3,803 research outputs found
Postquantum Br\`{e}gman relative entropies and nonlinear resource theories
We introduce the family of postquantum Br\`{e}gman relative entropies, based
on nonlinear embeddings into reflexive Banach spaces (with examples given by
reflexive noncommutative Orlicz spaces over semi-finite W*-algebras,
nonassociative L spaces over semi-finite JBW-algebras, and noncommutative
L spaces over arbitrary W*-algebras). This allows us to define a class of
geometric categories for nonlinear postquantum inference theory (providing an
extension of Chencov's approach to foundations of statistical inference), with
constrained maximisations of Br\`{e}gman relative entropies as morphisms and
nonlinear images of closed convex sets as objects. Further generalisation to a
framework for nonlinear convex operational theories is developed using a larger
class of morphisms, determined by Br\`{e}gman nonexpansive operations (which
provide a well-behaved family of Mielnik's nonlinear transmitters). As an
application, we derive a range of nonlinear postquantum resource theories
determined in terms of this class of operations.Comment: v2: several corrections and improvements, including an extension to
the postquantum (generally) and JBW-algebraic (specifically) cases, a section
on nonlinear resource theories, and more informative paper's titl
Towards Machine Wald
The past century has seen a steady increase in the need of estimating and
predicting complex systems and making (possibly critical) decisions with
limited information. Although computers have made possible the numerical
evaluation of sophisticated statistical models, these models are still designed
\emph{by humans} because there is currently no known recipe or algorithm for
dividing the design of a statistical model into a sequence of arithmetic
operations. Indeed enabling computers to \emph{think} as \emph{humans} have the
ability to do when faced with uncertainty is challenging in several major ways:
(1) Finding optimal statistical models remains to be formulated as a well posed
problem when information on the system of interest is incomplete and comes in
the form of a complex combination of sample data, partial knowledge of
constitutive relations and a limited description of the distribution of input
random variables. (2) The space of admissible scenarios along with the space of
relevant information, assumptions, and/or beliefs, tend to be infinite
dimensional, whereas calculus on a computer is necessarily discrete and finite.
With this purpose, this paper explores the foundations of a rigorous framework
for the scientific computation of optimal statistical estimators/models and
reviews their connections with Decision Theory, Machine Learning, Bayesian
Inference, Stochastic Optimization, Robust Optimization, Optimal Uncertainty
Quantification and Information Based Complexity.Comment: 37 page
Estimating input allocation for farm supply models
When building an economic model for supply analysis the aim is to model a decision making process of one or more agents which fits the observed practice as good as possible. Hereby the modeller is often confronted with incomplete information about the production process; particular crop specific input data are rarely available. The problem of defining activity related technology inputs coefficients is not new. A good deal of literature comes from the mathematical programming perspective, where input coefficients were estimated using a standard linear regression function to fully represent the mathematical program. However this approach is a pure technical device and may result in an inconsistent model. The author of the paper wants to investigate whether it is possible, employing proper estimation techniques, to simultaneously estimate all unknown coefficients of a mathematical farm supply model. This includes the estimation of parameters of the non linear cost function, used to calibrate and catch the simulation behaviour and the crop specific input coefficients. It is shown that a simultaneous estimation of all parameters improves the goodness of fit of the estimated parameters and that such an approach is technically feasible.farm supply model, input allocation, entropy, HDP, Research Methods/ Statistical Methods,
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