We investigate the discrete (finite) case of the Popper–Renyi theory of conditional probability, introducing discrete conditional probabilistic models for knowledge and conditional belief, and comparing them with the more standard plausibility models. We also consider a related notion, that of safe belief, which is a weak (non-negatively introspective) type of “knowledge”. We develop a probabilistic version of this concept (“degree of safety”) and we analyze its role in games. We completely axiomatize the logic of conditional belief, knowledge and (...) safe belief over conditional probabilistic models. We develop a theory of probabilistic dynamic belief revision, introducing probabilistic “action models” and proposing a notion of probabilistic update product, that comes together with appropriate reduction laws. (shrink)

In this paper I combine the dynamic epistemic logic ofGerbrandy (1999) with the probabilistic logic of Fagin and Halpern (1994). The resultis a new probabilistic dynamic epistemic logic, a logic for reasoning aboutprobability, information, and information change that takes higher orderinformation into account. Probabilistic epistemic models are defined, and away to build them for applications is given. Semantics and a proof systemis presented and a number of examples are discussed, including the MontyHall Dilemma.

This paper aims to extend in two directions the probabilistic dynamic epistemic logic provided in Kooi’s paper (J Logic Lang Inform 12(4):381–408, 2003) and to relate these extensions to ones made in van Benthem et al. (Proceedings of LOFT’06. Liverpool, 2006). Kooi’s probabilistic dynamic epistemic logic adds to probabilistic epistemic logic sentences that express consequences of public announcements. The paper (van Benthem et al., Proceedings of LOFT’06. Liverpool, 2006) extends (Kooi, J Logic Lang Inform 12(4):381–408, 2003) to (...) using action models, but in both papers, the probabilities are discrete, and are defined on trivial σ -algebras over finite sample spaces. The first extension offered in this paper is to add a previous-time operator to a probabilistic dynamic epistemic logic similar to Kooi’s in (J Logic Lang Inform 12(4):381–408, 2003). The other is to involve non-trivial σ -algebras and continuous probabilities in probabilistic dynamic epistemic logic. (shrink)

This paper studies Aumann’s agreeing to disagree theorem from the perspective of dynamic epistemic logic. This was first done by Dégremont and Roy (J Phil Log 41:735–764, 2012) in the qualitative framework of plausibility models. The current paper uses a probabilistic framework, and thus stays closer to Aumann’s original formulation. The paper first introduces enriched probabilistic Kripke frames and models, and various ways of updating them. This framework is then used to prove several agreement theorems, which are natural (...) formalizations of Aumann’s original result. Furthermore, a sound and complete axiomatization of a dynamic agreement logic is provided, in which one of these agreement theorems can be derived syntactically. These technical results are used to show the importance of explicitly representing the dynamics behind the agreement theorem, and lead to a clarification of some conceptual issues surrounding the agreement theorem, in particular concerning the role of common knowledge. The formalization of the agreement theorem thus constitutes a concrete example of the so-called dynamic turn in logic. (shrink)

Critically growing problems of fundamental science organisation and content are analysed with examples from physics and emerging interdisciplinary fields. Their origin is specified and new science structure (organisation and content) is proposed as a unified solution.

Two distinct conceptions for the relation between reversible, time-reversal invariant laws of nature and the irreversible behavior of physical systems are outlined. The standard, extrinsic concept of irreversibility is based on the notion of an open system interacting with its environment. An alternative, intrinsic concept of irreversibility does not explicitly refer to any environment at all. Basic aspects of the two concepts are presented and compared with each other. The significance of the terms extrinsic and intrinsic is discussed.

Successful applications of hierarchical complexity to the behaviors of organisms, animals and humans, and social entities evidence the scaling properties of self-similarity, thus the bounded fractal characteristics of orders of hierarchical complexity. The theory specifies an identical sequence of discrete-state transition steps required from each stage of performance to the next. It repeats at all scales. Tasks nested within the step sequence evidence self-similarity with the orders of complexity. This model introduces questions about noise categories when system tasks are (...) fully accounted for, dependent, self-similar, and measurable. Ubiquitous transition steps are inherent dynamics of evolution. (shrink)

