scholarly journals The Price Equation Program: Simple Invariances Unify Population Dynamics, Thermodynamics, Probability, Information and Inference

Entropy ◽  
2018 ◽  
Vol 20 (12) ◽  
pp. 978 ◽  
Author(s):  
Steven Frank
Keyword(s):  
Probability Distributions ◽  
Mathematical Structure ◽  
Basic Structure ◽  
Price Equation ◽  
Physical Work ◽  
Maximum Increase ◽  
Universal Structure ◽  
Powerful Constraint ◽  
Fundamental Equations ◽  
Probability Information

The fundamental equations of various disciplines often seem to share the same basic structure. Natural selection increases information in the same way that Bayesian updating increases information. Thermodynamics and the forms of common probability distributions express maximum increase in entropy, which appears mathematically as loss of information. Physical mechanics follows paths of change that maximize Fisher information. The information expressions typically have analogous interpretations as the Newtonian balance between force and acceleration, representing a partition between the direct causes of change and the opposing changes in the frame of reference. This web of vague analogies hints at a deeper common mathematical structure. I suggest that the Price equation expresses that underlying universal structure. The abstract Price equation describes dynamics as the change between two sets. One component of dynamics expresses the change in the frequency of things, holding constant the values associated with things. The other component of dynamics expresses the change in the values of things, holding constant the frequency of things. The separation of frequency from value generalizes Shannon’s separation of the frequency of symbols from the meaning of symbols in information theory. The Price equation’s generalized separation of frequency and value reveals a few simple invariances that define universal geometric aspects of change. For example, the conservation of total frequency, although a trivial invariance by itself, creates a powerful constraint on the geometry of change. That constraint plus a few others seem to explain the common structural forms of the equations in different disciplines. From that abstract perspective, interpretations such as selection, information, entropy, force, acceleration, and physical work arise from the same underlying geometry expressed by the Price equation.

scholarly journals Simple unity among the fundamental equations of science

2020 ◽  
Vol 375 (1797) ◽  
pp. 20190351 ◽  
Author(s):  
Steven A. Frank
Keyword(s):  
Information Theory ◽  
Natural Selection ◽  
Frame Of Reference ◽  
Price Equation ◽  
Physical Work ◽  
Total Probability ◽  
Theme Issue ◽  
Total Change ◽  
Fundamental Equations ◽  
Change Concerns

The Price equation describes the change in populations. Change concerns some value, such as biological fitness, information or physical work. The Price equation reveals universal aspects for the nature of change, independently of the meaning ascribed to values. By understanding those universal aspects, we can see more clearly why fundamental mathematical results in different disciplines often share a common form. We can also interpret more clearly the meaning of key results within each discipline. For example, the mathematics of natural selection in biology has a form closely related to information theory and physical entropy. Does that mean that natural selection is about information or entropy? Or do natural selection, information and entropy arise as interpretations of a common underlying abstraction? The Price equation suggests the latter. The Price equation achieves its abstract generality by partitioning change into two terms. The first term naturally associates with the direct forces that cause change. The second term naturally associates with the changing frame of reference. In the Price equation’s canonical form, total change remains zero because the conservation of total probability requires that all probabilities invariantly sum to one. Much of the shared common form for the mathematics of different disciplines may arise from that seemingly trivial invariance of total probability, which leads to the partitioning of total change into equal and opposite components of the direct forces and the changing frame of reference. This article is part of the theme issue ‘Fifty years of the Price equation’.

scholarly journals Structure of social attitudes based on lexical approach in Serbian-speaking area

Sociologija ◽  
2011 ◽  
Vol 53 (2) ◽  
pp. 195-212 ◽  
Author(s):  
Boban Petrovic ◽  
Janko Medjedovic
Keyword(s):  
Political Correctness ◽  
Basic Structure ◽  
Social Attitudes ◽  
Personality Psychology ◽  
Western Democracy ◽  
Sources Of Authority ◽  
English Speaking ◽  
Universal Structure ◽  
Lexical Approach ◽  
High Level

Although usually applied in the field of personality psychology, in the last decade there were attempts of applying the lexical paradigm in the studies social attitudes studies. One of those attempts was made by Saucier (2000), who included and analyzed all the words ending with the suffix ?-ism?. The product of this analyze is a two-form instrument called "Survey of Dictionary-Based Isms (SDI)", with its long, 40-item version, and brief, 28-item version. This instrument measures four main dimensions of basic social attitudes: alphaisms (traditional and religious sources of authority), betaisms (dismissing political correctness), gammaisms (believing in Western democracy) and deltaisms (personal spirituality). Until now, this instrument was only used in English-speaking area, and therefore the objective of this research was evaluation and validation of Saucier?s basic structure of social attitudes model in the Serbian-speaking area. In this research, conducted on the sample of 253 participants, both sexes, average age of 39.3 years (SD=14.9), a slightly shortened version of Saucier?s 24-item questionnaire was used. The results of this study contribute to the hypothesis of a universal structure of basic social attitudes: the factor analysis extracted four factors, which correlate from moderate to high level with the original dimensions. However, the structure and content of the factors pointed to a strong cultural influence on the forming and shaping the basic social attitudes. Practically, only the first factor, Religiosity, is a full replication of the original alpha factor. Other items built the factors somewhat different from the original: Hedonism, Rational Spirituality and Nationalism. The results show a better fitness of the model obtained in this study for the Serbian-speaking area compared to the original one.

