Probability Calculus Research Articles

Административные решения алгоритмов

Examples of algorithms being involved in administrative practice are multiplying. However, the process of their use remains unsettled. For the legal evaluation of the results of algorithms’ decisions and subsequent legal regulation, it is important to distinguish between automated decision-making systems and machine learning systems. But if the prospects for the use of algorithms by both courts and the executive branch seem obvious, such use should certainly take into account possible risks and disadvantages. This is especially true for machine learning algorithms. First, human decision-making inevitably leads to the problem of scoring, which has not only an ethical but also a legal dimension. Second, there is the problem of algorithm errors. A statistical decision, even based on a working algorithm, can be wrong. Statistics using probability calculus, even when applied to a mass of cases, leads to probable rather than certain conclusions. This means that the personalized decision of a machine learning algorithm cannot be final, but must be “sanctioned” by a human. Hence the inevitability of human control.

Vývoj a interpretace pojmu pravděpodobnost

Considerations related to randomness appear relatively late in European history, at the turn of the Middle Ages and the modern era. They initially concern to the chances of winning in various games or situations and later move on to introduce classical and geometric probabilities. From a mathematical point of view, the probability calculus is completed by Kolmogorov’s axiomatic theory. However, many open questions, problems and paradoxes remain in the way probability is perceived and interpreted. The four main directions in the concept of probability (logical, frequentist, subjective and propensity) are closely related to the way of perceiving randomness (epistemological or ontological). Reflections on the evolution of the perception of randomness and the interpretation of probability bring the ability to navigate the basic principles and findings of science and contributes to a deeper understanding of the entire issue.

Fracterm Calculus for Signed Common Meadows

A common meadow is an enrichment of a field with a division operator and an error value to make division total. A signed common meadow enriches a common meadow with a sign function that can be equationally axiomatised; the sign function can simulate an ordering on the underlying field but is not limited to orderings. In particular, of mathematical interest are the weakly signed common meadows. The prime example of a weakly signed common meadow is an expansion of a common meadow of complex numbers with a weak sign function. We show that all common meadows may be enlarged to a weakly signed common meadow. A special case is the 4-signed common meadows, which are precisely the enlargements of ordered fields. To illustrate the equational calculus for signed common meadows, we use it as a foundation for building a probability calculus and derive some classical formulae.

The Cultural Evolution of Games of Chance

Chance-based gambling has been a recurrent cultural activity throughout history and across many diverse human societies. In this paper, I combine quantitative and qualitative data and present a cultural evolutionary framework to explain why the odds in games of chance in premodern China appeared “designed” to ensure a moderate yet favorable house advantage. This is especially intriguing since extensive research in the history of probability has shown that, prior to the development of probability theory, people had very limited understanding of the nature of random events and were generally disinclined to think mathematically about the frequency of their occurrence. I argue that games of chance in the context of gambling may have culturally evolved into their documented forms via a process of selective imitation and retention, and neither the customers nor the gambling houses understood the probability calculus involved in these games.

A Bayesian network model to disentangle the effects of stand and climate factors on tree mortality of Chinese fir plantations

Tree mortality is a complex process that not only be affected by the various factors, such as stand and climate factors, but also the various long-term effects of the factors to each other. In this study, based on the long-term spacing trials of Chinese fir in four regions of southern China, a Bayesian network was used to model tree mortality in response to stand and climate factors, as well as comparing this approach with logistic regression and random forest method. The results showed that the Bayesian network method had the highest accuracy in predicting tree mortality. In addition, the Bayesian network approach could find the dependency in the relationship between data and provide a theoretical framework for modeling uncertainty by using probabilistic calculus and underlying graph structure. Sensitivity analysis showed relative diameter was the most important factor, and temperature was the most important climate factor. Furthermore, climate factors not only directly affected tree mortality, but also indirectly affected tree mortality through affecting relative diameter, stand density and Gini coefficient. We also found that stand competition, structural heterogeneity and age affected tree mortality under climate change, and a moderate level of competition condition and stand structure heterogeneity weakened the negative impact of climate factors on tree mortality. Old trees were more sensitive to climate change than young trees, especially under extreme climate conditions. Besides, we found that tree mortality was negatively correlated with moderate annual precipitation, winter mean minimum temperature, and stand structure (Gini), and low age, but positively correlated with low relative diameter, high density and age. The results will provide adaptive options for effective forest management of Chinese fir plantations under the backdrop of global climate change in the future.

