Conant & Ashby’s “__Every Good Regulator Of A System Must Be A Model Of That System__” opens with:

The design of a complex regulator often includes the making of a model of the system to be regulated. The making of such a model has hitherto been regarded as optional, as merely one of many possible ways.

In this paper a theorem is presented which shows, under very broad conditions, that any regulator that is maximally both successful and simple must be isomorphic with the system being regulated. (The exact assumptions are given.) Making a model is thus necessary.

This may be the most misleading title and summary I have ever seen on a math paper. If by “making a model” one means the sort of thing people usually do when model-making - i.e. reconstruct a system’s variables/parameters/structure from some information about them - then Conant & Ashby’s claim is simply false.

What they actually prove is that every regulator which is optimal and contains no unnecessary noise is *equivalent to* a regulator which first reconstructs the variable-values of the system it’s controlling, then chooses its output as a function of those values (ignoring the original inputs). This does *not* mean that every such regulator *actually* reconstructs the variable-values internally. And Ashby & Conant’s proof has several shortcomings even for this more modest claim.

This post presents a modification of the Good Regulator Theorem, and provides a reasonably-general condition under which any optimal minimal regulator must *actually* construct a model of the controlled system internally. The key idea is conceptually similar to __some of the pieces from Risks From Learned Optimization__. Basically: an information bottleneck can force the use of a model, in much the same way that an information bottleneck can force the use of a mesa-optimizer. Along the way, we’ll also review the original Good Regulator Theorem and a few minor variants which fix some other problems with the original theorem.

## The Original Good Regulator Theorem

We’re interested mainly in this causal diagram:

The main goal is to choose the regulator policy to minimize the entropy of outcome . Later sections will show that this is (roughly) equivalent to expected utility maximization.

After explaining this problem, Conant & Ashby replace it with a different problem, which is not equivalent, and they do not bother to point out that it is not equivalent. They just present roughly the diagram above, and then their actual math implicitly uses this diagram instead:

Rather than choosing a regulator policy , they instead choose a policy . In other words: they implicitly assume that the regulator has perfect information about the system state (and their proof does require this). Later, we’ll talk about how the original theorem generalizes to situations where the regulator does not have perfect information. But for now, I’ll just outline the argument from the paper.

We’ll use two assumptions:

- The entropy-minimizing distribution of is unique (i.e. if two different policies both achieve minimum entropy, they both produce the same -distribution). This assumption avoids a bunch of extra legwork which doesn’t really add any substance to the theorem.
- is a deterministic function of . Note that we can always make this hold by including any nondeterministic inputs to in itself (though that trick only works if we allow to have imperfect information about , which violates Conant & Ashby’s setup… more on that later).

The main lemma then says: **for any optimal regulator ****, **** is a deterministic function of **. Equivalently: all -values with nonzero probability (for a given -value ) must give the same .

Intuitive argument: if the regulator could pick two different -values (given ), then it can achieve strictly lower entropy by always picking whichever one has higher probability (unconditional on ). Even if the two have the same , always picking one or the other gives strictly lower entropy (since the one we pick will end up with higher once we pick it more often). If the regulator is optimal, then achieving strictly lower entropy is impossible, hence it must always pick the same -value given the same -value. For that argument unpacked into a formal proof, see the paper.

With the lemma nailed down, the last step in Conant & Ashby’s argument is that any remaining nondeterminism in is “unnecessary complexity”. All -values chosen with nonzero probability for a given -value must yield the same anyway, so there’s no reason to have more than one of them. We might as well make a deterministic function of .

Thus: **every “simplest” optimal regulator **(in the sense that it contains no unnecessary noise)** is a “model” of the system **(in the sense that the regulator output is a deterministic function of the system state ).

## The Problems

There are two immediate problems with this theorem:

- The notion of “model” is rather silly - e.g. the system could be quite complex, but the regulator could be an identity function, and it would count as a “model”
- The regulator is assumed to have perfect knowledge of the system state (i.e. second diagram rather than first)

Also, though I don’t consider it a “problem” so much as a choice which I think most people here will find more familiar:

- The theorem uses entropy-minimization as its notion of optimality, rather than expected-utility-maximization

We’ll address all of these in the next few sections. Making the notion of “model” less silly will take place in two steps - the first step to make it a little less silly while keeping around most of the original’s meaning, the second step to make it a lot less silly while changing the meaning significantly.

