Measuring and quantifying dependencies between random variables (RVs) can give critical insights into a dataset. Typical questions are: ‘Do underlying relationships exist?’, ‘Are some variables redundant?’, and ‘Is some target variable Y highly or weakly dependent on variable X?’ Interestingly, despite the evident need for a general-purpose measure of dependency between RVs, common practice is that most data analysts use the Pearson correlation coefficient to quantify dependence between RVs, while it is recognized that the correlation coefficient is essentially a measure for linear dependency only. Although many attempts have been made to define more generic dependency measures, there is no consensus yet on a standard, general-purpose dependency function. In fact, several ideal properties of a dependency function have been proposed, but without much argumentation. Motivated by this, we discuss and revise the list of desired properties and propose a new dependency function that meets all these requirements. This general-purpose dependency function provides data analysts with a powerful means to quantify the level of dependence between variables. To this end, we also provide Python code to determine the dependency function for use in practice.

We formulate and explore two basic axiomatic systems of type-free subjective probability. One of them explicates a notion of finitely additive probability. The other explicates a concept of infinitely additive probability. It is argued that the first of these systems is a suitable background theory for formally investigating controversial principles about type-free subjective probability.

The ergodic decomposition theorem is a cornerstone result of dynamical systems and ergodic theory. It states that every invariant measure on a dynamical system is a mixture of ergodic ones. Here, we formulate and prove the theorem in terms of string diagrams, using the formalism of Markov categories. We recover the usual measure-theoretic statement by instantiating our result in the category of stochastic kernels. Along the way, we give a conceptual treatment of several concepts in the theory of deterministic and stochastic dynamical systems. In particular, ergodic measures appear very naturally as particular cones of deterministic morphisms (in the sense of Markov categories), and the invariant $\sigma $ -algebra of a dynamical system can be seen as a colimit in the category of Markov kernels. In line with other uses of category theory, once the necessary structures are in place, our proof of the main theorem is much more intuitive than traditional approaches. In particular, it does not use any quantitative limiting arguments, and it does not rely on the cardinality of the group or monoid indexing the dynamics. We hope that this result paves the way for further applications of category theory to dynamical systems, ergodic theory, and information theory.

Scoring rules measure the deviation between a forecast, which assigns degrees of confidence to various events, and reality. Strictly proper scoring rules have the property that for any forecast, the mathematical expectation of the score of a forecast p by the lights of p is strictly better than the mathematical expectation of any other forecast q by the lights of p. Forecasts need not satisfy the axioms of the probability calculus, but Predd et al. [9] have shown that given a finite sample space and any strictly proper additive and continuous scoring rule, the score for any forecast that does not satisfy the axioms of probability is strictly dominated by the score for some probabilistically consistent forecast. Recently, this result has been extended to non-additive continuous scoring rules. In this paper, a condition weaker than continuity is given that suffices for the result, and the condition is proved to be optimal.

In the context of propositional logics, we apply semantics modulo satisfiability—a restricted semantics which comprehends only valuations that satisfy some specific set of formulas—with the aim to efficiently solve some computational tasks. Three possible such applications are developed.

We begin by studying the possibility of implicitly representing rational McNaughton functions in Łukasiewicz Infinitely-valued Logic through semantics modulo satisfiability. We theoretically investigate some approaches to such representation concept, called representation modulo satisfiability, and describe a polynomial algorithm that builds representations in the newly introduced system. An implementation of the algorithm, test results and ways to randomly generate rational McNaughton functions for testing are presented. Moreover, we propose an application of such representations to the formal verification of properties of neural networks by means of the reasoning framework of Łukasiewicz Infinitely-valued Logic.

Then, we move to the investigation of the satisfiability of joint probabilistic assignments to formulas of Łukasiewicz Infinitely-valued Logic, which is known to be an NP-complete problem. We provide an exact decision algorithm derived from the combination of linear algebraic methods with semantics modulo satisfiability. Also, we provide an implementation for such algorithm for which the phenomenon of phase transition is empirically detected.

Lastly, we study the game theory situation of observable games, which are games that are known to reach a Nash equilibrium, however, an external observer does not know what is the exact profile of actions that occur in a specific instance; thus, such observer assigns subjective probabilities to players actions. We study the decision problem of determining if a set of these probabilistic constraints is coherent by reducing it to the problems of satisfiability of probabilistic assignments to logical formulas both in Classical Propositional Logic and Łukasiewicz Infinitely-valued Logic depending on whether only pure equilibria or also mixed equilibria are allowed. Such reductions rely upon the properties of semantics modulo satisfiability. We provide complexity and algorithmic discussion for the coherence problem and, also, for the problem of computing maximal and minimal probabilistic constraints on actions that preserves coherence.

