Natural Latents: The Math

David Lorell

[-]Jeremy Gillen10mo3814Review for 2023 Review

This post deserves to be remembered as a LessWrong classic.

It directly tries to solve a difficult and important cluster of problems (whether it succeeds is yet to be seen).
It uses a new diagrammatic method of manipulating sets of independence relations.
It's a technical result! These feel like they're getting rarer on LessWrong and should be encouraged.

There are several problems that are fundamentally about attaching very different world models together and transferring information from one to the other.

Ontology identification involves taking a goal defined in an old ontology^[1] and accurately translating it into a new ontology.
High-level models and low-level models need to interact in a bounded agent. I.e. learning a high-level fact should influence your knowledge about low-level facts and vice versa.
Value identification is the problem of translating values from a human to an AI. This is much like ontology identification, with the added difficulty that we don't get as much detailed access or control over the human world model.
Interpretability is about finding recognisable concepts and algorithms in trained neural networks.

In general, we can solve these problems using shared variables and shared sub-structures that are present in both models.

We can stitch together very different world models along shared variables. E.g. if you have two models of molecular dynamics, one faster and simpler than the other. You want to simulate in the fast one, then switch to the slow one when particular interactions happen. To transfer the state from one to the other you identify variables present in both models (probably atom locations, velocities, some others), then just copy these values to the other model. Under-specified variables must be inferred from priors.
If you want to transfer a new concept from WM1 to a less knowledgeable WM2, you can do so by identifying the lower-level concepts that both WMs share, then constructing an "explanation" out of those concepts. An "explanation" would look like a WM fragment purely built out of variables and structures already in WM2.
An explanation is also a pointer. If you want to point at a very specific concept in someone else's WM, one way to do so is to explain that concept (in terms of lower level ideas that you are confident are shared).

Natural latents are a step toward solving all of these problems, via describing a subset of variables/structures that we should expect to find across all WMs (and more importantly, some of the conditions required for them to be present).

A natural latent should be extremely useful for any WM that contains variables which share redundant information. I think we can expect this to be common when highly redundant observations are compressed.

For example: If the ~same observation happens more than once, then any learner that generalizes well is going to notice this. It must somehow store the duplicated information (as well as each of the places where it is duplicated). That shared information is a natural latent. The result in this post suggests that this summary information should be isomorphic between agents, under the right conditions.^[2]

As far as I know, it's an open question which properties of environments&agents imply lots of natural latents.

The current state of this work has some limitations:

Both learners need access to the same or very similar low level observables X.
Both learners must have learned the same beliefs, otherwise their latents may be very different (although this is kinda fixed with an additional constraint on the latent).
John and David seem to have run into difficulties building useful applications of this theory.
John's posts aren't clear on how to identify and separate out the X variables from a general stream of data (although this seems fine for now).
With lots of compute, approximate models can be dropped in exchange for detailed models. One might drop the concept of "tree" in exchange for a complete categorization of types of trees.
- On the one hand, this is still tracking the same latent information. The theorems still work. But on the other hand, it isn't necessarily storing it in an easy-to-access way. This is fine for communication, but less fine for interpretability or manual joining of WMs.
- Perhaps there is some assumption we can make that guarantees all levels of abstraction will remain stored. Or perhaps we should expect interpretability of a WM to often involve some inferential work on the part of that WM.

If this line of research goes well, I hope that we will have theorems that say something like:

"For some bounded agent design, given observations of some complicated well-understood data-generating structure Z, and enough compute / attentional resources, the agent will learn a model of Z which contains parts x,y,w (each with predicable levels of approximate isomorphism to parts of the real Z). Upon observing more data, we can expect some parts (x,y) of this structure to remain unchanged."

^{^}
Think of an ontology the choice of variables in a particular Bayes net, for our current purposes.
^{^}
I'm leaning on the algorithmic definition of natural latents here.