Excessive use of social media is a developing concern in the twenty-first century. This issue needs to be addressed before it has any more significant consequences than what we are currently experiencing. As a preventive technique, advertisements and awareness-raising campaigns about the detrimental impact of digital technologies are used. The application of novel mathematical techniques and terminologies in this field of study will have significant potential to enhance healthy living by preventing certain ailments. This is the most compelling justification for (...) conducting a new study with the most up-to-date techniques at our disposal. This study investigates clear and concise transmission in order to generate a deterministic mathematical model of social media addiction S M A using the fractal-fractional derivative operator. Also, the analysis of the S M A model in terms of the invariant domain, the existence of a positive invariant solution, and equilibria assumptions are stated in a detailed manner. Besides that, the basic reproduction number ℝ 0 < 1 is computed, demonstrating that the proposed methodology is more efficacious. The Atangana-Baleanu FF differential operators are recently defined in FF differential operators that are applied to characterize the S M A model’s mathematical algorithm. We investigated the numerical behaviour of the S M A in three ways: changing the fractional-order α as well as the fractal-dimension ℘ ; changing α while keeping ℘ constant; and changing ℘ while keeping α constant. Our examined visualizations and simulation studies using MATLAB for the numerical modelling of the aforementioned system showed that the novel developed Atangana-Baleanu FF differential operators produce remarkable results when compared to the classical frame. (shrink)

We investigate a basic probabilistic dynamic semantics for a fragment containing conditionals, probability operators, modals, and attitude verbs, with the aim of shedding light on the prospects for adding probabilistic structure to models of the conversational common ground.

PROBABILISTIC APPROACH TO EPISTEMIC MODALS IN THE FRAMEWORK OF DYNAMIC SEMANTICS In dynamic semantics meaning of a statement is not equated with its truth conditions but with its context change potential. It has also been claimed that dynamic framework can automatically account for certain paradoxes that involve epistemic modals, such as the following one: it seems odd and incoherent to claim: (1) “It is raining and it might not rain”, whereas claiming (2) “It might not rain and it is (...) raining” does not seem equally odd (Yalcin, 2007). Nevertheless, it seems that it cannot capture the fact that statement (2) seems odd as well, even though not as odd as the statement (1) (Gauker, 2007). I will argue that certain probabilistic extensions to the dynamic model can account for this subtlety of our linguistic intuitions and represent if not an improved than at least an alternative framework for capturing the way contexts are updated and beliefs revised with uncertain information. (shrink)

Fractal dynamics is a unique, systems-based approach of looking at and thinking about organizations. In this view, the organization is viewed as part of a pervasive organizational fractal. Many astounding properties such as fractal completion, fractal influence and fractal universality emerge in fractal space. These offer new insight into the fundamental dynamics animating all organizations. Routine management tasks such as strategy formulation, cost cutting, process redesign, product formulation, management of change and leadership development can (...) be conducted anew with powerful and innovative insights, unavailable when management is approached traditionally through a linear and sequential thinking approach. This article focuses primarily on the rationale for employing fractal dynamics, and on the search for the pervasive organizational fractal that animates all organizations. A brief example of how fractal dynamics offers insights in the case of management of change is also provided. (shrink)

Climate dynamics and trends have significant environmental and socioeconomic impacts; however, in the Benin Republic, they are generally studied with diverse statistical methods ignoring the nonstationarity, nonlinearity, and self-similarity characteristics contained in precipitation time series. This can lead to erroneous conclusions and an unclear understanding of climatic dynamics. Based on daily precipitation data observed in the six synoptic stations of Benin Republic, in the period from 1951 to 2010, we have proposed determining the local trends of precipitations, investigating precipitation nonlinear (...) dynamics, and assessing climatic shift in the study period by Ensemble Empirical Mode Decomposition and Multifractal Detrended Fluctuation Analysis method. To overcome the detrending issue in the standard MFDFA method, the EEMD algorithm is embedded into the MFDFA. The study period is subdivided into three subperiods: 1951–1970, 1971–1990, and 1991–2010. Intrinsic Mode Functions are obtained according to the climatic region in which the stations are located. Results show that precipitation variation is significantly governed by the five first IMFs, in which oscillation periods vary from 1 to 25 days. The trend curves decrease at all the synoptic stations, and their slope values vary accordingly to the subperiods. Referring to the values of the multifractal spectrum parameters, α 0, and the width of the spectrum w, consistent changes are observed regardless of the subperiods and the concerned stations. The spatial and temporal variability of precipitation indicates that the multifractal properties are good indicators for assessing changes in precipitation dynamics, and the changes in their features could be explained by the global change than by the local climate variation. Despite the observed differences in multifractal spectra properties from the three subperiods, it is not possible to verify the subdivision of 1951–2010 in three subperiods as it is done by previous studies in West Africa. Our findings can be used in the validation of global and regional climate models since a valid model should explain empirically detected scaling properties in observed data. (shrink)