## Making The Notion Of “Model” A Little Less Silly

The notion of “model” basically says “ is a model of iff is a deterministic function of ” - the idea being that the regulator needs to reconstruct the value of from its inputs in order to choose its outputs. But the proof-as-written-in-the-paper assumes that takes as an input directly (i.e. the regulator chooses ), so really the regulator doesn’t need to “model” in any nontrivial sense in order for to be a deterministic function of . For instance, the regulator could just be the identity function: it takes in and returns . This does not sound like a “model”.

Fortunately, we can make the notion of “model” nontrivial quite easily:

- Assume that is a deterministic function of
- Assume that the regulator takes as input, rather than itself

The whole proof actually works just fine with these two assumptions, and I think this is what Conant & Ashby originally intended. The end result is that **the regulator output **** must be a deterministic function of ****, even if the regulator only takes **** as input, not **** itself** (assuming is a deterministic function of , i.e. the regulator has enough information to perfectly reconstruct ).

Note that this *still* does not mean that every optimal, not-unnecessarily-nondeterministic regulator *actually* reconstructs internally. It only shows that any optimal, not-unnecessarily-nondeterministic regulator is *equivalent to* one which reconstructs and then chooses its output as a deterministic function of (ignoring ).

## Minimum Entropy -> Maximum Expected Utility And Imperfect Knowledge

I think the theorem is simpler and more intuitive in a maximum expected utility framework, besides being more familiar.

We choose a policy function to maximize expected utility. Since there’s no decision-theoretic funny business in this particular setup, we can maximize for each -value independently:

Key thing to note: when two -values yield the same distribution function , the maximization problem

… is exactly the same for those two -values. So, we might as well choose the same optimal distribution , even if there are multiple optimal options. Using different optima for different , even when the maximization problems are the same, would be “unnecessary complexity” in exactly the same sense as Conant & Ashby’s theorem.

So: **every “simplest” **(in the sense that it does not have any unnecessary variation in decision distribution) **optimal **(in the sense that it maximizes expected utility)** regulator** **is a deterministic function of the posterior distribution of the system state **. In other words, there is some equivalent regulator which first calculates the Bayesian posterior of given , then throws away and computes its output just from that distribution.

This solves the “imperfect knowledge” issue for free. When input data is not sufficient to perfectly estimate the system state , our regulator output is a function of the posterior distribution of , rather than itself.

When system state *can* be perfectly estimated from inputs, the distribution is itself a deterministic function of , therefore the regulator output will also be a deterministic function of .

Important note: I am not sure whether this result holds for minimum entropy. It is a qualitatively different problem, and in some ways more interesting - it’s more like an embedded agency problem, since decisions for one input-value can influence the optimal choice given other -values.

## Making The Notion Of “Model” A Lot Less Silly

Finally, the main event. So far, we’ve said that regulators which are “optimal” and “simple” in various senses are *equivalent to* regulators which “use a model” - i.e. they first estimate the system state, then make a decision based on that estimate, ignoring the original input. Now we’ll see a condition under which “optimal” and “simple” regulators are not just *equivalent* to regulators which use a model, but in fact must use a model themselves.

Here’s the new picture:

Our regulator now receives two “rounds” of data (, then ) before choosing the output . In between, it chooses what information from to keep around - the retained information is the “model” . The interesting problem is to prove that, under certain conditions, will have properties which make the name “model” actually make sense.

Conceptually, “chooses which game” the regulator will play. In order to achieve optimal play across all “possible games” might choose, has to keep around any information relevant to any possible game. However, each game just takes as input (not directly), so at most has to keep around all the information relevant to . So: with a sufficiently rich “set of games” , we expect that will have to contain all information from relevant to .

On the flip side, we want this to be an information bottleneck: we want to contain as little information as possible (in an information-theoretic sense), while still achieving optimality. Combining this with the previous paragraph: we want to contain as little information as possible, while still containing all information from relevant to . That’s exactly the condition for the __Minimal Map Theorem__: must be (isomorphic to) the Bayesian distribution .

That’s what we’re going to prove: **if **** is a minimum-information optimal summary of ****, for a sufficiently rich “set of games”, then **** is isomorphic to the Bayesian posterior distribution on **** given ****, i.e. **. That’s the sense in which is a “model”.

As in the previous section, we can independently optimize for each -value:

Conceptually, our regulator sees the -value, then chooses a strategy , i.e. it chooses the distribution from which will be drawn for each value.