[-]Alexander Gietelink Oldenziel2y2421

It is a truth universally acknowledged that any article about the hidden nature of reality will not garner close to the number of comments as one about your sexy sexy homies. As the inaugural and possibly lone commenter, let me say I'm delighted to see finally this long-awaited post. I look forward to do a deep dive soon.

[-]Olli Järviniemi10mo80Review for 2023 Review

I sometimes use the notion of natural latents in my own thinking - it's useful in the same way that the notion of Bayes networks is useful.

A frame I have is that many real world questions consist of hierarchical latents: for example, the vitality of a city is determined by employment, number of companies, migration, free-time activities and so on, and "free-time activities" is a latent (or multiple latents?) on its own.

I sometimes get use of assessing whether a topic at hand is a high-level or low-level latent and orienting accordingly. For example: if the topic at hand is "what will the societal response to AI be like?", it's by default not a great conversational move to talk about one's interactions with ChatGPT the other day - those observations are likely too low-level^[1] to be informative about the high-level latent(s) under discussion. Conversely, if the topic at hand is low-level, then analyzing low-level observations is very sensible.

(One could probably have derived the same every-day lessons simply from Bayes nets, without the need for natural latent math, but the latter helped me clarify "hold on, what are the nodes of the Bayes net?")

But admittedly, while this is a fun perspective to think about, I haven't got that much value out of it so far. This is why I give this post +4 instead of +9 for the review.

^{^}
And, separately, too low sample size.

[-]tailcalled2y80

There's a thing I keep thinking of when it comes to natural latents. In social science one often speaks of statistical sex differences, e.g. women tend to be nicer than men. But these sex differences aren't deterministic, so one can't derive them from simply meeting 1 woman and 1 man. It feels like this is in tension with the insensitivity condition, like that this only permits universal generalizations (I guess stuff like anatomy?). But also I haven't really practiced natural latents at all so I sort of expect to be using them in a suboptimal way. Maybe you have a better idea?

Like I mean obviously one can deterministically recover it when one considers large samples of people, so it seems like maybe the "statistical man" and "statistical woman" concepts are different from the "universal man" and "universal woman" concepts. Not entirely sure this is a good approach though.

[-]johnswentworth2y90

It feels like this is in tension with the insensitivity condition, like that this only permits universal generalizations (I guess stuff like anatomy?)

Quite the opposite! In that example, what the insensitivity condition would say is: if I get a big sample of people (roughly 50/50 male/female), and quantify the average niceness of men and women in that sample, then I expect to get roughly the same numbers if I drop any one person (either man or woman) from the sample. It's the statistical average which has to be insensitive; any one "sample" can vary a lot.

That said, it does need to be more like a universal generalization if we impose a stronger invariance condition. The strongest invariance condition would say that we can recover the latent from any one "sample", which would be the sort of "universal generalization" you're imagining. Mathematically, the main thing that would give us is much stronger approximations, i.e. smaller 's.

[-]tailcalled2y20

Oops, my bad for focusing on the simplified version and then extrapolating incorrectly.

[-]Lorxus1y60

Probabilities of zero are extremely load-bearing for natural latents in the exact case...

Dumb question: Can you sketch out an argument for why this is the case and/or why this has to be the case? I agree that ideally/morally this should be true, but if we're already accepting a bounded degree of error elsewhere, what explodes if we accept it here?

[-]johnswentworth1y73

Consider the exact version of the redundancy condition for latent over $X_{1}, X_{2}$ :

$P [Λ, X_{1}, X_{2}] = P [Λ | X_{1}] P [X_{1}, X_{2}]$

and

$P [Λ, X_{1}, X_{2}] = P [Λ | X_{2}] P [X_{1}, X_{2}]$

Combine these two and we get, for all $Λ, X_{1}, X_{2}$ :

$P [Λ | X_{1}] = P [Λ | X_{2}]$ OR $P [X_{1}, X_{2}] = 0$

That's the foundation for a conceptually-simple method for finding the exact natural latent (if one exists) given a distribution $P [X_{1}, X_{2}]$ :

