. We point out the
form of the modified Legendre transform that should be used, instead, in the
non-additive thermodynamics induced by the Tsallis entropy. We comment on more
general implications of this proposal for the thermodynamics of "complex
systems".Comment: 23 pages. LaTeX2e. No figure
When Does a Mixture of Products Contain a Product of Mixtures?
We derive relations between theoretical properties of restricted Boltzmann
machines (RBMs), popular machine learning models which form the building blocks
of deep learning models, and several natural notions from discrete mathematics
and convex geometry. We give implications and equivalences relating
RBM-representable probability distributions, perfectly reconstructible inputs,
Hamming modes, zonotopes and zonosets, point configurations in hyperplane
arrangements, linear threshold codes, and multi-covering numbers of hypercubes.
As a motivating application, we prove results on the relative representational
power of mixtures of product distributions and products of mixtures of pairs of
product distributions (RBMs) that formally justify widely held intuitions about
distributed representations. In particular, we show that a mixture of products
requiring an exponentially larger number of parameters is needed to represent
the probability distributions which can be obtained as products of mixtures.Comment: 32 pages, 6 figures, 2 table
Characterizations of bivariate conic, extreme value, and Archimax copulas
Based on a general construction method by means of bivariate ultramodular copulas we construct, for particular settings, special bivariate conic, extreme value, and Archimax copulas. We also show that the sets of copulas obtained in this way are dense in the sets of all conic, extreme value, and Archimax copulas, respectively
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