We’ll start with a simplifying assumption: there is a *unique* optimal regulator . (Note that we’re assuming the full black-box optimal function of the regulator is unique; there can still be internally-different optimal regulators with the same optimal black-box function, e.g. using different maps .) This assumption is mainly to simplify the proof; the conclusion survives without it, but we would need to track *sets* of optimal strategies everywhere rather than just “the optimal strategy”, and the minimal-information assumption would ultimately substitute for uniqueness of the optimal regulator.

If two -values yield the same Bayesian posterior , then they must yield the same optimal strategy . Proof: the optimization problems are the same, and the optimum is unique, so the strategy is the same. (In the non-unique case, picking different strategies would force to contain strictly more information - i.e. - so the minimal-information optimal regulator will pick identical strategies whenever it can do so. Making this reasoning fully work with many optimal -values takes a bit of effort and doesn’t produce much useful insight, but it works.)

The next step is more interesting: given a sufficiently rich set of games, not only is the strategy a function of the posterior, the posterior is a function of the strategy. If two -values yield the same strategy , then they must yield the same Bayesian posterior . What do we mean by “sufficiently rich set of games”? Well, given two different distributions and , there must be *some* particular -value for which the optimal strategy is different from . The key is that we only need *one* -value for which the optimal strategies differ between and .

So: by “sufficiently rich set of games”, we mean that for every pair of -values with different Bayesian posteriors , there exists *some* -value for which the optimal strategy differs. Conceptually: **“sufficiently rich set of games” means that for each pair of two different possible posteriors ****, **** can pick at least one “game” (i.e. optimization problem) for which the optimal policy is different under the two posteriors**.

From there, the proof is easy. The posterior is a function of the strategy, the strategy is a function of , therefore the posterior is a function of : two different posteriors and must have two different “models” and . On the other hand, we already know that the optimal strategy is a function of , so in in order for to be information-minimal it must not distinguish between -values with the same posterior . Thus: if-and-only-if . **The “model” **** is isomorphic to the Bayesian posterior **.

## Takeaway

When should a regulator use a model internally? We have four key conditions:

- The regulator needs to make optimal decisions (in an expected utility sense)
- Information arrives in more than one timestep/chunk (, then ), and needs to be kept around until decision time
- Keeping/passing information is costly: the amount of information stored/passed needs to be minimized (while still achieving optimal control)
- Later information can “choose many different games” - specifically, whenever the posterior distribution of system-state given two possible values is different, there must be
*at least one*value under which optimal play differs for the two values.

Conceptually, because we don’t know what game we’re going to play, we need to keep around all the information potentially relevant to any possible game. The minimum information which can be kept, while still keeping all the information potentially relevant to any possible game, is the Bayesian posterior on system state . There’s still a degree of freedom in how we encode the posterior on (that’s the “isomorphism” part), but the “model” M definitely has to store exactly the posterior.

Status: strong opinions, weakly held. not a control theorist; not only

readyto eat my words, but I've already set the table.As I understand it, the original good regulator theorem seems even dumber than you point out.

First, the original paper doesn't make sense to me. Not surprising, old papers are often like that, and I don't know any control theory... but here's John Baez

alsogetting stuck, giving up, and deriving his own version of what he imagines the theorem should say:Anasks whether anyone has a rigorous proof:

unansweredStackExchange questionHello?? Isn't this one of the fundamental results of control theory? Where's a good proof of it? It's been cited 1,317 times and confidently brandished to make sweeping claims about how to e.g. organize society or make an ethical superintelligence.

It seems plausible that people just read the confident title (

Every good regulator of a system must be a model of that system- of course the paper proves the claim in its title...), saw the citation count / assumed other people had checked it out (yay information cascades!), and figured it must be correct...## Motte and bailey

The paper starts off by introducing the components of the regulatory system:

OK, so we're going to be talking about how regulators which ensurealso model their environments, right? Sounds good.

goodoutcomesWait...

Later...

We're talking about the

entire outcome spaceZ again. In the introduction we focused on regulators ensuring 'good' states, but we immediately gave that up to talk about entropy H(Z).Why does this matter? Well...

## The original theorem seems even dumber than John points out

John writes:

I suppose my intuition is that this is actually a significant problem.