Pick a value $X_{1}, X_{2}$ which has nonzero probability, and initialize a set $S$ containing that value. Then we must have $P [Λ | X \in S] = P [Λ | X_{1}] = P [Λ | X_{2}]$ for all $Λ$ .
Loop: add to $S$ a new value $X_{1}^{'}, X_{2}$ or $X_{1}, X_{2}^{'}$ where the value $X_{2}$ or $X_{1}$ (respectively) already appears in one of the pairs in $S$ . Then $P [Λ | X_{1}^{'}] = P [Λ | X \in S]$ or $P [Λ | X_{2}^{'}] = P [Λ | X \in S]$ , respectively. Repeat until there are no more candidate values to add to $S$ .
Pick a new pair and repeat with a new set, until all values of $X$ have been added to a set.
Now take the latent to be the equivalence class in which $X$ falls.

Does that make sense?

[-]Lorxus1y30

As I also said in person, very much so!

[-]yrimon2y60

Let's say every day at the office, we get three boxes of donuts, numbered 1, 2, and 3. I grab a donut from each box, plunk them down on napkins helpfully labeled X1, X2, and X3. The donuts vary in two aspects: size (big or small) and flavor (vanilla or chocolate). Across all boxes, the ratio of big to small donuts remains consistent. However, Boxes 1 and 2 share the same vanilla-to-chocolate ratio, which is different from that of Box 3.

Does the correlation between X1 and X2 imply that there is no natural latent? Is this the desired behavior of natural latents, despite the presence of the common size ratio? (and the commonality that I've only ever pulled out donuts; there has never been a tennis ball in any of the boxes!)

If so, why is this what we want from natural latents? If not, how does a natural latent arise despite the internal correlation?

[-]Thane Ruthenis2y*60

My take would be to split each "donut" variable into "donut size" $S_{i}$ and "donut flavour" $F_{i}$ . Then there a natural latent for the whole ${S_{i}}$ set of variables, and no natural latent for the whole ${F_{i}}$ set. ${F_{i}}$ basically becomes the "other stuff in the world" $Z$ variable relative to ${S_{i}}$ .

Granted, there's an issue in that we can basically do that for any set of variables $X_{i}$ , even entirely unrelated ones: deliberately search for some decomposition of $X_{i}$ into an $S_{i}$ and an $F_{i}$ such that there's a natural latent for $S_{i}$ . I think some more practical measures could be taken into account here, though, to enure that the abstractions we find are useful. For example, we can check the relative information contents/entropies of ${X_{i}}$ and ${S_{i}}$ , thereby measuring "how much" of the initial variable-set we're abstracting over. If it's too little, that's not a useful abstraction.^[1]

That passes my common-sense check, at least. It's essentially how we're able to decompose and group objects along many different dimensions. We can focus on objects' geometry (and therefore group all sphere-like objects, from billiard balls to planets to weather balloons) or their material (grouping all objects made out of rock) or their origin (grouping all man-made objects), etc.

Each grouping then corresponds to an abstraction, with its own generally-applicable properties. E. g., deriving a "sphere" abstraction lets us discover properties like "volume as a function of radius", and then we can usefully apply that to any spherical object we discover. Similarly, man-made objects tend to have a purpose/function (unlike natural ones), which likewise lets us usefully reason about that whole category in the abstract.

(Edit: On second thoughts, I think the obvious naive way of doing that just results in ${S_{i}}$ containing all mutual information between $X_{i}$ , with the "abstraction" then just being said mutual information. Which doesn't seem very useful. I still think there's something in that direction, but probably not exactly this.)

^{^}
Relevant: Finite Factored Sets, which IIRC offer some machinery for these sorts of decompositions of variables.

[-]yrimon2y50

This branch of research is aimed at finding a (nearly) objective way of thinking about the universe. When I imagine the end result, I imagine something that receives a distribution across a bunch of data, and finds a bunch of useful patterns within it. At the moment that looks like finding patterns in data via
find_natural_latent(get_chunks_of_data(data_distribution))
or perhaps showing that
find_top_n(n, (chunks, natural_latent(chunks)) for chunks in
all_chunked_subsets_of_data(data_distribution),
key=lambda chunks, latent: usefulness_metric(latent))
is a (convergent sub)goal of agents. As such, the notion that the donuts' data is simply poorly chunked - which needs to be solved anyway - makes a lot of sense to me.

