<p>I have a question about the free energy principle, taken from Friston et al.'s paper <a href="https://www.fil.ion.ucl.ac.uk/%7Ekarl/A%20free%20energy%20principle%20for%20the%20brain.pdf" rel="nofollow noreferrer">A Free Energy Principle For The Brain</a>:</p>
<p><span class="math-container">$\begin{align}F&amp;=-\int q(\theta)\ln\frac{p(\tilde y,\theta)}{q(\theta)}\text{d}\theta\\&amp;=-\langle\ln p(\tilde y,\theta)\rangle_q+\langle\ln q(\theta)\rangle_q\end{align}\tag{1}$</span></p>
<p>and equivalently</p>
<p><span class="math-container">$\begin{align}F=-\ln p(\tilde y)+D(q(\theta;\lambda)\|p(\theta|\tilde y))\end{align}\tag{2}$</span></p>
<p>or</p>
<p><span class="math-container">$\begin{align}F=-\langle\ln p(\tilde y(\alpha)|\theta)\rangle_q+D(q(\theta)\|p(\theta))\end{align}\tag{3}$</span></p>
<p>So <span class="math-container">$\tilde y$</span> is just short for <span class="math-container">$\tilde y(\alpha)$</span> and <span class="math-container">$q(\theta)$</span> is short for <span class="math-container">$q(\theta;\lambda)$</span>, I assume?</p>
<p>I don't understand the exact roles and causalities between the given variables.</p>
<ul>
<li>Does <span class="math-container">$\theta$</span> include external states of the organism, like its position or orientation?</li>
<li>Is <span class="math-container">$p$</span> some kind of objective probability? I understand that <span class="math-container">$p(\tilde y|\theta)$</span> can be seen as objective, i.e., sensors are noisy and given some fixed external state, the information produced by the sensor always follows a probability distribution. But what is <span class="math-container">$p(\theta)$</span>, how does <span class="math-container">$p$</span> assign probabilities to world states?  Are goals of the organism, like preferred temperature range, nutrition concentration, etc. encoded in this <span class="math-container">$p$</span>?  Is the <span class="math-container">$p$</span> some kind of equilibrium state in which the organism ends after minimizing free energy for enough time?  So does the FEP describe <em>how</em> an organism arrives in its preferred state (which is known to the principle), rather than <em>what</em> this preferred state looks like?</li>
<li>Is the dependency of <span class="math-container">$\tilde y$</span> on <span class="math-container">$\alpha$</span> deterministic, i.e., given some fixed <span class="math-container">$\theta$</span>, does a given action <span class="math-container">$\alpha$</span> always produce the same sensory input <span class="math-container">$\tilde y$</span>?  I assume not, but then I don't understand the formula, <span class="math-container">$\tilde y(\alpha)$</span> looks like a deterministic relationship, maybe <span class="math-container">$\tilde y(\alpha)$</span> is meant to be a random variable, but in that case we cannot write <span class="math-container">$\ln p(\tilde y(\alpha)|\theta)$</span> but rather <span class="math-container">$\langle\ln p(\tilde y(\alpha)|\theta)\rangle_a$</span> for some suitable probability density <span class="math-container">$a$</span>?</li>
<li>If an action can change the organism's external state, and the external state is included in <span class="math-container">$\theta$</span>, should we write <span class="math-container">$\theta(\alpha)$</span> to be precise? Or is the meaning of <span class="math-container">$\theta$</span> explicitly the external state before the action took place, and only <span class="math-container">$\tilde y$</span> updates with <span class="math-container">$\alpha$</span>?</li>
<li>Same with <span class="math-container">$p$</span>, does it depend on <span class="math-container">$\alpha$</span> (when <span class="math-container">$\tilde y$</span> depends on <span class="math-container">$\alpha$</span>)?  Does it update during the organism's evolution?</li>
</ul>
<p>My very vague understanding is that <span class="math-container">$q(\cdot;\lambda)$</span> represents the organism's internal model of the world (where <span class="math-container">$q(\theta)$</span> is short for <span class="math-container">$q(\theta;\lambda)$</span>) and that the action is chosen so that the resulting sensory input is likely to occur in the world states the organism expects itself to be in. In other words, the organism cannot optimize <span class="math-container">$\alpha$</span> on its own, since it has neither the information <span class="math-container">$\tilde y(\alpha)$</span>, i.e., it cannot predict the resulting sensory input, nor does it have access to <span class="math-container">$p$</span>. Therefore, the principle is more of an observed dynamic rather than something the organism can choose to follow.</p>
<p>BTW I'm not a psychologist, so please tell me if I posted this question in the wrong SE.</p>

CC BY-SA 4.0
2024-06-19T13:14:06.673Z
People who *are* psychologists have made some effort to understand Friston's free energy stuff and have come up empty. https://slatestarcodex.com/2018/03/04/god-help-us-lets-try-to-understand-friston-on-free-energy/ is an accessible for outsiders critique.
Bryan Krause
CC BY-SA 4.0
2024-06-19T13:19:11.167Z
Your question is just fine here; I was referring to your last sentence mentioning you're not a psychologist. My point is that people who are, that is, have a lot of experience in this field, often find Friston's reasoning impenetrable, his explanations opaque, and conclusions either tautological or nonsensical. So, you can try to understand the equations but it seems unlikely to come to much.
Bryan Krause
CC BY-SA 4.0
2024-06-19T13:20:10.750Z
If you're familiar with Bayesian statistics, the equations and concepts will likely be familiar, but the connections to the real world and the brain are still tenuous.
Bryan Krause
CC BY-SA 4.0
2024-06-19T13:22:53.500Z
I understand the math, it's more the interpretation of the variables which troubles me, and the hidden dependencies.  The Wikipedia article seems to try being more precise, but the formulas seem to be faulty (which is also noted in the article discussion).  In the formulas from the paper, I see nothing inherently bad (from a purely mathematical point of view).
fweth
CC BY-SA 4.0
2024-06-19T13:23:39.460Z
Yes, exactly the problem. That said, I think about math as a biologist rather than a mathematician, and from that perspective there's no meaning (for biology) in a purely mathematical point of view. Math with "hidden dependencies" and unclear interpretation of variables is faulty for those reasons, from a practical perspective.
Bryan Krause
<brain><mathematical-psychology>
Understanding Causal Relationship Between Variables In The Free Energy Principle
fweth
# Understanding Causal Relationship Between Variables In The Free Energy Principle