Abstract prepared by Sandro Márcio da Silva Preto.

E-mail: spreto@ime.usp.br

URL:https://doi.org/10.11606/T.45.2021.tde-17062021-163257

This paper generalises an argument for probabilism due to Lindley [9]. I extend the argument to a number of non-classical logical settings whose truth-values, seen here as ideal aims for belief, are in the set $\{0,1\}$ , and where logical consequence $\models $ is given the “no-drop” characterization. First I will show that, in each of these settings, an agent’s credence can only avoid accuracy-domination if its canonical transform is a (possibly non-classical) probability function. In other words, if an agent values accuracy as the fundamental epistemic virtue, it is a necessary requirement for rationality that her credence have some probabilistic structure. Then I show that for a certain class of reasonable measures of inaccuracy, having such a probabilistic structure is sufficient to avoid accuracy-domination in these non-classical settings.

Supra-Bayesianism is the Bayesian response to learning the opinions of others. Probability pooling constitutes an alternative response. One natural question is whether there are cases where probability pooling gives the supra-Bayesian result. This has been called the problem of Bayes-compatibility for pooling functions. It is known that in a common prior setting, under standard assumptions, linear pooling cannot be nontrivially Bayes-compatible. We show by contrast that geometric pooling can be nontrivially Bayes-compatible. Indeed, we show that, under certain assumptions, geometric and Bayes-compatible pooling are equivalent. Granting supra-Bayesianism its usual normative status, one upshot of our study is thus that, in a certain class of epistemic contexts, geometric pooling enjoys a normative advantage over linear pooling as a social learning mechanism. We discuss the philosophical ramifications of this advantage, which we show to be robust to variations in our statement of the Bayes-compatibility problem.

Several authors have investigated the question of whether canonical logic-based accounts of belief revision, and especially the theory of AGM revision operators, are compatible with the dynamics of Bayesian conditioning. Here we show that Leitgeb’s stability rule for acceptance, which has been offered as a possible solution to the Lottery paradox, allows to bridge AGM revision and Bayesian update: using the stability rule, we prove that AGM revision operators emerge from Bayesian conditioning by an application of the principle of maximum entropy. In situations of information loss, or whenever the agent relies on a qualitative description of her information state—such as a plausibility ranking over hypotheses, or a belief set—the dynamics of AGM belief revision are compatible with Bayesian conditioning; indeed, through the maximum entropy principle, conditioning naturally generates AGM revision operators. This mitigates an impossibility theorem of Lin and Kelly for tracking Bayesian conditioning with AGM revision, and suggests an approach to the compatibility problem that highlights the information loss incurred by acceptance rules in passing from probabilistic to qualitative representations of belief.

This paper contributes to a recent research program that extends arguments supporting elementary conditionalization to arguments supporting conditionalization with general, measure-theoretic conditional probabilities. I begin by suggesting an amendment to the framework that Rescorla (2018) has used to characterize regular conditional probabilities in terms of avoiding Dutch book. If we wish to model learning scenarios in which an agent gains complete membership knowledge about some subcollection of the events of interest to her, then we should focus on updating policies that are what I shall call proper. I go on to characterize regular conditional probabilities in proper learning scenarios using what van Fraassen (1999) calls The General Reflection Principle.

It is known that the Fourier–Stieltjes coefficients of a nonatomic coin-tossing measure may not vanish at infinity. However, we show that they could vanish at infinity along some integer subsequences, including the sequence ${\{b^{n}\}}_{n\geq 1}$ where $b$ is multiplicatively independent of 2 and the sequence given by the multiplicative semigroup generated by 3 and 5. The proof is based on elementary combinatorics and lower-bound estimates for linear forms in logarithms from transcendental number theory.

Choices rarely deal with certainties; and, where assertoric logic and modal logic are insufficient, those seeking to be reasonable turn to one or more things called “probability.” These things typically have a shared mathematical form, which is an arithmetic construct. The construct is often felt to be unsatisfactory for various reasons. A more general construct is that of a preordering, which may even be incomplete, allowing for cases in which there is no known probability relation between two propositions or between two events. Previous discussion of incomplete preorderings has been as if incidental, with researchers focusing upon preorderings for which quantifications are possible. This article presents formal axioms for the more general case. Challenges peculiar to some specific interpretations of the nature of probability are brought to light in the context of these propositions. A qualitative interpretation is offered for probability differences that are often taken to be quantified. A generalization of Bayesian updating is defended without dependence upon coherence. Qualitative hypothesis testing is offered as a possible alternative in cases for which quantitative hypothesis testing is shown to be unsuitable.