The fraternal birth-order effect as a statistical artefact: convergent evidence from probability calculus, simulated data, and multiverse meta-analysis.

The fraternal-birth order effect (FBOE) is a research claim which states that each older brother increases the odds of homosexual orientation in men via an immunoreactivity process known as the maternal immune hypothesis. Importantly, older sisters supposedly either do not affect these odds, or affect them to a lesser extent. Consequently, the fraternal birth-order effect predicts that the association between the number of older brothers and homosexual orientation in men is greater in magnitude than any association between the number of older sisters and homosexual orientation. This difference in magnitude represents the main theoretical estimand of the FBOE. In addition, no comparable effects should be observable among homosexual vs heterosexual women. Here, we triangulate the empirical foundations of the FBOE from three distinct, informative perspectives, complementing each other: first, drawing on basic probability calculus, we deduce mathematically that the body of statistical evidence used to make inferences about the main theoretical estimand of the FBOE rests on incorrect statistical reasoning. In particular, we show that throughout the literature researchers ascribe to the false assumptions that effects of family size should be adjusted for and that this could be achieved through the use of ratio variables. Second, using a data-simulation approach, we demonstrate that by using currently recommended statistical practices, researchers are bound to frequently draw incorrect conclusions. And third, we re-examine the empirical evidence of the fraternal birth-order effect in men and women by using a novel specification-curve and multiverse approach to meta-analysis (64 male and 17 female samples, N = 2,778,998). When analyzed correctly, the specific association between the number of older brothers and homosexual orientation is small, heterogenous in magnitude, and apparently not specific to men. In addition, existing research evidence seems to be exaggerated by small-study effects.

Learning from experience and conditionalization

Bayesianism can be characterized as the following twofold position: (i) rational credences obey the probability calculus; (ii) rational learning, i.e., the updating of credences, is regulated by some form of conditionalization. While the formal aspect of various forms of conditionalization has been explored in detail, the philosophical application to learning from experience is still deeply problematic. Some philosophers have proposed to revise the epistemology of perception; others have provided new formal accounts of conditionalization that are more in line with how we learn from perceptual experience. The current investigation argues that Bayesian epistemology is still incomplete; the epistemology of perception and the epistemology of rational reasoning have not been reconciled.

A calculus of probability or belief? Neural underpinnings of social decision-making in a card game

For decades, the prefrontal cortex (PFC) has been the focus of social neuroscience research, specifically regarding its role in competitive social decision-making. However, the distinct contributions of PFC subregions when making strategic decisions involving multiple types of information (social, non-social, and mixed information) remain unclear. This study investigates decision-making strategies (pure probability calculation vs. mentalizing) and their neural representations using functional near-infrared spectroscopy (fNIRS) data collected during a two-person card game. We observed individual differences in information processing strategy, indicating that some participants relied more on probability than others. Overall, the use of pure probability decreased over time in favor of other types of information (e.g., mixed information), with this effect being more pronounced within-round trials than across rounds. In the brain, (1) the lateral PFC activates when decisions are driven by probability calculations; (2) the right lateral PFC responds to trial difficulty; and (3) the anterior medial PFC is engaged when decision-making involves mentalizing. Furthermore, neural synchrony, which reflects the real-time interplay between individuals' cognitive processes, did not consistently contribute to correct decisions and fluctuated throughout the experiment, suggesting a hierarchical mentalizing mechanism at work.

NECESSARY AND SUFFICIENT CONDITIONS FOR DOMINATION RESULTS FOR PROPER SCORING RULES

AbstractScoring 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.

What in Fact Proves the Violation of the Bell-Type Inequalities?

A. Peres constructed an example of particles entangled in the state of spin singlet. He claimed to have obtained the CHSH inequality and concluded that the violation of this inequality shows that in a measurement in which some variables are tested, other variables, not tested, have no defined value. In the present paper is proved that the correct conclusion of the violation of the CHSH inequality is different. It is proved that the classical calculus of probabilities of test results, obeying the Kolmogorov axioms, is unfit for the quantum formalism, dominated by probability amplitudes.

