I’ve been trying to delve deeper into predictive processing theories of the brain, and I keep coming across Karl Friston’s work on “free energy”.

At first I felt bad for not understanding this. Then I realized I wasn’t alone. There’s an entire not-understanding-Karl-Friston internet fandom, complete with its own parody Twitter account and Markov blanket memes.

From the journal Neuropsychoanalysis (which based on its name I predict is a center of expertise in not understanding things):

At Columbia’s psychiatry department, I recently led a journal club for 15 PET and fMRI researhers, PhDs and MDs all, with well over $10 million in NIH grants between us, and we tried to understand Friston’s 2010 Nature Reviews Neuroscience paper – for an hour and a half. There was a lot of mathematical knowledge in the room: three statisticians, two physicists, a physical chemist, a nuclear physicist, and a large group of neuroimagers – but apparently we didn’t have what it took. I met with a Princeton physicist, a Stanford neurophysiologist, a Cold Springs Harbor neurobiologist to discuss the paper. Again blanks, one and all.

Normally this is the point at which I give up and say “screw it”. But almost all the most interesting neuroscience of the past decade involves this guy in one way or another. He’s the most-cited living neuroscientist, invented large parts of modern brain imaging, and received of the prestigious Golden Brain Award for excellence in neuroscience, which is somehow a real thing. His Am I Autistic – An Intellectual Autobiography short essay, written in a weirdly lucid style and describing hijinks like deriving the Schrodinger equation for fun in school, is as consistent with genius as anything I’ve ever read.

As for free energy, it’s been dubbed “a unified brain theory” (Friston 2010), a key through which “nearly every aspect of [brain] anatomy and physiology starts to make sense” (Friston 2009), “[the source of] the ability of biological systems to resist a natural tendency to disorder” (Friston 2012), an explanation of how life “inevitably and emergently” arose from the primordial soup (Friston 2013), and “a real life version of Isaac Asimov’s psychohistory” (description here of Allen 2018).

I continue to hope some science journalist takes up the mantle of explaining this comprehensively. Until that happens, I’ve been working to gather as many perspectives as I can, to talk to the few neuroscientists who claim to even partially understand what’s going on, and to piece together a partial understanding. I am not at all the right person to do this, and this is not an attempt to get a gears-level understanding – just the kind of pop-science-journalism understanding that gives us a slight summary-level idea of what’s going on. My ulterior motive is to get to the point where I can understand Friston’s recent explanation of depression, relevant to my interests as a psychiatrist.

Sources include Dr. Alianna Maren’s How To Read Karl Friston (In The Original Greek), Wilson and Golonka’s Free Energy: How the F*ck Does That Work, Ecologically?, Alius Magazine’s interview with Friston, Observing Ideas, and the ominously named Wo’s Weblog.

From these I get the impression that part of the problem is that “free energy” is a complicated concept being used in a lot of different ways.

First, free energy is a specific mathematical term in certain Bayesian equations.

I’m getting this from here, which goes into much more detail about the math than I can manage. What I’ve managed to extract: Bayes’ theorem, as always, is the mathematical rule for determining how much to weigh evidence. The brain is sometimes called a Bayesian machine, because it has to create a coherent picture of the world by weighing all the different data it gets – everything from millions of photoreceptors’ worth of vision, to millions of cochlear receptors worth of hearing, to all the other sense, to logical reasoning, to past experience, and so on. But actually using Bayes on all this data quickly gets computationally intractable.

Free energy is a quantity used in “variational Bayesian methods”, a specific computationally tractable way of approximating Bayes’ Theorem. Under this interpretation, Friston is claiming that the brain uses this Bayes-approximation algorithm. Minmizing the free energy quantity in this algorithm is equivalent-ish to trying to minimize prediction error, trying to minimize the amount you’re surprised by the world around you, and trying to maximize accuracy of mental models. This sounds in line with standard predictive processing theories. Under this interpretation, the brain implements predictive processing through free energy minimization.

Second, free energy minimization is an algorithm-agnostic way of saying you’re trying to approximate Bayes as accurately as possible.

