Journal articles: 'Measure Theoretic Probability Theory'

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KUYPER, RUTGER, and SEBASTIAAN A. TERWIJN. "MODEL THEORY OF MEASURE SPACES AND PROBABILITY LOGIC." Review of Symbolic Logic 6, no. 3 (April 8, 2013): 367–93. http://dx.doi.org/10.1017/s1755020313000063.

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AbstractWe study the model-theoretic aspects of a probability logic suited for talking about measure spaces. This nonclassical logic has a model theory rather different from that of classical predicate logic. In general, not every satisfiable set of sentences has a countable model, but we show that one can always build a model on the unit interval. Also, the probability logic under consideration is not compact. However, using ultraproducts we can prove a compactness theorem for a certain class of weak models.

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Khrennikov, Andrei, Shinichi Yamada, and Arnoud van Rooij. "The measure-theoretical approach to p-adic probability theory." Annales mathématiques Blaise Pascal 6, no. 1 (1999): 21–32. http://dx.doi.org/10.5802/ambp.112.

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Siegmund-Schultze, Reinhard. "Mathematicians Forced to Philosophize: An Introduction to Khinchin's Paper on von Mises' Theory of Probability." Science in Context 17, no. 3 (September 2004): 373–90. http://dx.doi.org/10.1017/s0269889704000171.

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What follows shall provide an introduction to a predominantly philosophical and polemical, but historically revealing, paper on the foundations of the theory of probability. The leading Russian probabilist Aleksandr Yakovlevich Khinchin (1894–1959) (see fig. 1) wrote the paper in the late 1930s, commenting on a slightly older, but still competing approach to probability theory by Richard von Mises. Together with the even more influential Andrey Nikolayevich Kolmogorov (1903–1987), who was nine years his junior, Khinchin had revolutionized probability theory around 1930 by introducing the modern measure-theoretic approach, which is still standard today and which allowed for a sufficiently general treatment of important new notions such as “stochastic processes.” This development had its first culmination in Kolmogorov's booklet, Grundbegriffe der Wahrscheinlichkeitsrechnung, written in German in 1933, which has exerted an enormous influence world wide.

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Pei, Han Ru, Zhi Jian Wang, and Yu Wang. "Bayesian Inference of Information Theoretic Metrics of Anonymity." Advanced Materials Research 989-994 (July 2014): 4680–83. http://dx.doi.org/10.4028/www.scientific.net/amr.989-994.4680.

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Information theoretic metrics is popular theory to measure anonymity. However the difficulty in getting the probability distribution of subjects hampers its practical usage. In this paper we propose a Bayesian inference method to tackle this problem. Our method makes it possible to compare the anonymity of different anonymous systems. We use this method to analyze Threshold Mix and point out different system parameters which do and do not have influence on anonymity.

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Bingham, N. H. "Doob: A Half-Century on." Journal of Applied Probability 42, no. 01 (March 2005): 257–66. http://dx.doi.org/10.1017/s0021900200000206.

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Probability theory, and its dynamic aspect stochastic process theory, is both a venerable subject, in that its roots go back to the mid-seventeenth century, and a young one, in that its modern formulation happened comparatively recently - well within living memory. The year 2003 marked the seventieth anniversary of Kolmogorov's Grundbegriffe der Wahrscheinlichkeitsrechnung, usually regarded as having inaugurated modern (measure-theoretic) probability theory. It also marked the fiftieth anniversary of Doob's Stochastic Processes. The profound and continuing influence of this classic work prompts the present piece.

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Bingham, N. H. "Doob: A Half-Century on." Journal of Applied Probability 42, no. 1 (March 2005): 257–66. http://dx.doi.org/10.1239/jap/1110381385.

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Probability theory, and its dynamic aspect stochastic process theory, is both a venerable subject, in that its roots go back to the mid-seventeenth century, and a young one, in that its modern formulation happened comparatively recently - well within living memory. The year 2003 marked the seventieth anniversary of Kolmogorov'sGrundbegriffe der Wahrscheinlichkeitsrechnung, usually regarded as having inaugurated modern (measure-theoretic) probability theory. It also marked the fiftieth anniversary of Doob'sStochastic Processes. The profound and continuing influence of this classic work prompts the present piece.

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Cho, Kenta, and Bart Jacobs. "Disintegration and Bayesian inversion via string diagrams." Mathematical Structures in Computer Science 29, no. 7 (March 13, 2019): 938–71. http://dx.doi.org/10.1017/s0960129518000488.

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AbstractThe notions of disintegration and Bayesian inversion are fundamental in conditional probability theory. They produce channels, as conditional probabilities, from a joint state, or from an already given channel (in opposite direction). These notions exist in the literature, in concrete situations, but are presented here in abstract graphical formulations. The resulting abstract descriptions are used for proving basic results in conditional probability theory. The existence of disintegration and Bayesian inversion is discussed for discrete probability, and also for measure-theoretic probability – via standard Borel spaces and via likelihoods. Finally, the usefulness of disintegration and Bayesian inversion is illustrated in several examples.

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Fritz, Tobias, and Eigil Fjeldgren Rischel. "Infinite products and zero-one laws in categorical probability." Compositionality 2 (August 11, 2020): 3. http://dx.doi.org/10.32408/compositionality-2-3.

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Markov categories are a recent category-theoretic approach to the foundations of probability and statistics. Here we develop this approach further by treating infinite products and the Kolmogorov extension theorem. This is relevant for all aspects of probability theory in which infinitely many random variables appear at a time. These infinite tensor products ⨂i∈JXi come in two versions: a weaker but more general one for families of objects (Xi)i∈J in semicartesian symmetric monoidal categories, and a stronger but more specific one for families of objects in Markov categories.As a first application, we state and prove versions of the zero--one laws of Kolmogorov and Hewitt--Savage for Markov categories. This gives general versions of these results which can be instantiated not only in measure-theoretic probability, where they specialize to the standard ones in the setting of standard Borel spaces, but also in other contexts.