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 generated 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 beliefs. (shrink)

The aim of the present paper is to give a purely probabilistic account for the approach to equilibrium of classical and quantum gas. The probability function used is classical. The probabilistic dynamics describes the evolution of the state of the gas due to unary and binary collisions. A state change amounts to a destruction in a state and the creation in another state. Transitions probabilities are splittled into destructions terms, denoting the random choice of the colliding particle(s), and (...) creation terms, describing the allocation of the same particle(s) on the final state(s). While the destruction term is the same for all types of particles, the creation one depends upon a parameter bound to the interpraticle correlation. The transition probabilities give rise to a homogeneous Markov chain. The equilibrium distributions satisfy the principle of detailed balance. Relaxation times depend upon the interparticle correlation. Relationships with the Ehrenfest urn model, Brillouin unified method, ensemble interpretation, and quantum H-theorem are considered too. (shrink)

Our universe, science reveals, began in utter simplicity, then evolved into burgeoning complexity. Starting with subatomic particles, dissimilar entities formed associations—binding, bonding, growing, branching, catalyzing, cooperating—as “self” joined “other” following universal laws with names such as gravity, chemical attraction, and natural selection. Ultimately life arose in a world of dynamic organic chemistry, and complexity exploded with wondrous new potential. Fast forward to human evolution, and a tension that had existed for billions of years now played out in an unprecedented arena (...) of conscious calculation and cultural diversity. Cooperation interleaving with competition; intimacy oscillating with integrity—we dwell in a world where yin meets yang in human affairs on many levels. In The Fractal Self, John Culliney and David Jones uncover surprising intersections between science and philosophy. Connecting evidence from evolutionary science with early insights of Daoist and Buddhist thinkers, among others, they maintain that sagely behavior, envisioned in these ancient traditions, represents a pinnacle of human achievement emerging out of our evolutionary heritage. They identify an archetype, “the fractal self,” a person in any walk of life who cultivates a cooperative spirit. A fractal self is a sage in training, who joins others in common cause, leads from within, and achieves personal satisfaction in coordinating smooth performance of the group, team, or institution in which he or she is embedded. Fractal selves commonly operate with dedication and compassionate practice in the service of human society or in conserving our planet. But the competitive side of human nature is susceptible to greed and aggression. Self-aggrandizement, dictatorial power, and ego-driven enforcement of will are the goals of those following a self-serving path—individuals the authors identify as antisages. Terrorist leaders are an especially murderous breed, but aggrandizers can be found throughout business, religion, educational institutions, and governments. Humanity has reached an existential tipping point: will the horizon already in view expand with cooperative progress toward godlike emergent opportunities or contract in the thrall of corrupt oligarchs and tribal animosities? We have brought ourselves to a chaotic edge between immense promise and existential danger and are even now making our greatest choice. (shrink)

This paper is the third and final one in a sequence of three. All three papers emphasize that a proposition can be justified by an infinite regress, on condition that epistemic justification is interpreted probabilistically. The first two papers showed this for one-dimensional chains and for one-dimensional loops of propositions, each proposition being justified probabilistically by its precursor. In the present paper we consider the more complicated case of two-dimensional nets, where each "child" proposition is probabilistically justified by two "parent" (...) propositions. Surprisingly, it turns out that probabilistic justification in two dimensions takes on the form of Mandelbrot's iteration. Like so many patterns in nature, probabilistic reasoning might in the end be fractal in character. (shrink)

