Marginal flow lemma derivation

Mathematical derivation of the Marginal Flow Lemma from the Ao Decomposition and the Markov Blanket Condition; forming in the last formulation the formular for Self-evidencing.

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$fe(x)={\mathbb{E}}_p(e|b)[f_e(x)]=({\Gamma }_{e,e}-Q_{e,e}){\nabla }_e\ln p^{\ast }(e)-Q_{e,b}{\nabla }_{b}p(e,b)$

using the marginal flow lemma, while $\alpha =(a,i)$:

$f_{\alpha }(x)=(\Gamma \alpha \alpha -Q_{\alpha \alpha }){\nabla }_{\alpha }\ln p^{\ast }(i,s,a)$

series of mathematical ‘inflationary devices’ (adding and subtracting the same quantity,so the total is 0, in order to introduce it to the equation) to express this surprisal in terms of its interaction with the external states beyond the blanket.

$-\ln p^{\ast }(i,s,a)=E_{p^{\ast }(e|i,s,a)}[-\ln p^{\ast }(i,s,a)]$ $=E_{p^{\ast }(e|i,s,a)}[\ln p^{\ast }(e|i,s,a)-\ln p^{\ast }(e,i,s,a)]$ $=E_{p^{\ast }(e|i,s,a)}[\ln p^{\ast }(e|i,s,a)-\ln p^{\ast }(i,s,a|e)-\ln p^{\ast }(e)]$ $=\underset{\text{Inaccuracy}}{\underbrace{E_{p^{\ast }(e|i,s,a)}[-\ln p^{\ast }(i,s,a|e)]}}+\underset{\text{Complexity}}{\underbrace{D_{KL}[p\ast (e|i,s,a)\Vert p(e)]}}$

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Evidence

Supporting

Handle	Title	Summary of relevant evidence	Link
Friston 2019	A Free Energy Principle for a Particular Physics
Millidge et al. 2021	A Mathematical Walkthrough and Discussion of the Free Energy Principle	p. 10

Counter

Handle	Title	Summary of relevant evidence	Link

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Connections

Derrivation on the Ao Decomposition after addition of the Markov Blanket Condition Formulation of The Free-Energy Principle Is traceable by Variational Inference Can be used to derive Active Inference reduces Inaccuracy

• Add Assumptions from Millidge et al. Appendix 10.3.2

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Metadata

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Sentence

Derivation

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