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Nowak, Piotr, and Olgierd Hryniewicz. "On MV-Algebraic Versions of the Strong Law of Large Numbers." Entropy 21, no. 7 (July 19, 2019): 710. http://dx.doi.org/10.3390/e21070710.

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Many-valued (MV; the many-valued logics considered by Łukasiewicz)-algebras are algebraic systems that generalize Boolean algebras. The MV-algebraic probability theory involves the notions of the state and observable, which abstract the probability measure and the random variable, both considered in the Kolmogorov probability theory. Within the MV-algebraic probability theory, many important theorems (such as various versions of the central limit theorem or the individual ergodic theorem) have been recently studied and proven. In particular, the counterpart of the Kolmogorov strong law of large numbers (SLLN) for sequences of independent observables has been considered. In this paper, we prove generalized MV-algebraic versions of the SLLN, i.e., counterparts of the Marcinkiewicz–Zygmund and Brunk–Prokhorov SLLN for independent observables, as well as the Korchevsky SLLN, where the independence of observables is not assumed. To this end, we apply the classical probability theory and some measure-theoretic methods. We also analyze examples of applications of the proven theorems. Our results open new directions of development of the MV-algebraic probability theory. They can also be applied to the problem of entropy estimation.

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Tonn, B. E., and A. Schaffhauser. "Towards a More General Theoretical and Mathematical Model of Probability for Policy Analysis." Environment and Planning A: Economy and Space 24, no. 9 (September 1992): 1337–53. http://dx.doi.org/10.1068/a241337.

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Uncertainty pervades policy analysis in ways that transcend classical concepts of probability. To benefit policy analysis, the concept of probability must be considerably broadened. It is argued that probability can be conceptualized with respect to the characteristics of policy problems that produce inherent uncertainty. Problems that encompass uncertainty can be characterized according to their: (1) fundamental requirements, for example forecasting, knowledge creation, fact establishment; (2) system properties such as disorderly versus orderly systems; (3) problem-solution strategy, for example subjective judgement, model-based analysis, data analysis; (4) problem-solution data requirements—from numerous and hard-to-measure variables to few and easy-to-measure variables, and (5) problem-solution frame—ranging from unbounded solution spaces to small and discrete solution spaces. The theory of lower probability is presented as a generalization of classical additive probability that can handle this generalized conceptualization of probability. Information-theoretic methods for integrating the two generalizations of probability are considered.

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Shen, Jinghua, and Yun Zhao. "Entropy of a Flow on Non-compact Sets." Open Systems & Information Dynamics 19, no. 02 (June 2012): 1250015. http://dx.doi.org/10.1142/s1230161212500151.

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Based on the theory of Carathéodory structure, this paper introduces the topological entropy of a flow on non-compact sets. Moreover, we introduce the definition of measure-theoretic entropy of a flow. It is shown that this entropy is equivalent to the one defined by Sun in [10]. The variational principle between topological entropy and measure-theoretic entropy of a flow is established. We also get the Brin-Katok's entropy formula for a flow.

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COLETTI, GIULIANELLA, and ROMANO SCOZZAFAVA. "CHARACTERIZATION OF COHERENT CONDITIONAL PROBABILITIES AS A TOOL FOR THEIR ASSESSMENT AND EXTENSION." International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 04, no. 02 (April 1996): 103–27. http://dx.doi.org/10.1142/s021848859600007x.

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A major purpose of this paper is to show the broad import and applicability of the theory of probability as proposed by de Finetti, which differs radically from the usual one (based on a measure-theoretic framework). In particular, with reference to a coherent conditional probability, we prove a characterization theorem, which provides also a useful algorithm for checking coherence of a given assessment. Moreover it allows to deepen and generalise in useful directions de Finetti’s extension theorem (dubbed as “the fundamental theorem of probability”), emphasising its operational aspects in many significant applications.

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Parsian, Ali. "Finding a measure, a ring theoretical approach." International Journal of Applied Mathematical Research 6, no. 1 (December 7, 2016): 1. http://dx.doi.org/10.14419/ijamr.v6i1.6864.

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Let $S$ be a nonempty set and $F$ consists of all $Z_{2}$ characteristic functions defined on $S$. We are supposed to introduce a ring isomorphic to $(P(S),\triangle,\cap)$, whose set is $F$. Then, assuming a finitely additive function $m$ defined on $P(S)$, we change $P(S)$ to a pseudometric space $(P(S),d_{m})$ in which its pseudometric is defined by $m$. Among other things, we investigate the concepts of convergence and continuity in the induced pseudometric space. Moreover, a theorem on the measure of some kinds of elements in $(P(S),m)$ will be established. At the end, as an application in probability theory, the probability of some events in the space of permutations with uniform probability will be determined. Some illustrative examples are included to show the usefulness and applicability of results.

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Tomaževič, Dejan, Boštjan Likar, and Franjo Pernuš. "MULTI-FEATURE MUTUAL INFORMATION IMAGE REGISTRATION." Image Analysis & Stereology 31, no. 1 (March 15, 2012): 43. http://dx.doi.org/10.5566/ias.v31.p43-53.

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Nowadays, information-theoretic similarity measures, especially the mutual information and its derivatives, are one of the most frequently used measures of global intensity feature correspondence in image registration. Because the traditional mutual information similarity measure ignores the dependency of intensity values of neighboring image elements, registration based on mutual information is not robust in cases of low global intensity correspondence. Robustness can be improved by adding spatial information in the form of local intensity changes to the global intensity correspondence. This paper presents a novel method, by which intensities, together with spatial information, i.e., relations between neighboring image elements in the form of intensity gradients, are included in information-theoretic similarity measures. In contrast to a number of heuristic methods that include additional features into the generic mutual information measure, the proposed method strictly follows information theory under certain assumptions on feature probability distribution. The novel approach solves the problem of efficient estimation of multifeature mutual information from sparse high-dimensional feature space. The proposed measure was tested on magnetic resonance (MR) and computed tomography (CT) images. In addition, the measure was tested on positron emission tomography (PET) and MR images from the widely used Retrospective Image Registration Evaluation project image database. The results indicate that multi-feature mutual information, which combines image intensities and intensity gradients, is more robust than the standard single-feature intensity based mutual information, especially in cases of low global intensity correspondences, such as in PET/MR images or significant intensity inhomogeneity.