We examine two connectionist networks—a fractal learning neural network (FLNN) and a Simple Recurrent Network (SRN)—that are trained to process center-embedded symbol sequences. Previous work provides evidence that connectionist networks trained on infinite-state languages tend to form fractal encodings. Most such work focuses on simple counting recursion cases (e.g., anbn), which are not comparable to the complex recursive patterns seen in natural language syntax. Here, we consider exponential state growth cases (including mirror recursion), describe a new training scheme (...) that seems to facilitate learning, and note that the connectionist learning of these cases has a continuous metamorphosis property that looks very different from what is achievable with symbolic encodings. We identify a property—ragged progressive generalization—which helps make this difference clearer. We suggest two conclusions. First, the fractal analysis of these more complex learning cases reveals the possibility of comparing connectionist networks and symbolic models of grammatical structure in a principled way—this helps remove the black box character of connectionist networks and indicates how the theory they support is different from symbolic approaches. Second, the findings indicate the value of future, linked mathematical and empirical work on these models—something that is more possible now than it was 10 years ago. (shrink)

Feminists have long been aware that the notion of women is problematic and using it uncritically without further qualifications leads to exclusions. In the article, I argue that the source of these problems lies in the understanding of concepts as static and clearly defined. I deploy Hegel’s idea of syllogism to define dynamic concepts, which I term ‘fractal concepts’ because of their complexity and constant development. In such structures the balance between the universal, the particular and the individual is (...) maintained and, consequently, every woman is essential for the definition of such a non-essentialist notion of women. I propose that an epistemological interpretation of the Hegelian idea of recognition can be used to describe the development of fractal concepts in the confrontation of different perspectives. It also explains the obstruction of this process by the desire for domination and control, which results in the appropriation of group identities by their privileged members. I situate the logical and political framework developed in the paper in the context of feminist controversies around the concept of woman and argue that it provides us with a comprehensive account of intersectionality as well as an explicative solution to the regress problem. (shrink)

A summary of topics and theoretical approaches to dynamical systems and psychology. Includes chapters on physical systems, self-organization, state space and dimensionality, networks, neurodynamics, fractals and how such concepts help to explain cognition and the mind.

Brian Skyrms (1987, 1990, 1993, 1997) has discussed the role of dynamic coherence arguments in the theory of personal or subjective probability. In particular, Skryms (1997) both reviews and discusses the utility of martingale arguments in establishing the convergence of beliefs within the context of radical probabilism. The classical martingale converence theorem, however, assumes the countable additivity of the underlying probability measure; an assumption rejected by some subjectivists such as Bruno de Finetti (see, e.g., de Finetti 1930 and 1972). This (...) brief note has a very modest goal: to briefly consider the extent to which Skyrms’s argument can be extended to the finitely additive case. (shrink)

A formulation of probabilistic causality is given in terms of the theory of abstract dynamical systems. Causal factors are identified as invariants of motion of a system. Repetition of an experiment leads to the notion of stationarity, and causal factors yield a decomposition of the stationary probability law of the experiment into ergodic components. In these, statistical behaviour is uniform. Control of identified causal factors leads to a corresponding statistical law for the events, which is offered as a (...) notion of probabilistic causality. After a suggestion by Feller, randomization is identified as mixing, formulated in above terms. (shrink)

We propose a new model for forming and revising beliefs about unknown probabilities. To go beyond what is known with certainty and represent the agent’s beliefs about probability, we consider a plausibility map, associating to each possible distribution a plausibility ranking. Beliefs are defined as in Belief Revision Theory, in terms of truth in the most plausible worlds. We consider two forms of conditioning or belief update, corresponding to the acquisition of two types of information: learning observable evidence obtained by (...) repeated sampling from the unknown distribution; and learning higher-order information about the distribution. The first changes only the plausibility map, but leaves the given set of possible distributions essentially unchanged; the second rules out some distributions, thus shrinking the set of possibilities, without changing their plausibility ordering.. We look at stability of beliefs under either of these types of learning, defining two related notions, as well as a measure of the verisimilitude of a given plausibility model. We prove a number of convergence results, showing how our agent’s beliefs track the true probability after repeated sampling, and how she eventually gains in a sense knowledge of that true probability. Finally, we sketch the contours of a dynamic doxastic logic for statistical learning. (shrink)