Процессная семантика комплексных чисел

Perspective approaches to imitation of natural thinking build on waveand quantumlike models, overcoming the limitations of Boolean logic and classical probability calculus. Further development of these methods requires an explanation of the reasons for these advantages. This paper shows that such a reason is the affinity of fundamental principles of natural cognition to the algebra of complex numbers. The vector structure of these numbers, in particular, matches the universal cognitive model of processes in nature. First, different phases of the complex number correspond to the phases "novelty", "goal", "action", and "result" of the life cycle of the considered activity. Real and imaginary axes of the complex plane then describe the factual and potential dimensions of the process's result. Simplicity and universality allow this model to avoid the major difficulties of modern approaches to predictive coding of cognitive information. These features open ways to algorithmic imitation of process thinking and the development of naturelike computational systems.

An Application of Relative Entropy in Structural Safety Analysis of Elastoplastic Beam under Fire Conditions

The main aim of this work is to propose a new algorithm of reliability assessment for steel civil engineering structures subjected to fire temperatures. This new algorithm is based upon the relative probabilistic entropy concept elaborated by Bhattacharyya, and this probabilistic distance is sought in-between extreme and admissible deformations of some structural beam subjected to higher temperatures. Similar to the First Order Reliability Method, this approach uses the first two probabilistic characteristics of the structural response, when structural output may be modelled with the use of Gaussian distribution. The probabilistic structural response is found here using hybrid computational technique–the Finite Element Method system ABAQUS with its fully coupled thermo-elastic analysis with 3D solid elements and probabilistic modules implemented in the computer algebra system MAPLE. The probabilistic response is determined via a triple stochastic analysis, which is based on the classical Monte-Carlo simulation, iterative generalized stochastic perturbation technique, and also using semi-analytical probabilistic calculus. Final determination of the relative entropy in the Serviceability Limit State of the given structure and its comparison with the results obtained using the FORM analysis enables to calibrate this new technique to numerical values proposed in the engineering designing codes. Hence, a more accurate probabilistic method may use some experimental-based admissible values included in the existing design of legal provisions.

Gambling on Claim Frequency: Loss Expectation and the Spread of Risks in the Making of Modern Insurance

Abstract In this paper, I argue that the two main principles underlying modern insurance business, i. e., loss expectation and the spread of risks, were not unknown in late-medieval and early-modern insurance practice. The former was embedded into premium rates and explains why in late-medieval and early-modern insurance contracts, premiums were expressed not as a bare sum but in the form of a rate. The latter was clearly formulated by Benedetto Cotrugli as he recommended to underwrite “continuously, & upon every ship, because the one offsets the other, & by pooling, [insurers] cannot but make a profit”. I focus on the principle of loss expectation and argue that underlying this principle there was a kind of commodification of uncertainty that both propelled and was supported by the semantic distinction of risk and danger. I finally sketch out some of the reasons why the basic principles underlying insurance business could find full exploitation only in modern society, and argue that if a statistical calculus of probability for insurance purposes was first carried out between the mid-17th century and the mid-18th century, it was not because of a deficit of ideas but because of a deficit of social structures. Compared to modern insurance business, late-medieval and early-modern insurance agreements were, in the end, a form of gambling based on some estimate of claim frequency.

The Binomial Distribution: Historical Origin and Evolution of Its Problem Situations

The increase in available probabilistic information and its usefulness for understanding the world has made it necessary to promote probabilistic literate citizens. For this, the binomial distribution is fundamental as one of the most important distributions for understanding random phenomena and effective decision making, and as a facilitator for the understanding of mathematical and probabilistic notions such as the normal distribution. However, to understand it effectively, it is necessary to consider how it has developed throughout history, that is, the components that gave it the form and meaning that we know today. To address this perspective, we identify the problem situations that gave origin to the binomial distribution, the operational and discursive practices developed to find solutions, and the conflicts that caused a leap in mathematical and probability heuristics, culminating in what is now known as the binomial distribution formula. As a result, we present five historical links to the binomial phenomenon where problem situations of increasing complexity were addressed: a case study using informal means (such as direct counting), the formalization of numerical patterns and constructs related to counting cases, specific probability calculus, the study and modeling of probability in variable or complex phenomena, and the use of the distribution formula as a tool to approaching notions such as the normal distribution. The periods and situations identified correspond to a required step in the design of binomial distribution learning from a historical epistemological perspective and when solving conflicts.