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Gazor, Majid. "A Measure Theoretical Version of the Aleksandrov Theorem." gmj 15, no. 1 (March 2008): 53–61. http://dx.doi.org/10.1515/gmj.2008.53.

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Abstract In this paper a theorem analogous to the Aleksandrov theorem is presented in terms of measure theory. Furthermore, we introduce the condensation rank of Hausdorff spaces and prove that any ordinal number is associated with the condensation rank of an appropriate locally compact totally imperfect space. This space is equipped with a probability Borel measure which is outer regular, vanishes at singletons, and is also inner regular in the sense of closed sets.

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Kamihigashi, Takashi, and John Stachurski. "A unified stability theory for classical and monotone Markov chains." Journal of Applied Probability 56, no. 01 (March 2019): 1–22. http://dx.doi.org/10.1017/jpr.2019.2.

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AbstractIn this paper we integrate two strands of the literature on stability of general state Markov chains: conventional, total-variation-based results and more recent order-theoretic results. First we introduce a complete metric over Borel probability measures based on ‘partial’ stochastic dominance. We then show that many conventional results framed in the setting of total variation distance have natural generalizations to the partially ordered setting when this metric is adopted.

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Smith, Martin R. "Information theoretic generalized Robinson–Foulds metrics for comparing phylogenetic trees." Bioinformatics 36, no. 20 (July 3, 2020): 5007–13. http://dx.doi.org/10.1093/bioinformatics/btaa614.

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Abstract Motivation The Robinson–Foulds (RF) metric is widely used by biologists, linguists and chemists to quantify similarity between pairs of phylogenetic trees. The measure tallies the number of bipartition splits that occur in both trees—but this conservative approach ignores potential similarities between almost-identical splits, with undesirable consequences. ‘Generalized’ RF metrics address this shortcoming by pairing splits in one tree with similar splits in the other. Each pair is assigned a similarity score, the sum of which enumerates the similarity between two trees. The challenge lies in quantifying split similarity: existing definitions lack a principled statistical underpinning, resulting in misleading tree distances that are difficult to interpret. Here, I propose probabilistic measures of split similarity, which allow tree similarity to be measured in natural units (bits). Results My new information-theoretic metrics outperform alternative measures of tree similarity when evaluated against a broad suite of criteria, even though they do not account for the non-independence of splits within a single tree. Mutual clustering information exhibits none of the undesirable properties that characterize other tree comparison metrics, and should be preferred to the RF metric. Availability and implementation The methods discussed in this article are implemented in the R package ‘TreeDist’, archived at https://dx.doi.org/10.5281/zenodo.3528123. Supplementary information Supplementary data are available at Bioinformatics online.

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Kucherenko, Tamara, and Christian Wolf. "Entropy and rotation sets: A toy model approach." Communications in Contemporary Mathematics 18, no. 05 (July 18, 2016): 1550083. http://dx.doi.org/10.1142/s0219199715500832.

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Given a continuous dynamical system [Formula: see text] on a compact metric space [Formula: see text] and a continuous potential [Formula: see text], the generalized rotation set is the subset of [Formula: see text] consisting of all integrals of [Formula: see text] with respect to all invariant probability measures. The localized entropy at a point in the rotation set is defined as the supremum of the measure-theoretic entropies over all invariant measures whose integrals produce that point. In this paper, we provide an introduction to the theory of rotation sets and localized entropies. Moreover, we consider a shift map and construct a Lipschitz continuous potential, for which we are able to explicitly compute the geometric shape of the rotation set and its boundary measures. We show that at a particular exposed point on the boundary there are exactly two ergodic localized measures of maximal entropy.

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RICHARD, RODOLPHE. "RÉPARTITION GALOISIENNE D'UNE CLASSE D'ISOGÉNIE DE COURBES ELLIPTIQUES." International Journal of Number Theory 09, no. 02 (December 5, 2012): 517–43. http://dx.doi.org/10.1142/s1793042112501199.

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Dans cet article, on montre que les orbites sous Galois des invariants modulaires associés à des courbes elliptiques complexes sans multiplication complexe variant dans une même classe d'isogénie s'équidistribuent dans la courbe modulaire vers la probabilité hyperbolique. La démonstration repose sur des arguments de théorie ergodique, notamment le théorème de Ratner (cf. [A. Eskin et H. Oh, Ergodic theoretic proof of equidistribution of Hecke points, Ergodic Theory Dynam. Systems26(1) (2006) 163–167]), ainsi que sur le théorème de l'image ouverte de Serre [J.-P. Serre, Abelian l-Adic Representations and Elliptic Curves (W. A. Benjamin, New York, 1968); Propriétés Galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math.15(4) (1972) 259–331] dans le cas où les invariants modulaires considérés sont algébriques sur Q, et des résultats de G. Shimura dans le cas transcendant [Introduction to the Arithmetic Theory of Automorphic Functions, Publications of the Mathematical Society of Japan (Princeton University Press, Princeton, NJ, 1994)]. In this article, it is shown that Galois orbits of invariants associated with non-CM and pairwise isogeneous complex elliptic curves equidistribute in the classical modular curve towards the hyperbolic probability measure. The proof is based on arguments from ergodic theory, especially Ratner's theorem on unipotent flows (cf. [A. Eskin and H. Oh, Ergodic theoretic proof of equidistribution of Hecke points, Ergodic Theory Dynam. Systems26(1) (2006) 163–167]), as well as on Serre's open image theorem [J.-P. Serre, Abelian l-Adic Representations and Elliptic Curves (W. A. Benjamin, New York, 1968); Propriétés Galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math.15(4) (1972) 259–331] in case of algebraic invariants, and on G. Shimura's work in the transcendant case [Introduction to the Arithmetic Theory of Automorphic Functions, Publications of the Mathematical Society of Japan (Princeton University Press, Princeton, NJ, 1994)].