At first glance, Good Regulator seems to basically prove something like 'there's a deterministic optimal policy wrt the observations', but even

that'stoo generous - it proves that there's a deterministic way to minimize outcome entropy. But what does that guarantee us - how do we know that's a 'good regulator'? Like, imagine an environment with a strong "attractor" outcome, like the streets being full of garbage. The regulator can try to push against this, but they can't always succeed due to the influence of random latent variables (this cuts against the determinism assumption, but you note that this can be rectified by reincluding X). However, by sitting back, theycanensure that the streets are full of garbage.The regulator does so, minimizes the entropy over the unconditional outcome distribution Z, and is christened a 'good regulator' which has built a 'model' of the environment. In reality, we have a deterministic regulator which does nothing, and our streets are full of trash.

Now, I think it's possible I've misunderstood, so I'd appreciate correction if I have. But

ifI haven't, andifno control theorists have in fact repaired and expanded this theorem before John's post...If that's true, what the heck happened? Control theorists just left a $100 bill on the ground for decades? A quick !scholar search doesn't reveal any good critiques.

The reason I think entropy minimization is basically an ok choice here is that there's not much restriction on

whichvariable's entropy is minimized. There's enough freedom that we can transform an expected-utility-maximization problem into an entropy-minimization problem.In particular, suppose we have a utility variable U, and we want to maximize E[U]. As long as possible values of U are bounded above, we can subtract a constant without changing anything, making U strictly negative. Then, we define a new random variable Z, which is generated from U in such a way that its entropy (given U) is -U bits. For instance, we could let Z be a list of ⌊−U⌋ 50/50 coin flips, plus one biased coin flip with bias chosen so the entropy of the coin flip is −U−⌊−U⌋, i.e. the fractional part of U. Then, minimizing entropy of Z (unconditional on U) is equivalent to maximizing E[U].

Okay, I agree that if you remove their determinism & full observability assumption (as you did in the post), it seems like your construction should work.

I still think that the original paper seems awful (because it's their responsibility to justify choices like this in order to explain how their result captures the intuitive meaning of a 'good regulator').

Oh absolutely, the original is still awful and their proof does not work with the construction I just gave.

BTW, this got a huge grin out of me:

"Good regulator" is used here to mean that it is good at keeping the output "regular". That is, reducing the entropy instead of "a regulator which produces good".

On page 4 the authors acknowledge that there are numerous ways by which one could consider a regulator successful, then go on to say "In this paper we shall use the last measure, H(Z), and we define ‘successful regulation’ as equivalent, to ‘H(Z) is minimal’."

The fact that reducing entropy doesn't line up with maximizing utility is true, but the authors never claimed it did. Reducing entropy generalises to a lot of real world problems.

I'm really happy to see this. I've had similar thoughts about the Good Regulator Theorem, but didn't take the time to write them up or really pursue the fix.

Marginally related: my hope at some point was to fix the Good Regulator Theorem and then integrate it with other representation theorems, to come up with a representation theorem which derived several things simultaneously:

The most ambitious version I can think of would address several more questions:

A. Some justification for the basic algebra. (I think sigma-algebras are probably not the right algebra to base beliefs on. Something resembling linear logic might be better for reasons we've discussed privately; that's very speculative of course. Ideally the right algebra should be derived from considerations arising in construction of the representation theorem, rather than attempting to force any outcome top-down.) This is related to questions of what's the right logic, and should touch on questions of constructivism vs platonism. Due to point #3, it should also touch on formal theories of truth, particularly if we can manage a theorem related to embedded agency rather than cartesian (dualistic) agency.

B. It should be better than CCT in that it should represent the full preference ordering, rather than only the optimal policy. This may or may not lead to InfraBayesian beliefs /expected values (the InfraBayesian representation theorem being a generalization of CCT which represents the whole preference ordering).

C. It should ideally deal with logical uncertainty, not just the logically omniscient case. This is hard. (But your representation theorem for logical induction is a start.) Or failing that, it should at least deal with a logically omniscient version of Radical Probabilism, ie address the radical-probabilist critique of Bayesian updating. (See my post Radical Probabilism; currently typing on a phone, so getting links isn't convenient.

D. Obviously it would ideally also deal with questions of CDT vs EDT (ie present a solution to the problem of counterfactuals).

E. And deal with tiling problems, perhaps as part of the basic criteria.

I was impressed by this post. I don't have the mathematical chops to evaluate it as math -- probably it's fairly trivial -- but I think it's rare for math to tell us something so interesting and important about the world, as this seems to do. See this comment where I summarize my takeaways; is it not quite amazing that these conclusions about artificial neural nets are provable (or provable-given-plausible-conditions) rather than just conjectures-which-seem-to-be-borne-out-by-ANN-behavior-so-far? (E.g. conclusions like "Neural nets trained on very complex open-ended real-world tasks/environments will build, remember, and use internal models of their environments... for something which resembles expected utility maximization!") Anyhow, I guess I shouldn't focus on the provability because even that's not super important. What matters is that this seems to be a fairly rigorous argument for a conclusion which many people doubt, that is pretty relevant to this whole AGI safety thing.

