52 research outputs found
Algorithmic Identification of Probabilities
TThe problem is to identify a probability associated with a set of natural
numbers, given an infinite data sequence of elements from the set. If the given
sequence is drawn i.i.d. and the probability mass function involved (the
target) belongs to a computably enumerable (c.e.) or co-computably enumerable
(co-c.e.) set of computable probability mass functions, then there is an
algorithm to almost surely identify the target in the limit. The technical tool
is the strong law of large numbers. If the set is finite and the elements of
the sequence are dependent while the sequence is typical in the sense of
Martin-L\"of for at least one measure belonging to a c.e. or co-c.e. set of
computable measures, then there is an algorithm to identify in the limit a
computable measure for which the sequence is typical (there may be more than
one such measure). The technical tool is the theory of Kolmogorov complexity.
We give the algorithms and consider the associated predictions.Comment: 19 pages LaTeX.Corrected errors and rewrote the entire paper. arXiv
admin note: text overlap with arXiv:1208.500
Identification of probabilities
Within psychology, neuroscience and artificial intelligence, there has been increasing interest in the proposal that the brain builds probabilistic models of sensory and linguistic input: that is, to infer a probabilistic model from a sample. The practical problems of such inference are substantial: the brain has limited data and restricted computational resources. But there is a more fundamental question: is the problem of inferring a probabilistic model from a sample possible even in principle? We explore this question and find some surprisingly positive and general results. First, for a broad class of probability distributions characterized by computability restrictions, we specify a learning algorithm that will almost surely identify a probability distribution in the limit given a finite i.i.d. sample of sufficient but unknown length. This is similarly shown to hold for sequences generated by a broad class of Markov chains, subject to computability assumptions. The technical tool is the strong law of large numbers. Second, for a large class of dependent sequences, we specify an algorithm which identifies in the limit a computable measure for which the sequence is typical, in the sense of Martin-L\xc3\xb6f (there may be more than one such measure). The technical tool is the theory of Kolmogorov complexity. We analyze the associated predictions in both cases. We also briefly consider special cases, including language learning, and wider theoretical implications for psychology
Quantum theory from four of Hardy's axioms
In a recent paper [e-print quant-ph/0101012], Hardy has given a derivation of
"quantum theory from five reasonable axioms." Here we show that Hardy's first
axiom, which identifies probability with limiting frequency in an ensemble, is
not necessary for his derivation. By reformulating Hardy's assumptions, and
modifying a part of his proof, in terms of Bayesian probabilities, we show that
his work can be easily reconciled with a Bayesian interpretation of quantum
probability.Comment: 5 page
Wave functions for arbitrary operator ordering in the de Sitter minisuperspace approximation
We derive exact series solutions for the Wheeler-DeWitt equation
corresponding to a spatially closed Friedmann-Robertson-Walker universe with
cosmological constant for arbitrary operator ordering of the scale factor of
the universe. The resulting wave functions are those relevant to the
approximation which has been widely used in two-dimensional minisuperspace
models with an inflationary scalar field for the purpose of predicting the
period of inflation which results from competing boundary condition proposals
for the wave function of the universe. The problem that Vilenkin's tunneling
wave function is not normalizable for general operator orderings, is shown to
persist for other values of the spatial curvature, and when additional matter
degrees of freedom such as radiation are included.Comment: 12 pages, revTeX-3.
Process reconstruction from incomplete and/or inconsistent data
We analyze how an action of a qubit channel (map) can be estimated from the
measured data that are incomplete or even inconsistent. That is, we consider
situations when measurement statistics is insufficient to determine consistent
probability distributions. As a consequence either the estimation
(reconstruction) of the channel completely fails or it results in an unphysical
channel (i.e., the corresponding map is not completely positive). We present a
regularization procedure that allows us to derive physically reasonable
estimates (approximations) of quantum channels. We illustrate our procedure on
specific examples and we show that the procedure can be also used for a
derivation of optimal approximations of operations that are forbidden by the
laws of quantum mechanics (e.g., the universal NOT gate).Comment: 9pages, 5 figure
Multicriteria analysis under uncertainty with IANUS - method and empirical results
IANUS is a method for aiding public decision-making that supports efforts towards sustainable development and has a wide range of application. IANUS stands for Integrated Assessment of Decisions uNder Uncertainty for Sustainable Development. This paper introduces the main features of IANUS and illustrates the method using the results of a case study in the Torgau region (eastern Germany). IANUS structures the decision process into four steps: scenario derivation, criteria selection, modeling, evaluation. Its overall aim is to extract the information needed for a sound, responsible decision in a clear, transparent manner. The method is designed for use in conflict situations where environmental and socioeconomic effects need to be considered and so an interdisciplinary approach is required. Special emphasis is placed on a broad perception and consideration of uncertainty. Three types of uncertainty are explicitly taken into account by IANUS: development uncertainty (uncertainty about the social, economic and other developments that affect the consequences of decision), model uncertainty (uncertainty associated with the prediction of the effects of decisions), and weight uncertainty (uncertainty about the appropriate weighting of the criteria). The backbone of IANUS is a multicriteria method with the ability to process uncertain information. In the case study the multicriteria method PROMETHEE is used. Since PROMETHEE in its basic versions is not able to process uncertain information an extension of this method is developed here and described in detail. --
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