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Lee, Jaeha, and Izumi Tsutsui. "Uncertainty Relation for Errors Focusing on General POVM Measurements with an Example of Two-State Quantum Systems." Entropy 22, no. 11 (October 27, 2020): 1222. http://dx.doi.org/10.3390/e22111222.

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A novel uncertainty relation for errors of general quantum measurement is presented. The new relation, which is presented in geometric terms for maps representing measurement, is completely operational and can be related directly to tangible measurement outcomes. The relation violates the naïve bound ℏ/2 for the position-momentum measurement, whilst nevertheless respecting Heisenberg’s philosophy of the uncertainty principle. The standard Kennard–Robertson uncertainty relation for state preparations expressed by standard deviations arises as a corollary to its special non-informative case. For the measurement on two-state quantum systems, the relation is found to offer virtually the tightest bound possible; the equality of the relation holds for the measurement performed over every pure state. The Ozawa relation for errors of quantum measurements will also be examined in this regard. In this paper, the Kolmogorovian measure-theoretic formalism of probability—which allows for the representation of quantum measurements by positive-operator valued measures (POVMs)—is given special attention, in regard to which some of the measure-theory specific facts are remarked along the exposition as appropriate.

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Linton, Oliver B. "AN INTRODUCTION TO ECONOMETRIC THEORY." Econometric Theory 14, no. 6 (December 1998): 795–98. http://dx.doi.org/10.1017/s0266466698146054.

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This book is a modern introduction to measure theoretic probability and statistical inference well targeted for graduate students in econometrics at top institutions. It would make an excellent textbook for first year graduate students who intend to specialize in econometrics or who have an advanced mathematical background, and it would also be a useful part of any graduate econometrics course. It is concise and intensely focused on the key conceptual points, thus counteracting the tendency toward long-windedness apparent in some recent econometric texts. Nevertheless, it provides many valuable insights into difficult material. In particular, the discussions of sigma fields and conditional expectation given a sigma field are very helpful. The coverage of multivariate concepts alongside univariate ones is particularly useful to econometricians and something that is missing from most comparable statistical texts. The author has a mature attitude to proof, providing complete and illuminating proofs of some results but making liberal use of simplifications provided by special cases, for example in Theorems 4.1 and 4.5 and Section 5.2.2, to shorten and focus the arguments. The proofs themselves are very clear and well presented. Carefully chosen diagrams are given throughout the book that nicely illustrate many of the key concepts. In addition, each chapter contains a long list of problems of varying complexity, which will be useful to instructors.

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Zheng, Fang, Wenhu Wu, and Ditang Fang. "Center-distance continuous probability models and the distance measure." Journal of Computer Science and Technology 13, no. 5 (September 1998): 426–37. http://dx.doi.org/10.1007/bf02948501.

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Szuster, Janusz, Pawel Wlaź, and Jerzy Żurawiecki. "On Recognition of Shift Registers." Combinatorics, Probability and Computing 5, no. 3 (September 1996): 307–15. http://dx.doi.org/10.1017/s0963548300002066.

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This paper deals with infinite binary sequences. Each sequence is treated as generated by a nondeterministic shift register. A measure-theoretic criterion helpful in finding a deterministic generator of the set of sequences is proposed.

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Rhee, Joon Hee. "Theoretical Identifications of the Market Price of Risk under the Affine Interest Rate Model with Jumps." Journal of Derivatives and Quantitative Studies 13, no. 2 (November 30, 2005): 133–43. http://dx.doi.org/10.1108/jdqs-02-2005-b0006.

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Any finance models must specify the market prices of risk that determines the relationship between the two probability measures. Although the general form of the change of measure is well known, few papers have investigated the change of measure for interest rate models and their implications for the way a model can fit to empirical facts about the behaviour of interest rates. This paper demonstrates that arbitrary specifications of market price of risk in empirical studies under the two factor affine interest rate model with jumps are not compatible with the theory of original interest rate model. Particularly, the empirical models of Duffee (2002) and Duarte (2003) may be wrong specifications in some parts under a rigorous theoretical interest rate theory.

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TUCKER-DROB, ROBIN D. "Weak equivalence and non-classifiability of measure preserving actions." Ergodic Theory and Dynamical Systems 35, no. 1 (August 13, 2013): 293–336. http://dx.doi.org/10.1017/etds.2013.40.

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AbstractAbért and Weiss have shown that the Bernoulli shift ${s}_{\Gamma } $ of a countably infinite group $\Gamma $ is weakly contained in any free measure preserving action $\boldsymbol{a}$ of $\Gamma $. Proving a conjecture of Ioana, we establish a strong version of this result by showing that ${\boldsymbol{s}}_{\Gamma } \times \boldsymbol{a}$ is weakly equivalent to $\boldsymbol{a}$. Using random Bernoulli shifts introduced by Abért, Glasner, and Virag, we generalize this to non-free actions, replacing ${\boldsymbol{s}}_{\Gamma } $ with a random Bernoulli shift associated to an invariant random subgroup, and replacing the product action with a relatively independent joining. The result for free actions is used along with the theory of Borel reducibility and Hjorth’s theory of turbulence to show that, on the weak equivalence class of a free measure preserving action, the equivalence relations of isomorphism, weak isomorphism, and unitary equivalence are not classifiable by countable structures. This in particular shows that there are no free weakly rigid actions, that is, actions whose weak equivalence class and isomorphism class coincide, answering negatively a question of Abért and Elek. We also answer a question of Kechris regarding two ergodic theoretic properties of residually finite groups. A countably infinite residually finite group $\Gamma $ is said to have property ${\text{EMD} }^{\ast } $ if the action ${\boldsymbol{p}}_{\Gamma } $ of $\Gamma $ on its profinite completion weakly contains all ergodic measure preserving actions of $\Gamma $, and $\Gamma $ is said to have property $\text{MD} $ if $\boldsymbol{\iota} \times {\boldsymbol{p}}_{\Gamma } $ weakly contains all measure preserving actions of $\Gamma $, where $\boldsymbol{\iota} $ denotes the identity action on a standard non-atomic probability space. Kechris has shown that ${\text{EMD} }^{\ast } $ implies $\text{MD} $ and asked if the two properties are actually equivalent. We provide a positive answer to this question by studying the relationship between convexity and weak containment in the space of measure preserving actions.