This comes from the same source as above. It also ends up equivalent-ish to all those other things like trying to be correct in your understanding of the world, and to standard predictive processing.

Third, free energy minimization is a claim that the fundamental psychological drive is the reduction of uncertainty.

I get this claim from the Alius interview, where Friston says:

If you subscribe to the premise that that creatures like you and me act to minimize their expected free energy, then we act to reduce expected surprise or, more simply, resolve uncertainty. So what’s the first thing that we would do on entering a dark room — we would turn on the lights. Why? Because this action has epistemic affordance; in other words, it resolves uncertainty (expected free energy). This simple argument generalizes to our inferences about (hidden or latent) states of the world — and the contingencies that underwrite those states of affairs.

The discovery that the only human motive is uncertainty-reduction might come as a surprise to humans who feel motivated by things like money, power, sex, friendship, or altruism. But the neuroscientist I talked to about this says I am not misinterpreting the interview. The claim really is that uncertainty-reduction is the only game in town.

In a sense, it must be true that there is only one human motivation. After all, if you’re Paris of Troy, getting offered the choice between power, fame, and sex – then some mental module must convert these to a common currency so it can decide which is most attractive. If that currency is, I dunno, dopamine in the striatum, then in some reductive sense, the only human motivation is increasing striatal dopamine (don’t philosophize at me, I know this is a stupid way of framing things, but you know what I mean). Then the only weird thing about the free energy formulation is identifying the common currency with uncertainty-minimization, which is some specific thing that already has another meaning.

I think the claim (briefly mentioned eg here) is that your brain hacks eg the hunger drive by “predicting” that your mouth is full of delicious food. Then, when your mouth is not full of delicious food, it’s a “prediction error”, it sets off all sorts of alarm bells, and your brain’s predictive machinery is confused and uncertain. The only way to “resolve” this “uncertainty” is to bring reality into line with the prediction and actually fill your mouth with delicious food. On the one hand, there is a lot of basic neuroscience research that suggests something like this is going on. On the other, Wo’s writes about this further:

The basic idea seems to go roughly as follows. Suppose my internal probability function Q assigns high probability to states in which I’m having a slice of pizza, while my sensory input suggests that I’m currently not having a slice of pizza. There are two ways of bringing Q in alignment with my sensory input: (a) I could change Q so that it no longer assigns high probability to pizza states, (b) I could grab a piece of pizza, thereby changing my sensory input so that it conforms to the pizza predictions of Q. Both (a) and (b) would lead to a state in which my (new) probability function Q’ assigns high probability to my (new) sensory input d’. Compared to the present state, the sensory input will then have lower surprise. So any transition to these states can be seen as a reduction of free energy, in the unambitious sense of the term.

Action is thus explained as an attempt to bring one’s sensory input in alignment with one’s representation of the world.

This is clearly nuts. When I decide to reach out for the pizza, I don’t assign high probability to states in which I’m already eating the slice. It is precisely my knowledge that I’m not eating the slice, together with my desire to eat the slice, that explains my reaching out.

There are at least two fundamental problems with the simple picture just outlined. One is that it makes little sense without postulating an independent source of goals or desires. Suppose it’s true that I reach out for the pizza because I hallucinate (as it were) that that’s what I’m doing, and I try to turn this hallucination into reality. Where does the hallucination come from? Surely it’s not just a technical glitch in my perceptual system. Otherwise it would be a miraculous coincidence that I mostly hallucinate pleasant and fitness-increasing states. Some further part of my cognitive architecture must trigger the hallucinations that cause me to act. (If there’s no such source, the much discussed “dark room problem” arises: why don’t we efficiently minimize sensory surprise (and thereby free energy) by sitting still in a dark room until we die?)

The second problem is that efficient action requires keeping track of both the actual state and the goal state. If I want to reach out for the pizza, I’d better know where my arms are, where the pizza is, what’s in between the two, and so on. If my internal representation of the world falsely says that the pizza is already in my mouth, it’s hard to explain how I manage to grab it from the plate.