When we perceive the external world, our brain has to deal with the incompleteness and uncertainty associated with sensory inputs, memory and prior knowledge. In theoretical neuroscience probabilistic approaches have received a growing interest recently, as they account for the ability to reason with incomplete knowledge and to efficiently describe perceptive and behavioral tasks. How can the probability distributions that need to be estimated in these models be represented and processed in the brain, in particular at the single cell (...) level? We consider the basic function carried out by photoreceptor cells which consists in detecting the presence or absence of light. We give a system-level understanding of the process of phototransduction based on a bayesian formalism: we show that the process of phototransduction is equivalent to a temporal probabilistic inference in a Hidden Markov Model (HMM), for estimating the presence or absence of light. Thus, the biochemical mechanisms of phototransduction underlie the estimation of the current state probability distribution of the presence of light. A classical descriptive model describes the interactions between the different molecular messengers, ions, enzymes and channel proteins occurring within the photoreceptor by a set of nonlinear coupled differential equations. In contrast, the probabilistic HMM model is described by a discrete recurrence equation. It appears that the binary HMM has a general solution in the case of constant input. This allows a detailed analysis of the dynamics of the system. The biochemical system and the HMM behave similarly under steady-state conditions. Consequently a formal equivalence can be found between the biochemical system and the HMM. Numerical simulations further extend the results to the dynamic case and to noisy input. All in all, we have derived a probabilistic model equivalent to a classical descriptive model of phototransduction, which has the additional advantage of assigning a function to phototransduction. The example of phototransduction shows how simple biochemical interactions underlie simple probabilistic inferences. (shrink)

Current dynamic-epistemic logics model different types of information change in multi-agent scenarios. We generalize these logics to a probabilistic setting, obtaining a calculus for multi-agent update with three natural slots: prior probability on states, occurrence probabilities in the relevant process taking place, and observation probabilities of events. To match this update mechanism, we present a complete dynamic logic of information change with a probabilistic character. The completeness proof follows a compositional methodology that applies to a much larger class (...) of dynamic-probabilistic logics as well. Finally, we discuss how our basic update rule can be parameterized for different update policies, or learning methods. (shrink)

Famous results by David Lewis show that plausible-sounding constraints on the probabilities of conditionals or evaluative claims lead to unacceptable results, by standard probabilistic reasoning. Existing presentations of these results rely on stronger assumptions than they really need. When we strip these arguments down to a minimal core, we can see both how certain replies miss the mark, and also how to devise parallel arguments for other domains, including epistemic “might,” probability claims, claims about comparative value, and so on. (...) A popular reply to Lewis's results is to claim that conditional claims, or claims about subjective value, lack truth conditions. For this strategy to have a chance of success, it needs to give up basic structural principles about how epistemic states can be updated—in a way that is strikingly parallel to the commitments of the project of dynamic semantics. (shrink)

Current dynamic-epistemic logics model different types of information change in multi-agent scenarios. We generalize these logics to a probabilistic setting, obtaining a calculus for multi-agent update with three natural slots: prior probability on states, occurrence probabilities in the relevant process taking place, and observation probabilities of events. To match this update mechanism, we present a complete dynamic logic of information change with a probabilistic character. The completeness proof follows a compositional methodology that applies to a much larger class (...) of dynamic-probabilistic logics as well. Finally, we discuss how our basic update rule can be parameterized for different update policies, or learning methods. (shrink)

In recent years, a number of theorists have claimed that beliefs about probability are transparent. To believe probably p is simply to have a high credence that p. In this paper, I prove a variety of triviality results for theses like the above. I show that such claims are inconsistent with the thesis that probabilistic modal sentences have propositions or sets of worlds as their meaning. Then I consider the extent to which a dynamic semantics for probabilistic modals (...) can capture theses connecting belief, certainty, credence, and probability. I show that although a dynamic semantics for probabilistic modals does allow one to validate such theses, it can only do so at a cost. I prove that such theses can only be valid if probabilistic modals do not satisfy the axioms of the probability calculus. (shrink)

The scheme of abstract dynamical systems will represent repetitive experimentation: There is a basic space of events X1 and the denumerable product … contains all possible sequences of events x = (x1, x2, … ). There are projections qn which give the nth member of x: qn (x) = xn. A transformation T is defined over X by the equation qn (Tx)= q n+1 (x). It removes the sequence by one step, T(x1,x2,…) = (x2,x3,…) and is known as the (...) shift transformation. It comes as an abstraction of the dynamical transformations of classical theories. Here it represents the performance of ‘the next’ experiment. (shrink)