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Goncharenko, Andriy Viktorovich. "Two Entropy Theory Wings as a New Trend for the Modern Means of Air Transport Operational Reliability Measure." Transactions on Aerospace Research 2020, no. 3 (September 1, 2020): 64–74. http://dx.doi.org/10.2478/tar-2020-0017.

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AbstractThe paper deals with the uncertainty of the operated system’s possible states hybrid combined optional functions. Traditionally, the probabilities of the system’s possible states are treated as the reliability measures. However, in the framework of the proposed doctrine, the optimality (for example, the maximal probability of the system’s state) is determined based upon a plausible assumption of the intrinsic objectively existing parameters. The two entropy theory wings consider on one hand the subjective preferences functions in subjective analysis, concerning the multi-alternativeness of the operational situation at an individual’s choice problems, and on the other hand the objectively existing characteristics used in theoretical physics. The discussed in the paper entropy paradigm proceeds with the objectively presented phenomena of the state’s probability and the probability’s maximum. The theoretical speculations and mathematical derivations are illustrated with the necessary plotted diagrams.

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Hanaoka, Ryota, and Norio Konno. "Return probability and self-similarity of the Riesz walk." Quantum Information and Computation 21, no. 5&6 (May 2021): 405–22. http://dx.doi.org/10.26421/qic21.5-6-5.

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The quantum walk is a counterpart of the random walk. The 2-state quantum walk in one dimension can be determined by a measure on the unit circle in the complex plane. As for the singular continuous measure, results on the corresponding quantum walk are limited. In this situation, we focus on a quantum walk, called the Riesz walk, given by the Riesz measure which is one of the famous singular continuous measures. The present paper is devoted to the return probability of the Riesz walk. Furthermore, we present some conjectures on the self-similarity of the walk.

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Kitaoka, Norihide, Ichiro Akahori, and Seiichi Nakagawa. "Confidence measure and rejection based on correctness probability of recognition candidates." Systems and Computers in Japan 35, no. 11 (2004): 91–103. http://dx.doi.org/10.1002/scj.20046.

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Torrente, Maria-Laura, and Pierpaolo Uberti. "Connectedness versus diversification: two sides of the same coin." Mathematics and Financial Economics 15, no. 3 (February 9, 2021): 639–55. http://dx.doi.org/10.1007/s11579-021-00291-4.

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AbstractIn the financial framework, the concepts of connectedness and diversification have been introduced and developed respectively in the context of systemic risk and portfolio theory. In this paper we propose a theoretical approach to bring to light the relation between connectedness and diversification. Starting from the respective axiomatic definitions, we prove that a class of proper measures of connectedness verifies, after a suitable functional transformation, the axiomatic requirements for a measure of diversification. The core idea of the paper is that connectedness and diversification are so deeply related that it is possible to pass from one concept to the other. In order to exploit such correspondence, we introduce a function, depending on the classical notion of rank of a matrix, that transforms a suitable proper measure of connectedness in a measure of diversification. We point out general properties of the proposed transformation function and apply it to a selection of measures of connectedness, such as the well-known Variance Inflation Factor.

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Rastegin, Alexey E. "Renyi and Tsallis formulations of noise-disturbance trade-off relations." Quantum Information and Computation 16, no. 3&4 (March 2016): 313–31. http://dx.doi.org/10.26421/qic16.3-4-7.

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We address an information-theoretic approach to noise and disturbance in quantum measurements. Properties of corresponding probability distributions are characterized by means of both the R´enyi and Tsallis entropies. Related information-theoretic measures of noise and disturbance are introduced. These definitions are based on the concept of conditional entropy. To motivate introduced measures, some important properties of the conditional R´enyi and Tsallis entropies are discussed. There exist several formulations of entropic uncertainty relations for a pair of observables. Trade-off relations for noise and disturbance are derived on the base of known formulations of such a kind.

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Amaral, Sergio, Douglas Allaire, and Karen Willcox. "Optimal $$L_2$$-norm empirical importance weights for the change of probability measure." Statistics and Computing 27, no. 3 (March 14, 2016): 625–43. http://dx.doi.org/10.1007/s11222-016-9644-3.

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Muliere, Pietro, Giovanni Parmigiani, and Nicholas G. Polson. "A Note on the Residual Entropy Function." Probability in the Engineering and Informational Sciences 7, no. 3 (July 1993): 413–20. http://dx.doi.org/10.1017/s0269964800003016.

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Interest in the informational content of truncation motivates the study of the residual entropy function, that is, the entropy of a right truncated random variable as a function of the truncation point. In this note we show that, under mild regularity conditions, the residual entropy function characterizes the probability distribution. We also derive relationships among residual entropy, monotonicity of the failure rate, and stochastic dominance. Information theoretic measures of distances between distributions are also revisited from a similar perspective. In particular, we study the residual divergence between two positive random variables and investigate some of its monotonicity properties. The results are relevant to information theory, reliability theory, search problems, and experimental design.

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Dombi, József, and Tamás Jónás. "An Elementary Proof of the General Poincaré Formula for λ-additive Measures." Acta Cybernetica 24, no. 2 (November 3, 2019): 173–85. http://dx.doi.org/10.14232/actacyb.24.2.2019.1.

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In a previous paper of ours [4], we presented the general formula for lambda-additive measure of union of n sets and gave a proof of it. That proof is based on the fact that the lambda-additive measure is representable. In this study, a novel and elementary proof of the formula for lambda-additive measure of the union of n sets is presented. Here, it is also demonstrated that, using elementary techniques, the well-known Poincare formula of probability theory is just a limit case of our general formula.