I don't know how to think about the possibilities when it comes to decomposing . Why would it always be possible to decompose random variables to allow for a natural latent? Do you have an easy example of this?
Also, what do you mean by mutual information between $X_{i}$ , given that there are at least 3 of them? And why would just extracting said mutual information be useless?
If you get the chance to point me towards good resources about any of these questions, that would be great.

[-]johnswentworth2y53

Regarding chunking: a background assumption for me is that the causal structure of the world yields a natural chunking, with each chunk taking up a little local "voxel" of spacetime.

Some amount spacetime-induced chunking is "forced upon" an embedded agent, in some sense, since their sensors and actuators are localized in spacetime.

Now, there's still degrees of freedom in taking more or less coarse-grained chunkings, and more or less coarse-graining differentially along different spacetime directions or in different places. But I expect that spacetime locality mostly nails down what we need as a starting point for convergent chunking.

[-]Thane Ruthenis2y30

Also, what do you mean by mutual information between , given that there are at least 3 of them?

You can generalize mutual information to N variables: interaction information.

Why would it always be possible to decompose random variables to allow for a natural latent?

Well, I suppose I overstated it a bit by saying "always"; you can certainly imagine artificial setups where the mutual information between a bunch of variables is zero. In practice, however, everything in the world is correlated with everything else, so in a real-world setting you'll likely find such a decomposition always, or almost always.

And why would just extracting said mutual information be useless?

Well, not useless as such – it's a useful formalism – but it would basically skip everything John and David's post is describing. Crucially, it won't uniquely determine whether a specific set of objects represents a well-abstracting category.

The abstraction-finding algorithm should be able to successfully abstract over data if and only if the underlying data actually correspond to some abstraction. If it can abstract over anything, however – any arbitrary bunch of objects – then whatever it is doing, it's not finding "abstractions". It may still be useful, but it's not what we're looking for here.

Concrete example: if we feed our algorithm 1000 examples of trees, it should output the "tree" abstraction. If we feed our algorithm 200 examples each of car tires, trees, hydrogen atoms, wallpapers, and continental-philosophy papers, it shouldn't actually find some abstraction which all of these objects are instances of. But as per the everything-is-correlated argument above, they likely have non-zero mutual information, so the naive "find a decomposition for which there's a natural latent" algorithm would fail to output nothing.

More broadly: We're looking for a "true name" of abstractions, and mutual information is sort-of related, but also clearly not precisely it.

[-]yrimon2y61

We could remove information from For instance, $Λ^{'}$ could be a bit indicating whether the temperature is above 100°C

I don't understand how this is less information than a bit indicating whether the temperature is above 50C. Specifically, given a bit telling you whether the temperature is above 50C, how do you know whether the temperature is above 100C or between 50C and 100C?

[-]johnswentworth2y50

Y'know, in hindsight I didn't think that one through correctly when writing the example.

[-]J Bostock2y53

I know very, very little about category theory, but some of this work regarding natural latents seem to absolutely smack of it. There seems to be a fairly important three-way relationship between causal models, finite factored sets, and Bayes nets.

To be precise, any causal model consisting of root sets , downstream sets $X$ , and functions mapping sets to downstream sets like $f_{4} : (B_{1} \otimes B_{3} \otimes X_{2}) \to X_{4}$ must, when equipped with a set of independent probability distributions over B, create a joint probability distribution compatible with the Bayes net that's isomorphic to the causal model in the obvious way. (So in the previous example, there would be arrows from only $B_{1}$ , $B_{3}$ , and $X_{2}$ to $X_{4}$ ) The proof of this seems almost trivial but I don't trust myself not to balls it up somehow when working with probability theory notation.