Dynamic approaches to understanding probability in the non-fundamental sciences turn on certain properties of physical processes that are apt to produce “probabilistically patterned” outcomes. The dynamic properties on their own, however, seem not quite sufficient to explain the patterns; in addition, some sort of assumption about initial conditions must be made, an assumption that itself typically takes a probabilistic form. How should such a posit be understood? That is the problem of initial conditions. Reichenbach, in his doctoral dissertation, floated (...) a Kantian solution to the problem. In this paper I provide a Reichenbachian alternative. (shrink)

The complex-systems approach to cognitive science seeks to move beyond the formalism of information exchange and to situate cognition within the broader formalism of energy flow. Changes in cognitive performance exhibit a fractal (i.e., power-law) relationship between size and time scale. These fractal fluctuations reflect the flow of energy at all scales governing cognition. Information transfer, as traditionally understood in the cognitive sciences, may be a subset of this multiscale energy flow. The cognitive system exhibits not just a (...) single power-law relationship between fluctuation size and time scale but actually exhibits many power-law relationships, whether over time or space. This change in fractal scaling, that is, multifractality, provides new insights into changes in energy flow through the cognitive system. We survey recent findings demonstrating the role of multifractality in (a) understanding atypical developmental outcomes, and (b) predicting cognitive change. We propose that multifractality provides insights into energy flows driving the emergence of cognitive structure. (shrink)

Does the philosophy of Radical Probabilism have enough structure to enable it to address fundamental epistemological questions? The requirement of dynamic coherence provides the structure for radical probabilist epistemology. This structure is sufficient to establish (i) the value of knowledge and (ii) long run convergence of degrees of belief.

In this paper, chaotic dynamics of a mixed Rayleigh–Liénard oscillator driven by parametric periodic damping and external excitations is investigated analytically and numerically. The equilibrium points and their stability evolutions are analytically analyzed, and the transitions of dynamical behaviors are explored in detail. Furthermore, from the Melnikov method, the analytical criterion for the appearance of the homoclinic chaos is derived. Analytical prediction is tested against numerical simulations based on the basin of attraction of initial conditions. As a result, it (...) is found that for ω = ν, the chaotic region decreases and disappears when the amplitude of the parametric periodic damping excitation increases. Moreover, increasing of F 1 and F 0 provokes an erosion of the basin of attraction and a modification of the geometrical shape of the chaotic attractors. For ω ≠ ν and η = 0.8, the fractality of the basin of attraction increases as the amplitude of the external periodic excitation and constant term increase. Bifurcation structures of our system are performed through the fourth-order Runge–Kutta ode 45 algorithm. It is found that the system displays a remarkable route to chaos. It is also found that the system exhibits monostable and bistable oscillations as well as the phenomenon of coexistence of attractors. (shrink)

Although its roots can be traced to the 19th century, progress in the study of nonlinear dynamical systems has taken off in the last 30 years. While pertinent source material exists, it is strewn about the literature in mathematics, physics, biology, economics, and psychology at varying levels of accessibility. A compendium research methods reflecting the expertise of major contributors to NDS psychology, Nonlinear Dynamical Systems Analysis for the Behavioral Sciences Using Real Data examines the techniques proven to be (...) the most useful in the behavioral sciences. The editors have brought together constructive work on new practical examples of methods and application built on nonlinear dynamics. They cover dynamics such as attractors, bifurcations, chaos, fractals, catastrophes, self-organization, and related issues in time series analysis, stationarity, modeling and hypothesis testing, probability, and experimental design. The analytic techniques discussed include several variants of the fractal dimension, several types of entropy, phase-space and state-space diagrams, recurrence analysis, spatial fractal analysis, oscillation functions, polynomial and Marquardt nonlinear regression, Markov chains, and symbolic dynamics. The book outlines the analytic requirements faced by social scientists and how they differ from those of mathematicians and natural scientists. It includes chapters centered on theory and procedural explanations for running the analyses with pertinent examples and others that illustrate applications where a particular form of analysis is seen in the context of a research problem. This combination of approaches conveys theoretical and practical knowledge that helps you develop skill and expertise in framing hypotheses dynamically and building viable analytic models to test them. (shrink)