A closer look at Friston’s papers suggests that the above rough proposal isn’t quite what he has in mind. Recall that minimizing free energy can be seen as an approximate method for bringing one probability function Q close to another function P. If we think of Q as representing the system’s beliefs about the present state, and P as a representation of its goals, then we have the required two components for explaining action. What’s unusual is only that the goals are represented by a probability function, rather than (say) a utility function. How would that work?

Here’s an idea. Given the present probability function Q, we can map any goal state A to the target function Q^A, which is Q conditionalized on A — or perhaps on certain sensory states that would go along with A. For example, if I successfully reach out for the pizza, my belief function Q will change to a function Q^A that assigns high probability to my arm being outstretched, to seeing and feeling the pizza in my fingers, etc. Choosing an act that minimizes the difference between my belief function and Q^A is then tantamount to choosing an act that realizes my goal.

This might lead to an interesting empirical model of how actions are generated. Of course we’d need to know more about how the target function Q^A is determined. I said it comes about by (approximately?) conditionalizing Q on the goal state A, but how do we identify the relevant A? Why do I want to reach out for the pizza? Arguably the explanation is that reaching out is likely (according to Q) to lead to a more distal state in which I eat the pizza, which I desire. So to compute the proximal target probability Q^A we presumably need to encode the system’s more distal goals and then use techniques from (stochastic) control theory, perhaps, to derive more immediate goals.

That version of the story looks much more plausible, and much less revolutionary, than the story outlined above. In the present version, perception and action are not two means to the same end — minimizing free energy. The free energy that’s minimized in perception is a completely different quantity than the free energy that’s minimized in action. What’s true is that both tasks involve mathematically similar optimization problems. But that isn’t too surprising given the well-known mathematical and computational parallels between conditionalizing and maximizing expected utility.

It’s tempting to throw this out entirely. But part of me does feel like there’s a weird connection between curiosity and every other drive. For example, sex seems like it should be pretty basic and curiosity-resistant. But how often do people say that they’re attracted to someone “because he’s mysterious”? And what about the Coolidge Effect (known in the polyamory community as “new relationship energy”)? After a while with the same partner, sex and romance lose their magic – only to reappear if the animal/person hooks up with a new partner. Doesn’t this point to some kind of connection between sexuality and curiosity?

What about the typical complaint of porn addicts – that they start off watching softcorn porn, find after a while that it’s no longer titillating, move on to harder porn, and eventually have to get into really perverted stuff just to feel anything at all? Is this a sort of uncertainty reduction?

The only problem is that this is a really specific kind of uncertainty reduction. Why should “uncertainty about what it would be like to be in a relationship with that particular attractive person” be so much more compelling than “uncertainty about what the middle letter of the Bible is”, a question which almost no one feels the slightest inclination to resolve? The interviewers ask Friston something sort of similar, referring to some experiments where people are happiest not when given easy things with no uncertainty, nor confusing things with unresolvable uncertainty, but puzzles – things that seem confusing at first, but actually have a lot of hidden order within them. They ask Friston whether he might want to switch teams to support a u-shaped theory where people like being in the middle between too little uncertainty or too much uncertainty. Friston…does not want to switch teams.

I do not think that “different laws may apply at different levels”. I see a singular and simple explanation for all the apparent dialectics above: they are all explained by minimization of expected free energy, expected surprise or uncertainty. I feel slightly puritanical when deflating some of the (magical) thinking about inverted U curves and “sweet spots”. However, things are just simpler than that: there is only one sweet spot; namely, the free energy minimum at the bottom of a U-shaped free energy function […]

This means that any opportunity to resolve uncertainty itself now becomes attractive (literally, in the mathematical sense of a random dynamical attractor) (Friston, 2013). In short, as nicely articulated by (Schmidhuber, 2010), the opportunity to answer “what would happen if I did that” is one of the most important resolvers of uncertainty. Formally, the resolution of uncertainty (aka intrinsic motivation, intrinsic value, epistemic value, the value of information, Bayesian surprise, etc. (Friston et al., 2017)) corresponds to salience. Note that in active inference, salience becomes an attribute of an action or policy in relation to the lived world. The mathematical homologue for contingencies (technically, the parameters of a generative model) corresponds to novelty. In other words, if there is an action that can reduce uncertainty about the consequences of a particular behavior, it is more likely to be expressed.

