Lévy Walk in Swarm Models Based on Bayesian and Inverse Bayesian Inference

Power Levy motion. I. Diffusion

Recently introduced and explored, power Brownian motion (PBM) is a versatile generalization of Brownian motion: it is Markovian on the one hand and it displays a variety of anomalous-diffusion behaviors on the other hand. Brownian motion is the universal scaling-limit of finite-variance random walks. Shifting from the finite-variance realm to the infinite-variance realm, the counterpart of Brownian motion is Levy motion: the stable and symmetric Levy process. This pair of papers introduces and explores power Levy motion (PLM), which is to Levy motion what PBM is to Brownian motion. This first part of the pair constructs PLM and explains its emergence and rationale. Taking on a "diffusion perspective," this part addresses the following facets and features of PLM: increments and their Fourier structure, selfsimilarity and Hurst exponent, sub-diffusion and super-diffusion, aging and anti-aging, and Holder exponent. Taking on an "evolution perspective," the second part will continue the investigation of PLM.

Levy Noise Affects Ornstein-Uhlenbeck Memory

This paper investigates the memory of the Ornstein-Uhlenbeck process (OUP) via three ratios of the OUP increments: signal-to-noise, noise-to-noise, and tail-to-tail. Intuition suggests the following points: (1) changing the noise that drives the OUP from Gauss to Levy will not affect the memory, as both noises share the common 'independent increments' property; (2) changing the auto-correlation of the OUP from exponential to slowly decaying will affect the memory, as the change yields a process with long-range correlations; and (3) with regard to Levy driving noise, the greater the noise fluctuations, the noisier the prediction of the OUP increments. This paper shows that intuition is plain wrong. Indeed, a detailed analysis establishes that for each of the three above-mentioned points, the very converse holds. Hence, Levy noise has a significant and counter-intuitive effect on Ornstein-Uhlenbeck memory.

Functional duality in group criticality via ambiguous interactions

Critical phenomena are wildly observed in living systems. If the system is at criticality, it can quickly transfer information and achieve optimal response to external stimuli. Especially, animal collective behavior has numerous critical properties, which are related to other research regions, such as the brain system. Although the critical phenomena influencing collective behavior have been extensively studied, two important aspects require clarification. First, these critical phenomena never occur on a single scale but are instead nested from the micro- to macro-levels (e.g., from a Lévy walk to scale-free correlation). Second, the functional role of group criticality is unclear. To elucidate these aspects, the ambiguous interaction model is constructed in this study; this model has a common framework and is a natural extension of previous representative models (such as the Boids and Vicsek models). We demonstrate that our model can explain the nested criticality of collective behavior across several scales (considering scale-free correlation, super diffusion, Lévy walks, and 1/f fluctuation for relative velocities). Our model can also explain the relationship between scale-free correlation and group turns. To examine this relation, we propose a new method, applying partial information decomposition (PID) to two scale-free induced subgroups. Using PID, we construct information flows between two scale-free induced subgroups and find that coupling of the group morphology (i.e., the velocity distributions) and its fluctuation power (i.e., the fluctuation distributions) likely enable rapid group turning. Thus, the flock morphology may help its internal fluctuation convert to dynamic behavior. Our result sheds new light on the role of group morphology, which is relatively unheeded, retaining the importance of fluctuation dynamics in group criticality.

Connecting the free energy principle with quantum cognition

It appears that the free energy minimization principle conflicts with quantum cognition since the former adheres to a restricted view based on experience while the latter allows deviations from such a restricted view. While free energy minimization, which incorporates Bayesian inference, leads to a Boolean lattice of propositions (classical logic), quantum cognition, which seems to be very dissimilar to Bayesian inference, leads to an orthomodular lattice of propositions (quantum logic). Thus, we address this challenging issue to bridge and connect the free energy minimization principle with the theory of quantum cognition. In this work, we introduce “excess Bayesian inference” and show that this excess Bayesian inference entails an underlying orthomodular lattice, while classic Bayesian inference entails a Boolean lattice. Excess Bayesian inference is implemented by extending the key idea of Bayesian inference beyond classic Bayesian inference and its variations. It is constructed by enhancing the idea of active inference and/or embodied intelligence. The appropriate lattice structure of its logic is obtained from a binary relation transformed from a distribution of the joint probabilities of data and hypotheses by employing a rough-set lattice technique in accordance with quantum cognition logic.

Concept Formation and Quantum-like Probability from Nonlocality in Cognition

Human decision-making is relevant for concept formation and cognitive illusions. Cognitive illusions can be explained by quantum probability, while the reason for introducing quantum mechanics is based on ad hoc bounded rationality (BR). Concept formation can be explained in a set-theoretic way, although such explanations have not been extended to cognitive illusions. We naturally expand the idea of BR to incomplete BR and introduce the key notion of nonlocality in cognition without any attempts on quantum theory. We define incomplete bounded rationality and nonlocality as a binary relation, construct a lattice from the relation by using a rough-set technique, and define probability in concept formation. By using probability defined in concept formation, we describe various cognitive illusions, such as the guppy effect, conjunction fallacy, order effect, and so on. It implies that cognitive illusions can be explained by changes in the probability space relevant to concept formation.

Amoebic Foraging Model of Metastatic Cancer Cells

The Lévy walk is a pattern that is often seen in the movement of living organisms; it has both ballistic and random features and is a behavior that has been recognized in various animals and unicellular organisms, such as amoebae, in recent years. We proposed an amoeba locomotion model that implements Bayesian and inverse Bayesian inference as a Lévy walk algorithm that balances exploration and exploitation, and through a comparison with general random walks, we confirmed its effectiveness. While Bayesian inference is expressed only by P(h) = P(h|d), we introduce inverse Bayesian inference expressed as P(d|h) = P(d) in a symmetry fashion. That symmetry contributes to balancing contracting and expanding the probability space. Additionally, the conditions of various environments were set, and experimental results were obtained that corresponded to changes in gait patterns with respect to changes in the conditions of actual metastatic cancer cells.

Power Laws Derived from a Bayesian Decision-Making Model in Non-Stationary Environments

The frequency of occurrence of step length in the migratory behaviour of various organisms, including humans, is characterized by the power law distribution. This pattern of behaviour is known as the Lévy walk, and the reason for this phenomenon has been investigated extensively. Especially in humans, one possibility might be that this pattern reflects the change in self-confidence in one’s chosen behaviour. We used simulations to demonstrate that active assumptions cause changes in the confidence level in one’s choice under a situation of lack of information. More specifically, we presented an algorithm that introduced the effects of learning and forgetting into Bayesian inference, and simulated an imitation game in which two decision-making agents incorporating the algorithm estimated each other’s internal models. For forgetting without learning, each agents’ confidence levels in their own estimation remained low owing to a lack of information about the counterpart, and the agents changed their hypotheses about the opponent frequently, and the frequency distribution of the duration of the hypotheses followed an exponential distribution for a wide range of forgetting rates. Conversely, when learning was introduced, high confidence levels occasionally occurred even at high forgetting rates, and exponential distributions universally turned into power law distribution.