# Some Interesting Observations on the Free Energy Principle

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## Abstract

**:**

## 1. Introduction

## 2. The Free Energy Principle in Brief

## 3. Observation One

## 4. Observation Two

#### Exact or Approximate?

## 5. Observation Three

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

**FEP_fluctuations.m**).

## Conflicts of Interest

## Appendix A

## Appendix B

**Definition**

**A1**

**.**a dissipative partition is a partition into external, blanket (i.e., sensory and active) and internal states, where internal and external states do not influence each other—and one or more subsets of states are dissipative, i.e., the leading diagonal elements of the associated Jacobian are large and negative.

**Lemma**

**A1**

**.**The sensory and active states of a dissipative partition constitute a Markov blanket $b=\left(s,a\right)$ that renders external and internal states conditionally independent:

**Proof.**

**Figure A1.**Dissipation and conditional independence: numerical analyses that show the conditional independence between internal and external states depends upon dissipation, as quantified by the average value of the leading diagonal Jacobians. In this example, a system with 24 states was divided equally into external, blanket and internal states. The panels above report the variance of the solenoidal term, the Jacobian and Hessian, based on 512 random samples where each element of the Jacobian was sampled from a unit Gaussian distribution and values of 4, 4 and 32 were added to the leading diagonal for the external, blanket and internal states, respectively. The black blocks on the lower left (and upper right) show that an absence of coupling in the Jacobian—between the external and internal states—precludes solenoidal coupling and renders the external and internal states conditionally independent.

## Appendix C

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**Figure 1.**Markov blankets. This schematic (reproduced from [3]) illustrates the partition of states into internal states (µ, in blue) and hidden or external states (η, in cyan) that are separated by a Markov blanket (b) comprising sensory (s, in magenta) and active states (a, in red). The upper panel shows this partition as it would be applied to action and perception in a brain. Note that the only missing influences are between internal and external states—and directed influences from external (respectively internal) to active (respectively sensory) states. The surviving directed influences are highlighted with dotted connectors. In this setting, the self-organisation of internal states then corresponds to perception, while active states couple internal states back to external states. The lower panel shows the same partition but rearranged so that the internal states are associated with the intracellular states of a Bacillus, where the sensory states become the surface states or cell membrane overlying active states (e.g., the actin filaments of the cytoskeleton). Here, the coupling between sensory and internal—and between active and external states—was suppressed to reveal a simple coupling architecture that leads to a Markov blanket. Autonomous states (α) are those states that are not influenced by external states, while particular states (π) constitute a particle; namely, autonomous and sensory states—or blanket and internal states. The equations of motion in the upper panel underwrite the conditional independencies of the Markov blanket, as described in the main text. By this, we do not mean that these equations of motion define a Markov blanket, or even that they are necessary for a Markov blanket—which can exist in static systems with no dynamics. Instead, these equations represent flows that, under certain assumptions, result in a Markov blanket at steady state. Please see (1) and the associated discussion.

**Figure 2.**Evidence bounds and gradient flows. This schematic tries to convey the intuition that the gradient flows on surprisal (pink)—as a function of some statistical manifold (here conditional expectations of internal states, given blanket states)—are the same as gradient flows on variational free energy (green); if, and only if, the KL divergence or evidence bound is conserved over the manifold. The left panel shows a view of the two functions from the side, while the right panel provides a view from the top.

**Figure 3.**Sentient dynamics and the representation of order. This figure illustrates approximate Bayesian inference that follows when associating the internal states of a system with a variational (i.e., approximate posterior) density over external states. This figure is based upon the simulation of a small rod-like particle used to illustrate different perspectives on self-organisation in [3], where the details of this system are specified. In brief, each macromolecule is defined by a set of electrochemical states modelled as stochastic Lorenz attractor. These attractors are coupled between pairs of macromolecules, where the coupling strength depends upon the distance between each pair. In addition, the position and velocity of each macromolecule are described by (stochastic) Newtonian equations of motion subject to forces based upon the difference in electrochemical states between a molecule and its neighbours. The upper panels illustrate a collection of simulated macromolecules, in terms of internal (blue) active (red) sensory (magenta) and external (cyan) states. The middle left panel shows the first canonical vector of motion over the external states (green arrows) that are represented by the internal states (blue dots). The blue and cyan dots are placed at the location of internal and external states, respectively. The colour level reflects the norm (sum of squares) of the first canonical vectors showing the greatest covariation between external and internal states. The middle panel illustrates a synchronisation manifold (conditioned upon the Markov blanket) that maps from the electrochemical states of internal macromolecules to the velocity of external macromolecules. The blue dots identify the manifold per se, while the cyan dots are the estimated expectations used to estimate the manifold (using a fifth-order polynomial regression). The lower panel shows the same information but plotted as a function of time during the last 512 s of the simulation. The conditional expectation is based upon the internal states, while the real motion is shown as a cyan line. The blue shaded areas correspond to 90% confidence intervals. The lower right panel illustrates simulated event-related potentials of the sort illustrated by the insert (lower right panel). The simulated evoked response potential (ERP) was obtained by time locking the internal electrochemical states to the six time points that showed the greatest expression of the first canonical variate (indicated by the vertical lines in the middle panel). The dotted lines are six trajectories around these points in time, while the solid lines correspond to the average. The blue lines are the responses of internal states, while the cyan lines correspond to the real motion associated with the first canonical vector. The timing in the lower panels was arbitrarily rescaled to match empirical peristimulus times—illustrated with an empirical example of event-related potentials in the middle right panel.

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Friston, K.J.; Da Costa, L.; Parr, T.
Some Interesting Observations on the Free Energy Principle. *Entropy* **2021**, *23*, 1076.
https://doi.org/10.3390/e23081076

**AMA Style**

Friston KJ, Da Costa L, Parr T.
Some Interesting Observations on the Free Energy Principle. *Entropy*. 2021; 23(8):1076.
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**Chicago/Turabian Style**

Friston, Karl J., Lancelot Da Costa, and Thomas Parr.
2021. "Some Interesting Observations on the Free Energy Principle" *Entropy* 23, no. 8: 1076.
https://doi.org/10.3390/e23081076