Given these imperatives, then the two ends of the inverted U become two extrema on different dimensions. In a world full of novelty and opportunity, we know immediately there is an opportunity to resolve reducible uncertainty and will immediately embark on joyful exploration — joyful because it reduces uncertainty or expected free energy (Joffily & Coricelli, 2013). Conversely, in a completely unpredictable world (i.e., a world with no precise sensory evidence, such as a dark room) there is no opportunity and all uncertainty is irreducible — a joyless world. Boredom is simply the product of explorative behavior; emptying a world of its epistemic value — a barren world in which all epistemic affordance has been exhausted through information seeking, free energy minimizing action.

Note that I slipped in the word “joyful” above. This brings something interesting to the table; namely, the affective valence of shifts in uncertainty — and how they are evaluated by our brains.

The only thing at all I am able to gather from this paragraph – besides the fact that apparently Karl Friston cites himself in conversation – is the Schmidhuber reference, which is actually really helpful. Schmidhuber is the guy behind eg the Formal Theory Of Fun & Creativity Explains Science, Art, Music, Humor, in which all of these are some form of taking a seemingly complex domain (in the mathematical sense of complexity) and reducing it to something simple (discovering a hidden order that makes it more compressible). I think Friston might be trying to hint that free energy minimization works in a Schmidhuberian sense where it applies to learning things that suddenly make large parts of our experience more comprehensible at once, rather than just “Here are some numbers: 1, 5, 7, 21 – now you have less uncertainty over what numbers I was about to tell you, isn’t that great?”

I agree this is one of life’s great joys, though maybe me and Karl Friston are not a 100% typical subset of humanity here. Also, I have trouble figuring out how to conceptualize other human drives like sex as this same kind complexity-reduction joy.

One more concern here – a lot of the things I read about this equivocate between “model accuracy maximization” and “surprise minimization”. These end really differently. Model accuracy maximization sounds like curiosity – you go out and explore as much of the world as possible to get a model that precisely matches reality. Surprise minimization sounds like locking yourself in a dark room with no stimuli, then predicting that you will be in a dark room with no stimuli, and never being surprised when your prediction turns out to be right. I understand Friston has written about the so-called “dark room problem”, but I haven’t had a chance to look into it as much as I should, and I can’t find anything that takes one or the other horn of the equivocation and says “definitely this one”.

Fourth, okay, all of this is pretty neat, but how does it explain all biological systems? How does it explain abiogenesis? And when do we get to the real-world version of psychohistory? In his Alius interview, Friston writes:

I first came up with a prototypical free energy principle when I was eight years old, in what I have previously called a “Gerald Durrell” moment (Friston, 2012). I was in the garden, during a gloriously hot 1960s British summer, preoccupied with the antics of some woodlice who were frantically scurrying around trying to find some shade. After half an hour of observation and innocent (childlike) contemplation, I realized their “scurrying” had no purpose or intent: they were simply moving faster in the sun — and slower in the shade. The simplicity of this explanation — for what one could artfully call biotic self-organization — appealed to me then and appeals to me now. It is exactly the same principle that underwrites the ensemble density dynamics of the free energy principle — and all its corollaries.

How do the wood lice have anything to do with any of the rest of this?