In the resulting Bayes net, one "minimal" natural latent which conditionally separates $X_{i}$ and $X_{j}$ is just the probabilities over just the root elements from $B$ which both $X_{i}$ and $X_{j}$ depend on. It might be possible to show that this "minimal" construction of $Λ$ satisfies a universal property, and so other $Λ^{'}$ which is also "minimal" in this way must be isomorphic to $Λ$ .

[-]XelaP6mo40

I think you could make the first theorem in the post (simplified fundamental theorem on two variables) easier to understand to a novice if you explicitly clarified that the conclusion diagram Λ' -> Λ -> X is the same as Λ' <- Λ -> X by the chain rerooting rule, and perhaps use the latter diagram in the picture as it more directly makes clear the idea of mediation/inducing independence.

[-]LawrenceC2yΩ440

Probabilities of zero are extremely load-bearing for natural latents in the exact case, and probabilities near zero are load-bearing in the approximate case; if the distribution is zero nowhere, then it can only have a natural latent if the ’s are all independent (in which case the trivial variable is a natural latent).

I'm a bit confused why this is the case. It seems like in the theorems, the only thing "near zero" is that D_KL (joint, factorized) < epsilon ~= 0 . But you. can satisfy this quite easily even with all probabilities > 0.

E.g. the trivial case where all variables are completely independents satisfies all the conditions of your theorem, but can clearly have every pair of probabilities > 0. Even in nontrivial cases, this is pretty easy (e.g. by mixing in irreducible noise with every variable).

[-]johnswentworth2yΩ220

Roughly speaking, all variables completely independent is the only way to satisfy all the preconditions without zero-ish probabilities.

This is easiest to see if we use a "strong invariance" condition, in which each of the must mediate between $X_{¯ i}$ and $Λ$ . Mental picture: equilibrium gas in a box, in which we can measure roughly the same temperature and pressure ( $Λ$ ) from any little spatially-localized chunk of the gas ( $X_{i}$ ). If I estimate a temperature of 10°C from one little chunk of the gas, then the probability of estimating 20°C from another little chunk must be approximately-zero. The only case where that doesn't imply near-zero probabilities is when all values of both chunks of gas always imply the same temperature, i.e. $Λ$ only ever takes on one value (and is therefore informationally empty). And in that case, the only way the conditions are satisfied is if the chunks of gas are unconditionally independent.

[-]LawrenceC2yΩ220

Hm, it sounds like you're claiming that if each pair of x, y, z are pairwise independent conditioned on the third variable, and p(x, y, z) =/= 0 for all x, y, z with nonzero p(x), p(y), p(z), then ?

I tried for a bit to show this but couldn't prove it, let alone the general case without strong invariance. My guess is I'm probably missing something really obvious.

[-]johnswentworth2yΩ220

Yeah, that's right.

The secret handshake is to start with " is independent of $Y$ given $Z$ " and " $X$ is independent of $Z$ given $Y$ ", expressed in this particular form:

$P [X, Y, Z] = P [X | Z] P [Y, Z] = P [X | Y] P [Y, Z]$

... then we immediately see that $P [X | Z] = P [X | Y]$ for all $X, Y, Z$ such that $P [Y, Z] > 0$ .

So if there are no zero probabilities, then $P [X | Z] = P [X | Y]$ for all $X, Y, Z$ .

That, in turn, implies that $P [X | Z]$ takes on the same value for all Z, which in turn means that it's equal to $P [X]$ . Thus $X$ and $Z$ are independent. Likewise for $X$ and $Y$ . Finally, we leverage independence of $Y$ and $Z$ given $X$ :

$P [X, Y, Z] = P [Y | X] P [Z | X] P [X]$

$= P [Y] P [Z] P [X]$

(A similar argument is in the middle of this post, along with a helpful-to-me visual.)

[-]LawrenceC2yΩ440

Right, the step I missed on was that P(X|Y) = P(X|Z) for all y, z implies P(X|Z) = P(X). Thanks!

