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

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## Abstract

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## 1. Introduction

^{−}

^{η}, where 1 < η ≤ 3, in a Lévy walk, while that occurring in a Brownian walk follows an exponential distribution, P(l)~e

^{−λl}. In other words, the former is sometimes accompanied by linear movements over long distances compared to the latter, and the reason for this pattern in the migration of organisms has been the subject of much discussion [7].

## 2. Materials and Methods

#### 2.1. Extended Bayesian Inference

_{k}are first defined, and a model for each hypothesis (the generation distribution of data d) is prepared in the form of conditional probability $P\left(d|{h}_{k}\right)$. This conditional probability is referred to as the likelihood that the data are fixed and is considered to be a function of the hypothesis. In addition, the confidence $P\left({h}_{k}\right)$ for each hypothesis is prepared as a prior probability.

^{t}, then the posterior probability ${P}^{t}\left({h}_{k}|{d}^{t}\right)$ is calculated using Bayes’ theorem as follows:

_{k}and the model for each hypothesis becomes invariant, as in Bayesian inference. Thus, the extended Bayesian inference adds the forgetting rate β and the learning rate γ to Bayesian inference, and agrees with Bayesian inference when β = γ = 0. The extended Bayesian inference updates the confidence of each hypothesis by Equation (9) each time the data are observed and modifies the model of the hypothesis with the maximum confidence by Equation (10).

#### 2.2. Imitation Game

^{t}from the generative model ${C}^{t}\left(d|{h}_{\mathrm{max}}^{t}\right)$ of the hypothesis ${h}_{\mathrm{max}}^{t}$, which it believes most at each time step, and presents it to its opponent. Each agent observes the real number presented by its counterpart and modifies its generative model and confidence values for hypotheses using the extended Bayesian inference described above. The simulation was conducted up to 2000 steps, and the data in the interval of 1000 ≤ t ≤ 2000 was used for the analysis and the display.

## 3. Results

#### 3.1. Simulation Results

#### 3.2. Comparison between with and without Learning

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Outline of imitation game with two decision-making agents using Bayesian inference. First, each agent estimates the generative model of the partner agent from the observation data and uses it as its own generative model. Next, each agent samples a real number from its own generative model and presents it to the other party. The agents repeat these procedures in each step.

**Figure 2.**Results of cases in which the forgetting rate was set to β = 0.005 or 0.1 without learning (γ = 0.0). The left and right columns represent the results in the case of β = 0.005 and the case of β = 0.1, respectively. (

**a**) The time evolution of the normal distribution mean value estimated by each agent. (

**b**) The time evolution of the hypothesis ${h}_{\mathrm{max}}^{t}$ that Agent1 believes the most. (

**c**) Cumulative distribution function (CDF) of duration T of ${h}_{\mathrm{max}}^{t}$. The exponent of the truncated power law distribution for β = 0.005 was η = 1.58. The exponents of the exponential distribution for β = 0.1 was λ = 0.075. The fitting ranges $\left[{\widehat{T}}_{\mathrm{min}},{\widehat{T}}_{\mathrm{max}}\right]$ for β = 0.005 and 0.1 were [5, 927] and [22, 116], respectively.

**Figure 3.**Examples of time evolution when the forgetting rate β = 0.3. The left and right columns represent the results in the case without learning (γ = 0.0) and the case with learning (γ = 0.1), respectively. (

**a**) The normal distribution mean value estimated by each agent. (

**b**) The hypothesis ${h}_{\mathrm{max}}^{t}$ that Agent1 believes the most. (

**c**) Agent1′s confidence in each hypothesis.

**Figure 4.**Results of cases in which the forgetting rate was set to β = 0.3, 0.5, or 0.7. (

**a**) Cumulative distribution function (CDF) of duration T of ${h}_{\mathrm{max}}^{t}$ in the case without learning (γ = 0.0). The exponents of the exponential distribution for β = 0.3, 0.5, and 0.7 were λ = 0.24, 0.42, and 0.60, respectively. The fitting ranges $\left[{\widehat{T}}_{\mathrm{min}},{\widehat{T}}_{\mathrm{max}}\right]$ for β = 0.3, 0.5, and 0.7 were [14, 61], [6, 27], and [5, 22], respectively. (

**b**) Cumulative distribution function (CDF) of duration T of ${h}_{\mathrm{max}}^{t}$ in the case with learning (γ = 0.1). The exponents of the truncated power law distribution for β = 0.3, 0.5, and 0.7 were η = 1.73, 2.00, and 2.44, respectively. 1 < η ≤ 3 was satisfied in all cases. The fitting ranges $\left[{\widehat{T}}_{\mathrm{min}},{\widehat{T}}_{\mathrm{max}}\right]$ for β = 0.3, 0.5, and 0.7 were [29, 920], [18, 890], and [8, 720], respectively.

**Figure 5.**Universally seen power laws. Power laws are found in a wide range of parameter regions; the exponents of the truncated power law distribution are close to 2. (

**a**) α = 0.3, that is, β = γ = 0.3. The exponent of the truncated power law distribution was η = 1.91. The fitting range $\left[{\widehat{T}}_{\mathrm{min}},{\widehat{T}}_{\mathrm{max}}\right]$ was [4, 808]. (

**b**) α = 0.5, that is, β = γ = 0.5. The exponent of the truncated power law distribution was η = 1.92. The fitting range $\left[{\widehat{T}}_{\mathrm{min}},{\widehat{T}}_{\mathrm{max}}\right]$ was [1, 942]. (

**c**) α = 0.7, that is, β = γ = 0.7. The exponent of the truncated power law distribution was η = 2.10. The fitting range $\left[{\widehat{T}}_{\mathrm{min}},{\widehat{T}}_{\mathrm{max}}\right]$ was [4, 970].

**Figure 6.**Comparison between cases with no learning (γ = 0.0) and learning (γ = 0.1). (

**a**) Change in the time average of confidence level with change in the forgetting rate. (

**b**) The relationship between duration T of ${h}_{\mathrm{max}}^{t}$ and confidence (β = 0.3). The vertical axis represents the average confidence during T indicated by the horizontal axis. As the duration increases, the confidence in the hypothesis increases. The duration on the horizontal axis is displayed up to 300 for clarity.

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**MDPI and ACS Style**

Shinohara, S.; Manome, N.; Nakajima, Y.; Gunji, Y.P.; Moriyama, T.; Okamoto, H.; Mitsuyoshi, S.; Chung, U.-i.
Power Laws Derived from a Bayesian Decision-Making Model in Non-Stationary Environments. *Symmetry* **2021**, *13*, 718.
https://doi.org/10.3390/sym13040718

**AMA Style**

Shinohara S, Manome N, Nakajima Y, Gunji YP, Moriyama T, Okamoto H, Mitsuyoshi S, Chung U-i.
Power Laws Derived from a Bayesian Decision-Making Model in Non-Stationary Environments. *Symmetry*. 2021; 13(4):718.
https://doi.org/10.3390/sym13040718

**Chicago/Turabian Style**

Shinohara, Shuji, Nobuhito Manome, Yoshihiro Nakajima, Yukio Pegio Gunji, Toru Moriyama, Hiroshi Okamoto, Shunji Mitsuyoshi, and Ung-il Chung.
2021. "Power Laws Derived from a Bayesian Decision-Making Model in Non-Stationary Environments" *Symmetry* 13, no. 4: 718.
https://doi.org/10.3390/sym13040718