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Balasubramanian, Vijay. "Statistical Inference, Occam's Razor, and Statistical Mechanics on the Space of Probability Distributions." Neural Computation 9, no. 2 (February 1, 1997): 349–68. http://dx.doi.org/10.1162/neco.1997.9.2.349.

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The task of parametric model selection is cast in terms of a statistical mechanics on the space of probability distributions. Using the techniques of low-temperature expansions, I arrive at a systematic series for the Bayesian posterior probability of a model family that significantly extends known results in the literature. In particular, I arrive at a precise understanding of how Occam's razor, the principle that simpler models should be preferred until the data justify more complex models, is automatically embodied by probability theory. These results require a measure on the space of model parameters and I derive and discuss an interpretation of Jeffreys' prior distribution as a uniform prior over the distributions indexed by a family. Finally, I derive a theoretical index of the complexity of a parametric family relative to some true distribution that I call the razor of the model. The form of the razor immediately suggests several interesting questions in the theory of learning that can be studied using the techniques of statistical mechanics.

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Bürger, Reinhard, and Immanuel M. Bomze. "Stationary distributions under mutation-selection balance: structure and properties." Advances in Applied Probability 28, no. 1 (March 1996): 227–51. http://dx.doi.org/10.2307/1427919.

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A general model for the evolution of the frequency distribution of types in a population under mutation and selection is derived and investigated. The approach is sufficiently general to subsume classical models with a finite number of alleles, as well as models with a continuum of possible alleles as used in quantitative genetics. The dynamics of the corresponding probability distributions is governed by an integro-differential equation in the Banach space of Borel measures on a locally compact space. Existence and uniqueness of the solutions of the initial value problem is proved using basic semigroup theory. A complete characterization of the structure of stationary distributions is presented. Then, existence and uniqueness of stationary distributions is proved under mild conditions by applying operator theoretic generalizations of Perron–Frobenius theory. For an extension of Kingman's original house-of-cards model, a classification of possible stationary distributions is obtained.

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Korec, Ivan. "Definability of Arithmetic Operations from the Order and a Random Relation1." Fundamenta Informaticae 18, no. 2-4 (April 1, 1993): 287–96. http://dx.doi.org/10.3233/fi-1993-182-414.

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For almost all binary relations R ⊆ N 2 the addition and multiplication on the set N of nonnegative integers (and hence all arithmetical relations) are first order definable in the structure (N; ⩽, R). The defining formulae can be chosen independently on R and the words “for almost all” mean “with probability 1” by a very natural probability measure.

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Frank, Lawrence R., and Vitaly L. Galinsky. "Dynamic Multiscale Modes of Resting State Brain Activity Detected by Entropy Field Decomposition." Neural Computation 28, no. 9 (September 2016): 1769–811. http://dx.doi.org/10.1162/neco_a_00871.

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The ability of functional magnetic resonance imaging (FMRI) to noninvasively measure fluctuations in brain activity in the absence of an applied stimulus offers the possibility of discerning functional networks in the resting state of the brain. However, the reconstruction of brain networks from these signal fluctuations poses a significant challenge because they are generally nonlinear and nongaussian and can overlap in both their spatial and temporal extent. Moreover, because there is no explicit input stimulus, there is no signal model with which to compare the brain responses. A variety of techniques have been devised to address this problem, but the predominant approaches are based on the presupposition of statistical properties of complex brain signal parameters, which are unprovable but facilitate the analysis. In this article, we address this problem with a new method, entropy field decomposition, for estimating structure within spatiotemporal data. This method is based on a general information field-theoretic formulation of Bayesian probability theory incorporating prior coupling information that allows the enumeration of the most probable parameter configurations without the need for unjustified statistical assumptions. This approach facilitates the construction of brain activation modes directly from the spatial-temporal correlation structure of the data. These modes and their associated spatial-temporal correlation structure can then be used to generate space-time activity probability trajectories, called functional connectivity pathways, which provide a characterization of functional brain networks.

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McRoberts, N., C. Hall, L. V. Madden, and G. Hughes. "Perceptions of Disease Risk: From Social Construction of Subjective Judgments to Rational Decision Making." Phytopathology® 101, no. 6 (June 2011): 654–65. http://dx.doi.org/10.1094/phyto-04-10-0126.

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Many factors influence how people form risk perceptions. Farmers' perceptions of risk and levels of risk aversion impact on decision-making about such things as technology adoption and disease management practices. Irrespective of the underlying factors that affect risk perceptions, those perceptions can be summarized by variables capturing impact and uncertainty components of risk. We discuss a new framework that has the subjective probability of disease and the cost of decision errors as its central features, which might allow a better integration of social science and epidemiology, to the benefit of plant disease management. By focusing on the probability and cost (or impact) dimensions of risk, the framework integrates research from the social sciences, economics, decision theory, and epidemiology. In particular, we review some useful properties of expected regret and skill value, two measures of expected cost that are particularly useful in the evaluation of decision tools. We highlight decision-theoretic constraints on the usefulness of decision tools that may partly explain cases of failure of adoption. We extend this analysis by considering information-theoretic criteria that link model complexity and relative performance and which might explain why users reject forecasters that impose even moderate increases in the complexity of decision making despite improvements in performance or accept very simple decision tools that have relatively poor performance.

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Bhatia, Munish. "Intelligent System of Game-Theory-Based Decision Making in Smart Sports Industry." ACM Transactions on Intelligent Systems and Technology 12, no. 3 (April 22, 2021): 1–23. http://dx.doi.org/10.1145/3447986.