[-]Gurkenglas2y40

Absent feedback, today I read further, to the premise of the maxent conjecture. Let X be 100 numbers up to 1 million, rerolled until the remainder of their sum modulo 1000000 ends up 0 or 1. (X' will have sum-remainder circa 50 or circa -50.) Given X', X1 has a 25%/50%/25% pattern around X'1. Given X2 through X100, X1 has a 50%/50% distribution. So the (First/Strong) Universal Natural Latent Conjecture fails, right?

[-]johnswentworth2y52

I believe no natural latent exists in that case?

The Universal Natural Latent Conjecture doesn't say X' is always a natural latent, it says X' is a natural latent if any natural latent exists.

[-]Oliver Sourbut2y42

One thing that initially stood out to me on the fundamental theorem was: where did the arrow come from? It 'gets introduced' in the first bookkeeping step (we draw $N \to M$ and then reorder the $(N, M, X_{¯ i})$ subgraph at each $i$ .

This seemed suspicious to me at first! It seemed like kind of a choice, so what if we just didn't add that arrow? Could we land at a conclusion of $N$ AND $M \to X$ ? That's way too strong! But I played with it a bit, and there's no obvious way to do the second frankenstitch which brings everything together unless you draw in that extra arrow and rearrange. You just can't get a globally consistent topological ordering without $N$ somehow becoming ancesterable to $X_{¯ i}$ . (Otherwise the glommed $X_{¯ i}$ variables interfere with each other when you try to find $N$ 's ancestors in the stitch.)

Still, this move seems quite salient - in particular that arrow-addition feels something like the 'lossiest' step in the proof (except for the final bookkeeping which gloms all the $X_{i}$ together, implicitly drawing a load of arrows between them all)?

[-]Nina Panickssery2y40

Is my understanding correct or am I missing something?

A latent variable is a variable you can sample that gives you some subset of the mutual information between all the different X's + possibly some independent extra "noise" unrelated to the X's
A natural latent is a variable that you can sample that at the limit of sampling will give you all the mutual information between the X's - nothing more or less

E.g. in the biased die example, every each die roll sample has, in expectation, the same information content, which is the die bias + random noise, and so the mutual info of n rolls is the die bias itself

(where "different X's" above can be thought of as different (or the same) distributions over the observation space of X corresponding to the different sampling instances, perhaps a non-standard framing)

[-]johnswentworth2y20

First bullet is correct, second bullet is close but not quite right. Just one sample of a natural latent will give you (approximately) all the mutual information between the X's, and can give you some additional "noise" as well.

E.g. in the biased die example with many rolls, we can sample the bias given the rolls. Because that distribution is very pointy the sample will be very close to the "true bias", that one sample will capture approximately-all of the mutual information between the rolls.

(Note: I did skip a subtle step there - our natural latents need a stronger condition than just "close to the true bias" in this example, since the low-order bits of the latent could in-principle contain a bunch of relevant information which the true bias doesn't; that would mess everything up. And indeed, that would mess everything up if we tried to use e.g. the empirical frequencies rather than a sample from P[bias | X]: given all but one die roll and the empirical frequencies calculated from all of the die rolls, we could exactly calculate the outcome of the remaining die roll. That's why we do the sampling thing; the little bit of noise introduced by sampling is load-bearing, since it wipes out info in those low-order bits.

... but that's a subtlety which you should not worry about until after the main picture makes sense conceptually.)

[-]RogerDearnaley2y*4-1

Speaking as an (ex-)scientist, the fact that natural latents exist, can be found, and are well defined and basically unique for a systems feels… blindingly obvious: we do this all the time. But I gather we never previously had a description of their necessary properties in term of Bayes nets before, nor a proof of in what sense they are unique. I don't think I was ever very worried that AI was going to discover an entirely different ontology for physics so different that pressure or temperature were not well-defined abstractions represented in it, but I also didn't have a uniqueness proof demonstrating that fact, and with the fate of the human race potentially on the line, that's a reassuring thing to have. Physicists often have a tendency to take on trust things that any mathematician would be aghast at, just because they seem to work in practice. [But then, neither have I ever figured out why anyone thinks the diamond-maximization problem is hard, as opposed to just tedious (you just have to list your purity specifications in isotopic and crystalographic terms, and then put it inside a standard causal wrapper).]