As best I can understand (and I’m drawing from here and here again), this is an ultimate meaning of “free energy” which is sort of like a formalization of homeostasis. It goes like this: consider a probability distribution of all the states an organism can be in. For example, your body can be at (90 degrees F, heart rate 10), (90 degrees F, heart rate 70), (98 degrees F, heart rate 10), (98 degrees F, heart rate 70), or any of a trillion other different combinations of possible parameters. But in fact, living systems successfully restrict themselves to tiny fractions of this space – if you go too far away from (98 degrees F, heart rate 70), you die. So you have two probability distributions – the maximum-entropy one where you could have any combination of heart rate and body temperature, and the one your body is aiming for with a life-compatible combination of heart rate and body temperature. Whenever you have a system trying to convert one probability distribution into another probability distribution, you can think of it as doing Bayesian work and following free energy principles. So free energy seems to be something like just a formal explanation of how certain systems display goal-directed behavior, without having to bring in an anthropomorphic or teleological concept of “goal-directedness”.

Friston mentions many times that free energy is “almost tautological”, and one of the neuroscientists I talked to who claimed to half-understand it said it should be viewed more as an elegant way of looking at things than as a scientific theory per se. From the Alius interview:

The free energy principle stands in stark distinction to things like predictive coding and the Bayesian brain hypothesis. This is because the free energy principle is what it is — a principle. Like Hamilton’s Principle of Stationary Action, it cannot be falsified. It cannot be disproven. In fact, there’s not much you can do with it, unless you ask whether measurable systems conform to the principle.

So we haven’t got a real-life version of Asimov’s psychohistory, is what you’re saying?

But also:

The Bayesian brain hypothesis is a corollary of the free energy principle and is realized through processes like predictive coding or abductive inference under prior beliefs. However, the Bayesian brain is not the free energy principle, because both the Bayesian brain hypothesis and predictive coding are incomplete theories of how we infer states of affairs.

This missing bit is the enactive compass of the free energy principle. In other words, the free energy principle is not just about making the best (Bayesian) sense of sensory impressions of what’s “out there”. It tries to understand how we sample the world and author our own sensations. Again, we come back to the woodlice and their scurrying — and an attempt to understand the imperatives behind this apparently purposeful sampling of the world. It is this enactive, embodied, extended, embedded, and encultured aspect that is lacking from the Bayesian brain and predictive coding theories; precisely because they do not consider entropy reduction […]

In short, the free energy principle fully endorses the Bayesian brain hypothesis — but that’s not the story. The only way you can change “the shape of things” — i.e., bound entropy production — is to act on the world. This is what distinguishes the free energy principle from predictive processing. In fact, we have now taken to referring to the free energy principle as “active inference”, which seems closer to the mark and slightly less pretentious for non-mathematicians.

So maybe the free energy principle is the unification of predictive coding of internal models, with the “action in the world is just another form of prediction” thesis mentioned above? I guess I thought that was part of the standard predictive coding story, but maybe I’m wrong?

Overall, the best I can do here is this: the free energy principle seems like an attempt to unify perception, cognition, homeostasis, and action.

“Free energy” is a mathematical concept that represents the failure of some things to match other things they’re supposed to be predicting.

The brain tries to minimize its free energy with respect to the world, ie minimize the difference between its models and reality. Sometimes it does that by updating its models of the world. Other times it does that by changing the world to better match its models.

Perception and cognition are both attempts to create accurate models that match the world, thus minimizing free energy.

Homeostasis and action are both attempts to make reality match mental models. Action tries to get the organism’s external state to match a mental model. Homeostasis tries to get the organism’s internal state to match a mental model. Since even bacteria are doing something homeostasis-like, all life shares the principle of being free energy minimizers.

So life isn’t doing four things – perceiving, thinking, acting, and maintaining homeostasis. It’s really just doing one thing – minimizing free energy – in four different ways – with the particular way it implements this in any given situation depending on which free energy minimization opportunities are most convenient. Or something. All of this might be a useful thing to know, or it might just be a cool philosophical way of looking at things, I’m still not sure.

Or something like this? Maybe? Somebody please help?

Discussion question for those of you on the subreddit – if the free energy principle were right, would it disprove the orthogonality thesis? Might it be impossible to design a working brain with any goal besides free energy reduction? Would anything – even a paperclip maximizer – have to start by minimizing uncertainty, and then add paperclip maximization in later as a hack? Would it change anything if it did?