It's possible that I'm making mountains out of molehills here so I'd be interested to hear pushback. But as it stands I feel like the ideas in this post deserve to be turned into a paper and more widely publicized.

Planned summary (edited in response to feedback):

Four things I'd change:

fromXrelevant toS, not the information that S contains (since it doesn't have access to all the information S contains).everychange in S changes the optimal strategy. With information coming in two steps, we have to keep around all the information from the first step which could be relevant toanyof the possible games; our "strategy" includes strategies for the sub-games resulting from each possible value of the second input, and a change in any one of those is enough.I was considering this, but the problem is that in your setup S is supposed to be derived from X (that is, S is a deterministic function of X), which is not true when X = training data and S = that which we want to predict.

If S is derived from X, then "information in S" = "information in X relevant to S"

Fair point. I kind of wanted to abstract away this detail in the operationalization of "relevant", but it does seem misleading as stated. Changed to "important for optimal performance".

I was hoping that this would come through via the neural net example, where Z obviously includes new information in the form of the new test inputs which have to be labeled. I've added the sentence "Note that it is usually important that Z contains some new input (e.g. test images to be classified) to prevent M from hardcoding solutions to Z without needing to look at S" to the second point to clarify.

(In general I struggled with keeping the summary short vs. staying true to the details of the causal model.)

That's an (implicit) assumption in Conant & Ashby's setup, I explicitly remove that constraint in the "Minimum Entropy -> Maximum Expected Utility and Imperfect Knowledge" section. (That's the "imperfect knowledge" part.)

Same here. Once we relax the "S is a deterministic function of X" constraint, the "information in X relevant to S" is exactly the posterior distribution (s↦P[S=s|X]), which is why that distribution comes up so much in the later sections.

Yeah, the number of necessary nontrivial pieces is... just a little to high to not have to worry about inductive distance.

... That'll teach me to skim through the math in posts I'm trying to summarize. I've edited the summary, lmk if it looks good now.

Good enough. I don't love it, but I also don't see easy ways to improve it without making it longer and more technical (which would mean it's not strictly an improvement). Maybe at some point I'll take the time to make a shorter and less math-dense writeup.

Appreciate this post! I had seen the good regulator theorem referenced every now and then, but wasn't sure what exactly the relevant claims were, and wouldn't have known how to go through the original proof myself. This is helpful.

(E.g. the result was cited by Frith & Metzinger as part of their argument that, as an agent seeks to avoid being punished by society, this constitutes an attempt to regulate society's behavior; and for the regulation be successful, the agent needs to internalize a model of the society's preferences, which once internalized becomes something like a subagent which then regulates the agent in turn and causes behaviors such as self-punishment. It sounds like the math of the theorem isn't very strongly relevant for that particular argument, though some form of the overall argument still sounds plausible to me regardless.)

Given your four conditions, I wonder if there's a result like "optimally power-seeking agents (minimizing information costs) must model the world." That is, I think power is about being able to achieve a wide range of different goals (to win at 'many different games' the environment could ask of you), and so if you want to be able to sufficiently accurately estimate the expected power provided by a course of action, you have to know how well you can win at all these different games.

Yes! That is exactly the sort of theorem I'd expect to hold. (Though you might need to be in POMDP-land, not just MDP-land, for it to be interesting.)

Thanks, this is really cool! I don't know much about this stuff so I may be getting over-hyped, but still.

I'd love to see a follow-up to this post that starts with the Takeaway and explains how this would work in practice for a big artificial neural net undergoing training. Something like this, I expect:

--There are lottery tickets that make pretty close to optimal decisions, and as training progresses they get increasingly more weight until eventually the network is dominated by one or more of the close-to-optimal lottery tickets.

--Because of key conditions 2 and 3, the optimal tickets will involve some sort of subcomponent that compresses information from X and stores it to later be combined with Y.