[-]Linda Linsefors11mo*Ω230

In standard form, a natural latent is always approximately a deterministic function of . Specifically: $Λ (X) \approx \prod_{i} (x^{'} \mapsto P [X_{i} = x_{i}^{'} | X_{¯ i}])$ .

What does the arrow mean in this expression?

[-]Dalcy1y30

I've been doing a deep dive on this post, and while the main theorems make sense I find myself quite confused about some basic concepts. I would really appreciate some help here!

So 'latents' are defined by their conditional distribution functions whose shape is implicit in the factorization that the latents need to satisfy, meaning they don't have to always look like , they can look like $P [Λ], P [X | Λ]$ , etc, right?
I don't get the 'standard form' business. It seems like a procedure to turn one latent variable $Λ$ into another relative to $X$ ? I don't get what the notation $Λ^{*} = (x \mapsto P [X = x ∣ Λ])$ means—does it mean that it takes $Λ$ defined by some conditional distribution function like $P [Λ], P [X | Λ], P [Λ | X]$ and converts it into $P [X | Λ]$ ? That doesn't seem so, the notation looks more like a likelihood function than a conditional distribution. But then what conditional distribution defines this latent $Λ^{*}$ ?

The Resampling stuff is a bit confusing too:

if we have a natural latent $Λ$ , then construct a new natural latent by resampling $Λ$ conditional on $X$ (i.e. sample from $P [Λ | X]$ ), independently of whatever other stuff $Λ^{'}$ we’re interested in.

I don't know what operation is being performed here - what CPDs come in, what CPDs leave.
"construct a new natural latent by resampling $Λ$ conditional on $X$ (i.e. sample from $P [Λ | X]$ ), independently of whatever other stuff $Λ^{'}$ we’re interested in." isn't this what we are already doing when stating a diagram like $X_{1} \leftarrow Λ \to X_{2}$ , which implies a factorization $P [Λ] P [X_{1} | Λ] P [X_{2} | Λ]$ , none of which have $Λ^{'}$ ! What changes when resampling? aaaaahhh I think I'm really confused here.
Also does all this imply that we're starting out assuming that $Λ$ shares a probability space with all the other possible latents, e.g. $P [X, Λ, Λ^{'}, Λ^{''}, \dots]$ ? How does this square with a latent variable being defined by the CPD implicit in the factorization?

And finally:

In standard form, a natural latent is always approximately a deterministic function of $X$ . Specifically: $Λ (X) \approx \prod_{i} (x^{'} \mapsto P [X_{i} = x_{i}^{'} | X_{¯ i}])$ .
...
Suppose there exists an approximate natural latent over $X_{1}, \dots, X_{n}$ . Construct a new random variable $X^{'},$ sampled from the distribution $(x^{'} \mapsto \prod_{i} P [X_{i} = x_{i}^{'} | X_{¯ i}])$ . (In other words: simultaneously resample each $X_{i}$ given all the others.) Conjecture: $X^{'}$ is an approximate natural latent (though the approximation may not be the best possible). And if so, a key question is: how good is the approximation?

Where is the top result proved, and how is this statement different from the Universal Natural Latent Conjecture below? Also is this post relevant to either of these statements, and if so, does that mean they only hold under strong redundancy?

[-]johnswentworth1y60

So 'latents' are defined by their conditional distribution functions whose shape is implicit in the factorization that the latents need to satisfy, meaning they don't have to always look like , they can look like $P [Λ], P [X | Λ]$ , etc, right?

The key idea here is that, when "choosing a latent", we're not allowed to choose $P [X]$ ; $P [X]$ is fixed/known/given, a latent is just a helpful tool for reasoning about or representing $P [X]$ . So another way to phrase it is: we're choosing our whole model $P [X, Λ]$ , but with a constraint on the marginal $P [X]$ . $P [Λ | X]$ then captures all of the degrees of freedom we have in choosing a latent.

