# Probabilistic Inference for Dynamical Systems

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## Abstract

**:**

## 1. Introduction

## 2. Why Bayesian Inference?

## 3. Dynamical Evolution of Probabilities

## 4. Fluid Theories in a Bayesian Formulation

## 5. Including Particular Knowledge into Our Models

- (1)
**Bayes’ theorem**: the posterior distribution $P\left(\mathit{u}\right|{\mathcal{I}}_{0},\mathcal{R})$ is given in terms of the prior $P\left(\mathit{u}\right|{\mathcal{I}}_{0})$ by$$P\left(\mathit{u}\right|{\mathcal{I}}_{0},\mathcal{R})=\frac{P\left(\mathit{u}\right|{\mathcal{I}}_{0})\xb7P\left(\mathcal{R}\right|\mathit{u},{\mathcal{I}}_{0})}{P\left(\mathcal{R}\right|{\mathcal{I}}_{0})}.$$This method is most useful when $\mathcal{R}$ is comprised of statements about the states $\mathit{u}$ (e.g., boundary conditions).- (2)
**Principle of maximum entropy**: the posterior distribution $p\left(\mathit{u}\right)$ is the one that maximizes$$\mathcal{S}[{p}_{0}\to p]=-{\int}_{V}d\mathit{u}\phantom{\rule{0.277778em}{0ex}}p\left(\mathit{u}\right)ln\frac{p\left(\mathit{u}\right)}{{p}_{0}\left(\mathit{u}\right)},$$

## 6. The Maximum Caliber Principle

## 7. An Illustration: Newtonian Mechanics of Charged Particles

## 8. Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Derivation of the Lorenz Force from the Lagrangian of a Particle in an Electromagnetic Field

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**MDPI and ACS Style**

Davis, S.; González, D.; Gutiérrez, G. Probabilistic Inference for Dynamical Systems. *Entropy* **2018**, *20*, 696.
https://doi.org/10.3390/e20090696

**AMA Style**

Davis S, González D, Gutiérrez G. Probabilistic Inference for Dynamical Systems. *Entropy*. 2018; 20(9):696.
https://doi.org/10.3390/e20090696

**Chicago/Turabian Style**

Davis, Sergio, Diego González, and Gonzalo Gutiérrez. 2018. "Probabilistic Inference for Dynamical Systems" *Entropy* 20, no. 9: 696.
https://doi.org/10.3390/e20090696