Computer Aided Tolerance Analysis of Parts and Assemblies

1989 ◽  
Author(s):  
R. T. Scott ◽  
G. A. Gabriele
Keyword(s):  
Real World ◽  
Probability Distributions ◽  
Three Dimensional ◽  
Tolerance Analysis ◽  
The Real ◽  
Assembly Tolerance ◽  
Computer Aided ◽  
Probability Information ◽  
Exact Constraint

Abstract An exact constraint scheme based on the physical contacting constraints of real part mating features is used to represent the process of assembling the parts. To provide useful probability information about how assembly dimensions are distributed when the parts are assembled as intended, the real world constraints that would prevent interference are ignored. This work addresses some limitations in the area of three dimensional assembly tolerance analysis. As a result of this work, the following were demonstrated: 1. Assembly of parts whose assembly mating features are subjected to variation; 2. Assemble parts using a real world set of exact constraints; 3. Provide probability distributions of assembly dimensions.

scholarly journals The Price equation and the unity of social evolution theory

2019 ◽  
Author(s):  
Jussi Lehtonen
Keyword(s):  
Group Selection ◽  
Social Evolution ◽  
Causal Structure ◽  
Mathematical Structure ◽  
Price Equation ◽  
Alternative Formulation ◽  
Behavioural Ecology ◽  
Evolution Theory ◽  
Weak Selection ◽  
Modelling Methods

The Price equation has been entangled with social evolution theory from the start. It has been used to derive the most general versions of kin selection theory, and Price himself produced a multilevel equation which provides an alternative formulation of social evolution theory, dividing selection into components between and within groups. In this sense, the Price equation forms a basis for both kin and group selection, so often pitted against each other in the literature. Contextual analysis and the neighbour approach are prominent alternatives for analysing group selection. I discuss these four approaches to social evolution theory and their connections to the Price equation, focusing on their similarities and common mathematical structure. Despite different notations and modelling traditions, all four approaches are ultimately linked by a common set of mathematical components, revealing their underlying unity in a transparent way. The Price equation can similarly be used in the derivation of streamlined, weak selection social evolution modelling methods. These weak selection models are practical and powerful methods for constructing models in evolutionary and behavioural ecology, they can clarify the causal structure of models, and can be easily converted between the four social evolution approaches just like their regression counterparts.

scholarly journals The Fundamental Equations of Change in Statistical Ensembles and Biological Populations

Entropy ◽  
2020 ◽  
Vol 22 (12) ◽  
pp. 1395
Author(s):  
Steven A. Frank ◽  
Frank J. Bruggeman
Keyword(s):  
Probability Distribution ◽  
General Equation ◽  
Recent Article ◽  
Rate Of Change ◽  
The Other ◽  
Price Equation ◽  
Abstract Form ◽  
Statistical Ensembles ◽  
Fundamental Equations ◽  
Theoretical Results

A recent article in Nature Physics unified key results from thermodynamics, statistics, and information theory. The unification arose from a general equation for the rate of change in the information content of a system. The general equation describes the change in the moments of an observable quantity over a probability distribution. One term in the equation describes the change in the probability distribution. The other term describes the change in the observable values for a given state. We show the equivalence of this general equation for moment dynamics with the widely known Price equation from evolutionary theory, named after George Price. We introduce the Price equation from its biological roots, review a mathematically abstract form of the equation, and discuss the potential for this equation to unify diverse mathematical theories from different disciplines. The new work in Nature Physics and many applications in biology show that this equation also provides the basis for deriving many novel theoretical results within each discipline.

scholarly journals The Price equation and the unity of social evolution theory

2020 ◽  
Vol 375 (1797) ◽  
pp. 20190362 ◽  
Author(s):  
Jussi Lehtonen
Keyword(s):  
Group Selection ◽  
Social Evolution ◽  
Causal Structure ◽  
Mathematical Structure ◽  
Price Equation ◽  
Alternative Formulation ◽  
Evolution Theory ◽  
Weak Selection ◽  
Theme Issue ◽  
Modelling Methods

The Price equation has been entangled with social evolution theory from the start. It has been used to derive the most general versions of kin selection theory, and Price himself produced a multilevel equation that provides an alternative formulation of social evolution theory, dividing selection into components between and within groups. In this sense, the Price equation forms a basis for both kin and group selection, so often pitted against each other in the literature. Contextual analysis and the neighbour approach are prominent alternatives for analysing group selection. I discuss these four approaches to social evolution theory and their connections to the Price equation, focusing on their similarities and common mathematical structure. Despite different notations and modelling traditions, all four approaches are ultimately linked by a common set of mathematical components, revealing their underlying unity in a transparent way. The Price equation can similarly be used in the derivation of streamlined, weak selection social evolution modelling methods. These weak selection models are practical and powerful methods for constructing models in evolutionary and behavioural ecology; they can clarify the causal structure of models, and can be easily converted between the four social evolution approaches just like their regression counterparts. This article is part of the theme issue ‘Fifty years of the Price equation’.