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Internet of Things (IoT) technology backed by Artificial Intelligence (AI) techniques has been increasingly utilized for the realization of the Industry 4.0 vision. Conspicuously, this work provides a novel notion of the smart sports industry for provisioning efficient services in the sports arena. Specifically, an IoT-inspired framework has been proposed for real-time analysis of athlete performance. IoT data is utilized to quantify athlete performance in the terms of probability parameters of Probabilistic Measure of Performance (PMP) and Level of Performance Measure (LoPM). Moreover, a two-player game-theory-based mathematical framework has been presented for efficient decision modeling by the monitoring officials. The presented model is validated experimentally by deployment in District Sports Academy (DSA) for 60 days over four players. Based on the comparative analysis with state-of-the-art decision-modeling approaches, the proposed model acquired enhanced performance values in terms of Temporal Delay, Classification Efficiency, Statistical Efficacy, Correlation Analysis, and Reliability.

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Dahlsten, O. C. O., and M. B. Plenio. "Entanglement probability distribution of bi-partite randomised stabilizer states." Quantum Information and Computation 6, no. 6 (September 2006): 527–38. http://dx.doi.org/10.26421/qic6.6-5.

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We study the entanglement properties of random pure stabilizer states in spin-1/2 particles. We obtain a compact and exact expression for the probability distribution of the entanglement values across any bipartite cut. This allows for exact derivations of the average entanglement and the degree of concentration of measure around this average. We also give simple bounds on these quantities. We find that for large systems the average entanglement is near maximal and the measure is concentrated around it.

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Wei, Jie, Wenxian Xie, and Yufeng Nie. "Shared Node and Its Improvement to the Theory Analysis and Solving Algorithm for the Loop Cutset." Mathematics 8, no. 9 (September 19, 2020): 1625. http://dx.doi.org/10.3390/math8091625.

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Bayesian Network is one of the famous network models, and the loop cutset is one of the crucial structures for Bayesian Inference. In the Bayesian Network and its inference, how to measure the relationship between nodes is very important, because the relationship between different nodes has significant influence on the node-probability of the loop cutset. To analyse the relationship between two nodes in a graph, we define the shared node, prove the upper and lower bounds of the shared nodes number, and affirm that the shared node influences the node-probability of the loop cutset according to the theorems and experiments. These results can explain the problems that we found in studying on the statistical node-probability belonging to the loop cutset. The shared nodes are performed not only to improve the theoretical analysis on the loop cutset, but also to the loop cutset solving algorithms, especially the heuristic algorithms, in which the heuristic strategy can be optimized by a shared node. Our results provide a new tool to gauge the relationship between different nodes, a new perspective to estimate the loop cutset, and it is helpful to the loop cutset algorithm and network analysis.

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Magori-Cohen, R., Y. Louzoun, and S. H. Kleinstein. "Mutation parameters from DNA sequence data using graph theoretic measures on lineage trees." Bioinformatics 22, no. 14 (July 15, 2006): e332-e340. http://dx.doi.org/10.1093/bioinformatics/btl239.

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43

Bürger, Reinhard, and Immanuel M. Bomze. "Stationary distributions under mutation-selection balance: structure and properties." Advances in Applied Probability 28, no. 01 (March 1996): 227–51. http://dx.doi.org/10.1017/s0001867800027348.

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A general model for the evolution of the frequency distribution of types in a population under mutation and selection is derived and investigated. The approach is sufficiently general to subsume classical models with a finite number of alleles, as well as models with a continuum of possible alleles as used in quantitative genetics. The dynamics of the corresponding probability distributions is governed by an integro-differential equation in the Banach space of Borel measures on a locally compact space. Existence and uniqueness of the solutions of the initial value problem is proved using basic semigroup theory. A complete characterization of the structure of stationary distributions is presented. Then, existence and uniqueness of stationary distributions is proved under mild conditions by applying operator theoretic generalizations of Perron–Frobenius theory. For an extension of Kingman's original house-of-cards model, a classification of possible stationary distributions is obtained.

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FLAMINIO, TOMMASO. "THREE CHARACTERIZATIONS OF STRICT COHERENCE ON INFINITE-VALUED EVENTS." Review of Symbolic Logic 13, no. 3 (October 4, 2019): 593–610. http://dx.doi.org/10.1017/s1755020319000546.

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AbstractThis article builds on a recent paper coauthored by the present author, H. Hosni and F. Montagna. It is meant to contribute to the logical foundations of probability theory on many-valued events and, specifically, to a deeper understanding of the notion of strict coherence. In particular, we will make use of geometrical, measure-theoretical and logical methods to provide three characterizations of strict coherence on formulas of infinite-valued Łukasiewicz logic.

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Binns, Stephen, Bjørn Kjos-Hanssen, Manuel Lerman, and Reed Solomon. "On a conjecture of Dobrinen and Simpson concerning almost everywhere domination." Journal of Symbolic Logic 71, no. 1 (March 2006): 119–36. http://dx.doi.org/10.2178/jsl/1140641165.

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Dobrinen and Simpson [4] introduced the notions of almost everywhere domination and uniform almost everywhere domination to study recursion theoretic analogues of results in set theory concerning domination in generic extensions of transitive models of ZFC and to study regularity properties of the Lebesgue measure on 2ω in reverse mathematics. In this article, we examine one of their conjectures concerning these notions.Throughout this article, ≤T denotes Turing reducibility and μ denotes the Lebesgue (or “fair coin”) probability measure on 2ω given byA property holds almost everywhere or for almost all X ∈ 2ω if it holds on a set of measure 1. For f, g ∈ ωω, f dominatesg if ∃m∀n < m(f(n) > g(n)).(Dobrinen, Simpson). A set A ∈ 2ωis almost everywhere (a.e.) dominating if for almost all X ∈ 2ω and all functions g ≤TX, there is a function f ≤TA such that f dominates g. A is uniformly almost everywhere (u.a.e.) dominating if there is a function f ≤TA such that for almost all X ∈ 2ω and all functions g ≤TX, f dominates g.There are several trivial but useful observations to make about these definitions. First, although these properties are stated for sets, they are also properties of Turing degrees. That is, a set is (u.)a.e. dominating if and only if every other set of the same degree is (u.)a.e. dominating. Second, both properties are closed upwards in the Turing degrees. Third, u.a.e. domination implies a.e. domination. Finally, if A is u.a.e. dominating, then there is a function f ≤TA which dominates every computable function.