‘Probability dynamics’ (PD) is a second-order probabilistic theory in which probability distribution d X = (P(X 1), . . . , P(X m )) on partition U m X of sample space Ω is weighted by ‘credence’ (c) ranging from −∞ to +∞. c is the relative degree of certainty of d X in ‘α-evidence’ α X =[c; d X ] on U m X . It is shown that higher-order probabilities cannot provide a theory of PD. PD applies (...) to both subjectivist and frequentist theories. ‘Straight PD’ (SPD) produces associative and commutative mergers of evidence, in which evidences of positive credence are mutually reinforcing. ‘Offsetting PD’ (OPD) sets off conflicting evidences against each other. Subjectivist PD is a quantified second-order logic of action. Frequentist PD relates to descriptions of physical states of affairs. Acceptance of evidence α 1 X = [c 1; d 1 X ] at t 1 updates α 0 X = [c 0; d 0 X ] at t 0 into SPD-merger or OPD-merger . Given ‘co-evidence’ at t 0 < t 1, ‘indirect’ PD accepts evidence at t 1 and produces support for update α 10 X = [c10;d 10 X α 0 X such that in SPD and in OPD. For binary X and Y, with α 0 X = [c 0; P 0(X)] at t 0 (short-hand for ) and the accepted evidence, the support is ; ; where ρ0(X, Y) is the correlation coefficient of X and Y, and update α 10 X of α 0 X is with ‘accord’ λ = 1 in SPD and in OPD. As , tends toward update P 1(Y) of P 0(Y), but P 10(X) does not converge toward ‘ ’ of ‘probability kinematics’. Therefore PD is not compatible with probability kinematics. A process of ‘normalization’ interprets ‘β-evidence’ [[c 1; q 1]& . . . &[c m ; q m ]; U m X ]] with [c j ; q j ] on and m ≥ 3, as an α-evidence on {X 1, . . . , X m }. It is shown that SPD- and OPD-updates can be derived from updated cumulative functions. Time-biased updates are discussed. A PD-based theory of confirmation (PDCT) is presented. (shrink)

In the literature there are at least two models for probabilistic belief revision: Bayesian updating and imaging [Lewis, D. K. (1973), Counterfactuals, Blackwell, Oxford; Gärdenfors, P. (1988), Knowledge in flux: modeling the dynamics of epistemic states, MIT Press, Cambridge, MA]. In this paper we focus on imaging rules that can be described by the following procedure: (1) Identify every state with some real valued vector of characteristics, and accordingly identify every probabilistic belief with an expected vector of characteristics; (...) (2) For every initial belief and every piece of information, choose the revised belief which is compatible with this information and for which the expected vector of characteristics has minimal Euclidean distance to the expected vector of characteristics of the initial belief. This class of rules thus satisfies an intuitive notion of minimal belief revision. The main result in this paper is to provide an axiomatic characterization of this class of imaging rules. (shrink)

The area of smart power grids needs to constantly improve its efficiency and resilience, to provide high quality electrical power in a resilient grid, while managing faults and avoiding failures. Achieving this requires high component reliability, adequate maintenance, and a studied failure occurrence. Correct system operation involves those activities and novel methodologies to detect, classify, and isolate faults and failures and model and simulate processes with predictive algorithms and analytics. In this paper, we showcase the application of a complex-adaptive, self-organizing (...) modeling method, and Probabilistic Boolean Networks, as a way towards the understanding of the dynamics of smart grid devices, and to model and characterize their behavior. This work demonstrates that PBNs are equivalent to the standard Reinforcement Learning Cycle, in which the agent/model has an interaction with its environment and receives feedback from it in the form of a reward signal. Different reward structures were created to characterize preferred behavior. This information can be used to guide the PBN to avoid fault conditions and failures. (shrink)

This paper presents an w-completeness theorem for a new propositional probabilistic logic, namely, the dynamic propositional logic of qualitative probabilities (DQP), which has been introduced by the author as a dynamic extension of the logic of qualitative probabilities (Q P) introduced by Segerberg.