scholarly journals Towards a Pricean foundation for cultural evolutionary theory

2021 ◽  
Author(s):  
Lorenzo Baravalle ◽  
Victor J. Luque
Keyword(s):  
Evolutionary Theory ◽  
Cultural Change ◽  
Explanatory Power ◽  
Fundamental Equation ◽  
Price Equation ◽  
Dual Inheritance Theory ◽  
Heuristic Device ◽  
Cultural Evolutionary ◽  
Fundamental Equations ◽  
Dual Inheritance

The Price equation is currently considered one of the fundamental equations – or even the fundamental equation – of evolution. In this article, we explore the role of this equation within cultural evolutionary theory. More specifically, we use it to account for the explanatory power and the theoretical structure of a certain generalised version of dual-inheritance theory. First, we argue that, in spite of not having a definite empirical content, the Price equation offers a suitable formalisation of the processes of cultural evolution, and provides a powerful heuristic device for discovering the actual causes of cultural change and accumulation. Second, we argue that, as a consequence of this, a certain version of the Price equation is the fundamental law of cultural evolutionary theory. In order to support this claim, we sketch the ideal structure of dual-inheritance theory and we stress the unificatory role that the Price equation plays in it. 

scholarly journals Probabilistic and Nonprobabilistic Sensitivity Analyses of Uncertain Parameters

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Sheng-En Fang ◽  
Qiu-Hu Zhang ◽  
Bao Zhang ◽  
Xiao-Hua Zhang
Keyword(s):  
Probability Distributions ◽  
Probabilistic Method ◽  
Sensitivity Analyses ◽  
Uncertain Parameters ◽  
Stochastic Response ◽  
First Order ◽  
Simple Implementation ◽  
Probability Information ◽  
Main Effects ◽  
Mathematical Derivation

Parameter sensitivity analyses have been widely applied to industrial problems for evaluating parameter significance, effects on responses, uncertainty influence, and so forth. In the interest of simple implementation and computational efficiency, this study has developed two sensitivity analysis methods corresponding to the situations with or without sufficient probability information. The probabilistic method is established with the aid of the stochastic response surface and the mathematical derivation proves that the coefficients of first-order items embody the parameter main effects on the response. Simultaneously, a nonprobabilistic interval analysis based method is brought forward for the circumstance when the parameter probability distributions are unknown. The two methods have been verified against a numerical beam example with their accuracy compared to that of a traditional variance-based method. The analysis results have demonstrated the reliability and accuracy of the developed methods. And their suitability for different situations has also been discussed.

scholarly journals On New Families of Integrals in Analytical Studies of Superconductors within the Conformal Transformation Method

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Ryszard Gonczarek ◽  
Mateusz Krzyzosiak ◽  
Adam Gonczarek ◽  
Lucjan Jacak
Keyword(s):  
Conformal Transformation ◽  
Mathematical Structure ◽  
Transformation Method ◽  
Variable Density ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Quantitative Characteristics ◽  
Analytical Formulas ◽  
Fundamental Equations ◽  
New Generation

We show that, by applying the conformal transformation method, strongly correlated superconducting systems can be discussed in terms of the Fermi liquid with a variable density of states function. Within this approach, it is possible to formulate and carry out purely analytical study based on a set of fundamental equations. After presenting the mathematical structure of thes-wave superconducting gap and other quantitative characteristics of superconductors, we evaluate and discuss integrals inherent in fundamental equations describing superconducting systems. The results presented here extend the approach formulated by Abrikosov and Maki, which was restricted to the first-order expansion. A few infinite families of integrals are derived and allow us to express the fundamental equations by means of analytical formulas. They can be then exploited in order to find quantitative characteristics of superconducting systems by the method of successive approximations. We show that the results can be applied in studies of high-Tcsuperconductors and other superconducting materials of the new generation.

New classes of integrals inherent in the mathematical structure of extended equations describing superconducting systems

2015 ◽  
Vol 29 (17) ◽  
pp. 1550117 ◽  
Author(s):  
Ryszard Gonczarek ◽  
Mateusz Krzyzosiak ◽  
Adam Gonczarek ◽  
Lucjan Jacak
Keyword(s):  
Mathematical Structure ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Superconducting Gap ◽  
S Wave ◽  
First Order ◽  
Order Expansion ◽  
Quantitative Characteristics ◽  
Fundamental Equations ◽  
New Generation

In this paper, we discuss the mathematical structure of the s-wave superconducting gap and other quantitative characteristics of superconducting systems. In particular, we evaluate and discuss integrals inherent in fundamental equations describing superconducting systems. The results presented here extend the approach formulated by Abrikosov and Maki, which was restricted to the first-order expansion. A few infinite families of integrals are derived and allow us to express the fundamental equations by means of analytic formulas. They can be then exploited in order to find some quantitative characteristics of superconducting systems by the method of successive approximations. We show that the results can be applied to some modern formalisms in order to study high-Tc superconductors and other superconducting materials of the new generation.

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