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COHEN, SHAY B., ROBERT J. SIMMONS, and NOAH A. SMITH. "Products of weighted logic programs." Theory and Practice of Logic Programming 11, no. 2-3 (January 28, 2011): 263–96. http://dx.doi.org/10.1017/s1471068410000529.

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AbstractWeighted logic programming, a generalization of bottom-up logic programming, is a well-suited framework for specifying dynamic programming algorithms. In this setting, proofs correspond to the algorithm's output space, such as a path through a graph or a grammatical derivation, and are given a real-valued score (often interpreted as a probability) that depends on the real weights of the base axioms used in the proof. The desired output is a function over all possible proofs, such as a sum of scores or an optimal score. We describe the product transformation, which can merge two weighted logic programs into a new one. The resulting program optimizes a product of proof scores from the original programs, constituting a scoring function known in machine learning as a “product of experts.” Through the addition of intuitive constraining side conditions, we show that several important dynamic programming algorithms can be derived by applying product to weighted logic programs corresponding to simpler weighted logic programs. In addition, we show how the computation of Kullback–Leibler divergence, an information-theoretic measure, can be interpreted using product.

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Planton, Samuel, Timo van Kerkoerle, Leïla Abbih, Maxime Maheu, Florent Meyniel, Mariano Sigman, Liping Wang, Santiago Figueira, Sergio Romano, and Stanislas Dehaene. "A theory of memory for binary sequences: Evidence for a mental compression algorithm in humans." PLOS Computational Biology 17, no. 1 (January 19, 2021): e1008598. http://dx.doi.org/10.1371/journal.pcbi.1008598.

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Working memory capacity can be improved by recoding the memorized information in a condensed form. Here, we tested the theory that human adults encode binary sequences of stimuli in memory using an abstract internal language and a recursive compression algorithm. The theory predicts that the psychological complexity of a given sequence should be proportional to the length of its shortest description in the proposed language, which can capture any nested pattern of repetitions and alternations using a limited number of instructions. Five experiments examine the capacity of the theory to predict human adults’ memory for a variety of auditory and visual sequences. We probed memory using a sequence violation paradigm in which participants attempted to detect occasional violations in an otherwise fixed sequence. Both subjective complexity ratings and objective violation detection performance were well predicted by our theoretical measure of complexity, which simply reflects a weighted sum of the number of elementary instructions and digits in the shortest formula that captures the sequence in our language. While a simpler transition probability model, when tested as a single predictor in the statistical analyses, accounted for significant variance in the data, the goodness-of-fit with the data significantly improved when the language-based complexity measure was included in the statistical model, while the variance explained by the transition probability model largely decreased. Model comparison also showed that shortest description length in a recursive language provides a better fit than six alternative previously proposed models of sequence encoding. The data support the hypothesis that, beyond the extraction of statistical knowledge, human sequence coding relies on an internal compression using language-like nested structures.

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AISTLEITNER, CHRISTOPH. "Normal Numbers and the Normality Measure." Combinatorics, Probability and Computing 22, no. 3 (April 5, 2013): 342–45. http://dx.doi.org/10.1017/s0963548313000084.

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In a paper published in this journal, Alon, Kohayakawa, Mauduit, Moreira and Rödl proved that the minimal possible value of the normality measure of an N-element binary sequence satisfies \begin{equation*} \biggl( \frac{1}{2} + o(1) \biggr) \log_2 N \leq \min_{E_N \in \{0,1\}^N} \mathcal{N}(E_N) \leq 3 N^{1/3} (\log N)^{2/3} \end{equation*} for sufficiently large N, and conjectured that the lower bound can be improved to some power of N. In this note it is observed that a construction of Levin of a normal number having small discrepancy gives a construction of a binary sequence EN with (EN) = O((log N)2), thus disproving the conjecture above.

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Zenil, Hector, Fernando Soler-Toscano, Jean-Paul Delahaye, and Nicolas Gauvrit. "Two-dimensional Kolmogorov complexity and an empirical validation of the Coding theorem method by compressibility." PeerJ Computer Science 1 (September 30, 2015): e23. http://dx.doi.org/10.7717/peerj-cs.23.

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We propose a measure based upon the fundamental theoretical concept in algorithmic information theory that provides a natural approach to the problem of evaluatingn-dimensional complexity by using ann-dimensional deterministic Turing machine. The technique is interesting because it provides a natural algorithmic process for symmetry breaking generating complexn-dimensional structures from perfectly symmetric and fully deterministic computational rules producing a distribution of patterns as described by algorithmic probability. Algorithmic probability also elegantly connects the frequency of occurrence of a pattern with its algorithmic complexity, hence effectively providing estimations to the complexity of the generated patterns. Experiments to validate estimations of algorithmic complexity based on these concepts are presented, showing that the measure is stable in the face of some changes in computational formalism and that results are in agreement with the results obtained using lossless compression algorithms when both methods overlap in their range of applicability. We then use the output frequency of the set of 2-dimensional Turing machines to classify the algorithmic complexity of the space-time evolutions of Elementary Cellular Automata.

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Li, Siman, Li Shi, and Wei Gao. "Two modified Zagreb indices for random structures." Main Group Metal Chemistry 44, no. 1 (January 1, 2021): 150–56. http://dx.doi.org/10.1515/mgmc-2021-0013.

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Abstract Random structure plays an important role in the composition of compounds, and topological index is an important index to measure indirectly the properties of compounds. The Zagreb indices and its revised versions (or redefined versions) are frequently used chemical topological indices, which provide the theoretical basis for the determination of various physical-chemical properties of compounds. This article uses the tricks of probability theory to determine the reduced second Zagreb index and hyper-Zagreb index of two kinds of vital random graphs: G(n, p) and G(n, m).

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Journal articles on the topic 'Measure Theoretic Probability Theory'

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