This paper provides a functional analogue of the recently initiated dual Orlicz–Brunn–Minkowski theory for star bodies. We first propose the Orlicz addition of measures, and establish the dual functional Orlicz–Brunn–Minkowski inequality. Based on a family of linear Orlicz additions of two measures, we provide an interpretation for the famous $f$-divergence. Jensen’s inequality for integrals is also proved to be equivalent to the newly established dual functional Orlicz–Brunn–Minkowski inequality. An optimization problem for the $f$-divergence is proposed, and related functional affine isoperimetric inequalities are established.

We discuss the reconciliation problem between probability measures: given n⩾2 probability spaces $(\Omega,{\mathcal{F}}_1,{\mathbb{P}}_1),\ldots,(\Omega,{\mathcal{F}}_n,{\mathbb{P}}_n)$ with a common sample space, does there exist an overall probability measure ${\mathbb{P}} \ \text{on} \ {\mathcal{F}} = \sigma({\mathcal{F}}_1,\ldots,{\mathcal{F}}_n)$ such that, for all i, the restriction of ${\mathbb{P}} \ \text{to} \ {\mathcal{F}}_i$ coincides with ${\mathbb{P}}_i$? General criteria for the existence of a reconciliation are stated, along with some counterexamples that highlight some delicate issues. Connections to earlier (recent and far less recent) work are discussed, and elementary self-contained proofs for the various results are given.

The theory of influence and sharp threshold is a key tool in probability and probabilistic combinatorics, with numerous applications. One significant aspect of the theory is directed at identifying the level of generality of the product probability space that accommodates the event under study. We derive the influence inequality for a completely general product space, by establishing a relationship to the Lebesgue cube studied by Bourgain, Kahn, Kalai, Katznelson and Linial (BKKKL) in 1992. This resolves one of the assertions of BKKKL. Our conclusion is valid also in the setting of the generalized influences of Keller.

We will propose an alternative condition for stochastic domination. This condition differs in an essential way from the strong likelihood ratio property. We also show an example which satisfies the new condition, but does not satisfy the strong likelihood ratio property.

In this paper we introduce certain Hankel matrices that can be used to study ME (matrix exponential) distributions, in particular to compute their ME orders. The Hankel matrices for a given ME probability distribution can be constructed if one of the following five types of information about the distribution is available: (i) an ME representation, (ii) its moments, (iii) the derivatives of its distribution function, (iv) its Laplace-Stieltjes transform, or (v) its distribution function. Using the Hankel matrices, a necessary and sufficient condition for a probability distribution to be an ME distribution is found and a method of computing the ME order of the ME distribution developed. Implications for the PH (phase-type) order of PH distributions are examined. The relationship between the ME order, the PH order, and a lower bound on the PH order given by Aldous and Shepp (1987) is discussed in numerical examples.

The literature on subjective probabilities contains a number of functions that have been proposed as ‘scoring functions’. The principal requirement is that, with several events that may occur in the future and to which subjective ‘probabilities’ are assigned, the expected score given by these ‘probabilities’ will be extremised if the values assigned equal the ‘true probabilities’ of the various outcomes. This article discusses the question of what other scoring functions might be used (beyond those so far proposed).

The accountant Nigrini remarked that in tables of data distributed according to Benford's law, the sum of all elements with first digit d (d = 1, 2,· ··, 9) is approximately constant. In this note, a mathematical formulation of Nigrini's observation is given and it is shown that Benford's law is the unique probability distribution such that the expected sum of all elements with first digits d1, · ··, dk is constant for every fixed k.

A basic theory for probabilistic convergence spaces based on filter convergence is introduced. As in Florescu's previous theory of probabilistic convergence structures based on nets, one is able to assign a probability that a given filter converges to a given point. Various concepts and theorems pertaining to convergence spaces are extended to the realm of probabilistic convergence spaces, and illustrated by means of examples based on convergence in probability and convergence almost everywhere. Diagonal axioms due to Kowalsky and Fischer are also studied, first for convergence spaces and then in the setting of probabilistic convergence spaces.

Probabilistic quasi-metric spaces are introduced and used to define ordered probabilistic metric spaces. The latter spaces arise naturally in the study of probability and statistics; they closely resemble the uniform ordered spaces of L. Nachbin. A procedure is described for constructing ordered probabilistic metric spaces from quasi-simple spaces, and a completions theory is developed simultaneously for probabilistic quasi-metric spaces and ordered probabilistic metric spaces.