--Key condition 4 might not be strictly true in practice; it might not be that our dataset of training examples is so diverse that literally every way the distribution over S could vary corresponds to a different optimal behavior. And even if our dataset was that diverse, it might take a long time to learn our way through the entire dataset. However, what we CAN say is that the subcomponent that compresses information from X and stores it to later be combined with Y will increasingly (as training continues) store "all and only the relevant information," i.e. all and only the information that thus-far-in-training has mattered to performance. Moreover, intuitively there is an

extent to whichY can "choose many different games," anextent to whichthe training data so far has "made relevant" information about various aspects of S. To the extent that Y can choose many different games -- that is, to the extent that the training data makes aspects of S relevant -- the network will store information about those aspects of S.--Thus, for a neural network being trained on some very complex open-ended real-world task/environment, it's plausible that Y can "choose many different games" to a large extent, such that the close-to-optimal lottery tickets will have an information-compressing-and-retaining subcomponent that contains lots of information about S but not much information about X. In particular it in some sense "represents all the aspects of S that are likely to be relevant to decision-making."

--Intuitively, this is sufficient for us to be confident in saying: Neural nets trained on very complex open-ended real-world tasks/environments

will build, remember, and use internal models of their environments. (Maybe we even add something like...for something which resembles expected utility maximization!)Anyhow. I think this is important enough that there should be a published paper laying all this out. It should start with the theorem you just proved/fixed, and then move on to the neural net context like I just sketched. This is important because it's something a bunch of people would want to cite as a building block for arguments about agency, mesa-optimizers, human modelling, etc. etc.

Your bullet points are basically correct. In practice, applying the theorem to any particular NN would require some careful setup to make the causal structure match - i.e. we have to designate the right things as "system", "regulator", "map", "inputs X & Y", and "outcome", and that will vary from architecture to architecture. But I expect it can be applied to most architectures used in practice.

I'm probably not going to turn this into a paper myself soon. At the moment, I'm pursuing threads which I think are much more promising - in particular, thinking about when a "regulator's model" mirrors the

structureof the system/environment, not just its black-box functionality. This was just a side-project within that pursuit. If someone else wants to turn this into a paper, I'd be happy to help, and there's enough technical work to be done in applying it to NNs that you wouldn't just be writing up this post.Doesn't sound like a job for me, but would you consider e.g. getting a grant to hire someone to coauthor this with you? I think the "getting a grant" part would not be the hard part.

Yeah, "get a grant" is definitely not the part of that plan which is a hard sell. Hiring people is a PITA. If I ever get to a point where I have enough things like this, which could relatively-easily be offloaded to another person, I'll probably do it. But at this point, no.

What is the type signature of the regulator? It's a policy on state space S, and it returns states as well? Are those its "actions"? (From the point of view of the causal graph, I suppose Z just depends on whatever the regulator outputs, and the true state S, so maybe it's not important

whatthe regulator outputs. Just that by the original account, any deterministic regulator could be "optimal", even if it doesn't do meaningful computation.)That's the right question to ask. Conant & Ashby intentionally leave both the type signature and the causal structure of the regulator undefined - they have a whole spiel about how it can apply to multiple different setups (though they fail to mention that in some of those setups - e.g. feedback control - the content of the theorem is trivial).

For purposes of my version of the theorem, the types of the variables themselves don't particularly matter, as long as the causal structure applies. The proofs implicitly assumed that the variables have finitely many values, but of course we can get around that by taking limits, as long as we're consistent about our notion of "minimal information".

Note on notation...

You can think of something like x↦f(x,Y) as a python dictionary mapping x-values to the corresponding f(x,Y) values. That whole dictionary would be a function of Y. In the case of something like (y,r↦P∗[R=r|X,Y=y]), it's a partial policy mapping each second-input-value y and regulator output value r to the probability that the regulator chooses that output value on that input value, and we're thinking of that whole partial policy as a function of the first input value X. So, it's a function which is itself a random variable constructed from X.

The reason I need something like this is because sometimes I want to say e.g. "two policies (x,r↦P[R=r|X=x]) are identical" (i.e. P[R=r|X=x] is the the same for all r, x), sometimes I want to say "two distributions (r↦P[R=r|X]) are identical" (i.e. two X-values yield the same output distribution), etc, and writing it all out in terms of quantifiers makes it hard to see what's going on conceptually.

I've been trying to figure out a good notation for this, and I haven't settled on one, so I'd be interested in peoples' thoughts on it. Thankyou to TurnTrout for some good advice already; I've updated the post based on that. The notation remains somewhat cumbersome and likely confusing for people not accustomed to dense math notation; I'm interested in suggestions to improve both of those problems.