Empirical evidence for moral Bayesianism

ABSTRACT Many philosophers in the field of meta-ethics believe that rational degrees of confidence in moral judgments should have a probabilistic structure, in the same way as do rational degrees of belief. The current paper examines this position, termed “moral Bayesianism,” from an empirical point of view. To this end, we assessed the extent to which degrees of moral judgments obey the third axiom of the probability calculus, if P A ∩ B = 0 then P A ∪ B = P A + P B , known as finite additivity, as compared to degrees of beliefs on the one hand and degrees of desires on the other. Results generally converged to show that degrees of moral judgment are more similar to degrees of belief than to degrees of desire in this respect. This supports the adoption of a Bayesian approach to the study of moral judgments. To further support moral Bayesianism, we also demonstrated its predictive power. Finally, we discuss the relevancy of our results to the meta-ethical debate between moral cognitivists and moral non-cognitivists.

IS CAUSAL REASONING HARDER THAN PROBABILISTIC REASONING?

AbstractMany tasks in statistical and causal inference can be construed as problems of entailment in a suitable formal language. We ask whether those problems are more difficult, from a computational perspective, for causal probabilistic languages than for pure probabilistic (or “associational”) languages. Despite several senses in which causal reasoning is indeed more complex—both expressively and inferentially—we show that causal entailment (or satisfiability) problems can be systematically and robustly reduced to purely probabilistic problems. Thus there is no jump in computational complexity. Along the way we answer several open problems concerning the complexity of well-known probability logics, in particular demonstrating the ${\exists \mathbb {R}}$ -completeness of a polynomial probability calculus, as well as a seemingly much simpler system, the logic of comparative conditional probability.

A Note on Confluence in Typed Probabilistic Lambda Calculi

On the topic of probabilistic rewriting, there are several works studying both termination and confluence of different systems. While working with a lambda calculus modelling quantum computation, we found a system with probabilistic rewriting rules and strongly normalizing terms. We examine the effect of small modifications in probabilistic rewriting, affine variables, and strategies on the overall confluence in this strongly normalizing probabilistic calculus.

Rational Norms for Degreed Intention (and the Discrepancy between Theoretical and Practical Reason)

ABSTRACT Given the success of the formal approach, within contemporary epistemology, to understanding degreed belief, some philosophers have recently considered its extension to the challenge of understanding intention. According to them, (1) intentions can also admit of degrees, as beliefs do, and (2) these degreed states are all governed by the norms of the probability calculus, such that the rational norms for belief and for intention exhibit certain structural similarity. This paper, however, raises some worries about (2). It considers two schemes for representing degreed intention, and casts some doubt on applying probabilistic norms to degreed intention on each scheme. And it argues that the norms of Intention–Belief Consistency and Enkrasia cannot plausibly be turned into norms prescribing a simple correspondence relation between one’s degrees of belief and intention. The results suggest a potential structural discrepancy between the norms of theoretical and practical rationality—at least for degreed belief and intention. This potential discrepancy not only raises interesting questions about the intention–belief relation and rational norms for degreed intention. It also poses a new challenge for intention-based expressivism about normative judgment and cognitivism about practical rationality.

An application of probabilistic calculus to a problem of genetics

An application of probabilistic calculus to a problem of genetics

Carnap’s Logical Probability and Free Will Dilemma

Pondering the question of free will in the context of probability allows us to take a fresh look at a number of old problems. We are able to avoid deterministic entrapments and attempt to look at free will as an outcome of the entire decision-making system. In my paper, I will argue that free will should be considered in the context of a complex system of decisions, not individual cases. The proposed system will be probabilistic in character, so it will be embedded in the calculus of probability. To achieve the stated goal, I will refer to two areas of Carnap’s interest: the relationship between free will and determinism, and the probability-based decision-making system. First, I will present Carnap’s compatibilist position. On this basis, I will show how free will can be examined on deterministic grounds. Then I will present Carnap’s probabilistic project—the so-called logical interpretation of probability. In addition to presenting its characteristics and functionality, I will argue for its usefulness in the context of decision analysis and its immunity to problems associated with determinism. Finally, I will show how the two mentioned elements can be combined, as a result of which I will present a concept for a probabilistic analysis of free will. In this context, I will identify free will with the individual characteristics of the system. My main aim is to present the theme of free will in the light of a formal analysis based on probability rather than metaphysical assumptions.