(Asked by: username_0 on 2024-06-19)

I have a question about the free energy principle, taken from Friston et al.'s paper [A Free Energy Principle For The Brain](https://www.fil.ion.ucl.ac.uk/%7Ekarl/A%20free%20energy%20principle%20for%20the%20brain.pdf):

$\begin{align}F&=-\int q(\theta)\ln\frac{p(\tilde y,\theta)}{q(\theta)}\text{d}\theta\\&=-\langle\ln p(\tilde y,\theta)\rangle\_q+\langle\ln q(\theta)\rangle\_q\end{align}\tag{1}$

and equivalently

$\begin{align}F=-\ln p(\tilde y)+D(q(\theta;\lambda)\|p(\theta|\tilde y))\end{align}\tag{2}$

or

$\begin{align}F=-\langle\ln p(\tilde y(\alpha)|\theta)\rangle\_q+D(q(\theta)\|p(\theta))\end{align}\tag{3}$

So $\tilde y$ is just short for $\tilde y(\alpha)$ and $q(\theta)$ is short for $q(\theta;\lambda)$, I assume?

I don't understand the exact roles and causalities between the given variables.

* Does $\theta$ include external states of the organism, like its position or orientation?
* Is $p$ some kind of objective probability? I understand that $p(\tilde y|\theta)$ can be seen as objective, i.e., sensors are noisy and given some fixed external state, the information produced by the sensor always follows a probability distribution. But what is $p(\theta)$, how does $p$ assign probabilities to world states? Are goals of the organism, like preferred temperature range, nutrition concentration, etc. encoded in this $p$? Is the $p$ some kind of equilibrium state in which the organism ends after minimizing free energy for enough time? So does the FEP describe *how* an organism arrives in its preferred state (which is known to the principle), rather than *what* this preferred state looks like?
* Is the dependency of $\tilde y$ on $\alpha$ deterministic, i.e., given some fixed $\theta$, does a given action $\alpha$ always produce the same sensory input $\tilde y$? I assume not, but then I don't understand the formula, $\tilde y(\alpha)$ looks like a deterministic relationship, maybe $\tilde y(\alpha)$ is meant to be a random variable, but in that case we cannot write $\ln p(\tilde y(\alpha)|\theta)$ but rather $\langle\ln p(\tilde y(\alpha)|\theta)\rangle\_a$ for some suitable probability density $a$?
* If an action can change the organism's external state, and the external state is included in $\theta$, should we write $\theta(\alpha)$ to be precise? Or is the meaning of $\theta$ explicitly the external state before the action took place, and only $\tilde y$ updates with $\alpha$?
* Same with $p$, does it depend on $\alpha$ (when $\tilde y$ depends on $\alpha$)? Does it update during the organism's evolution?

My very vague understanding is that $q(\cdot;\lambda)$ represents the organism's internal model of the world (where $q(\theta)$ is short for $q(\theta;\lambda)$) and that the action is chosen so that the resulting sensory input is likely to occur in the world states the organism expects itself to be in. In other words, the organism cannot optimize $\alpha$ on its own, since it has neither the information $\tilde y(\alpha)$, i.e., it cannot predict the resulting sensory input, nor does it have access to $p$. Therefore, the principle is more of an observed dynamic rather than something the organism can choose to follow.

BTW I'm not a psychologist, so please tell me if I posted this question in the wrong SE.

**Tags:** brain, mathematical-psychology

> **Comments:**
> **username_1**: People who *are* psychologists have made some effort to understand Friston's free energy stuff and have come up empty. https://slatestarcodex.com/2018/03/04/god-help-us-lets-try-to-understand-friston-on-free-energy/ is an accessible for outsiders critique.  
> **username_1**: Your question is just fine here; I was referring to your last sentence mentioning you're not a psychologist. My point is that people who are, that is, have a lot of experience in this field, often find Friston's reasoning impenetrable, his explanations opaque, and conclusions either tautological or nonsensical. So, you can try to understand the equations but it seems unlikely to come to much.  
> **username_1**: If you're familiar with Bayesian statistics, the equations and concepts will likely be familiar, but the connections to the real world and the brain are still tenuous.  
> **username_0**: I understand the math, it's more the interpretation of the variables which troubles me, and the hidden dependencies.  The Wikipedia article seems to try being more precise, but the formulas seem to be faulty (which is also noted in the article discussion).  In the formulas from the paper, I see nothing inherently bad (from a purely mathematical point of view).  
> **username_1**: Yes, exactly the problem. That said, I think about math as a biologist rather than a mathematician, and from that perspective there's no meaning (for biology) in a purely mathematical point of view. Math with "hidden dependencies" and unclear interpretation of variables is faulty for those reasons, from a practical perspective.  

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## 0 answers