Now, we won't typically represent the latent explicitly as $P [Λ | X]$ . Typically we'll choose latents such that $P [X, Λ]$ satisfies some factorization(s), and those factorizations will provide a more compact representation of the distribution than two giant tables for $P [X]$ , $P [Λ | X]$ . For instance, insofar as $P [Λ, X]$ factors as $P [Λ] \prod_{i} P [X_{i} | Λ]$ , we might want to represent the distribution as $P [Λ]$ and ${P [X_{i} | Λ]}$ (both for analytic and computational purposes).

I don't get the 'standard form' business.

We've largely moved away from using the standard form anyway, I recommend ignoring it at this point.

Also is this post relevant to either of these statements, and if so, does that mean they only hold under strong redundancy?

Yup, that post proves the universal natural latent conjecture when strong redundancy holds (over 3 or more variables). Whether the conjecture does not hold when strong redundancy fails is an open question. But since the strong redundancy result we've mostly shifted toward viewing strong redundancy as the usual condition to look for, and focused less on weak redundancy.

Resampling

Also does all this imply that we're starting out assuming that $Λ$ shares a probability space with all the other possible latents, e.g. $P [X, Λ, Λ^{'}, Λ^{''}, \dots]$ ? How does this square with a latent variable being defined by the CPD implicit in the factorization?

We conceptually start with the objects $P [X]$ , $P [Λ | X]$ , and $P [Λ^{'} | X]$ . (We're imagining here that two different agents measure the same distribution $P [X]$ , but then they each model it using their own latents.) Given only those objects, the joint distribution $P [X, Λ, Λ^{'}]$ is underdefined - indeed, it's unclear what such a joint distribution would even mean! Whose distribution is it?

One simple answer (unsure whether this will end up being the best way to think about it): one agent is trying to reason about the observables $X$ , their own latent $Λ$ , and the other agent's latent $Λ^{'}$ simultaneously, e.g. in order to predict whether the other agent's latent is isomorphic to $Λ$ (as would be useful for communication).

Since $Λ$ and $Λ^{'}$ are both latents, in order to define $P [X, Λ, Λ^{'}]$ , the agent needs to specify $P [Λ, Λ^{'} | X]$ . That's where the underdefinition comes in: only $P [Λ | X]$ and $P [Λ^{'} | X]$ were specified up-front. So, we sidestep the problem: we construct a new latent $Λ^{''}$ such that $P [Λ^{''} | X]$ matches $P [Λ | X]$ , but $Λ^{''}$ is independent of $Λ^{'}$ given $X$ . Then we've specified the whole distribution $P [X, Λ^{'}, Λ^{''}] = P [X] P [Λ^{'} | X] P [Λ^{''} | X]$ .

Hopefully that clarifies what the math is, at least. It's still a bit fishy conceptually, and I'm not convinced it's the best way to handle the part it's trying to handle.

[-]Dalcy1y30

Thank you, that is very clarifying!

[-]Towards_Keeperhood2y*30

First a note:

the two chunks are independent given the pressure and temperature of the gas

I'd be careful here: If the two chunks of gas are in a (closed) room which e.g. was previously colder on one side and warmer on the other and then equilibriated to same temperature everywhere, the space of microscopic states it can have evolved into is much smaller than the space of microscopic states that meet the temperature and pressure requirements (since the initial entropy was lower and physics is deterministic). Therefore in this case (or generally in cases in our simple universe rather than thought experiments where states are randomly sampled) a hypercomputer could see more mutual information between the chunks of gas than just pressure and temperature. I wouldn't call the chunks approximately independent either, the point is that we with our bounded intellects are not able to keep track of the other mutual information.

Main comment:

(EDIT: I might've misunderstood the motivation behind natural latents in what I wrote below.)

I assume you want to use natural latents to formalize what a natural abstraction is.

The " induces independence between all $X_{i}$ " criterion seems too strong to me.

IIUC you want that if we have an abstraction like "human", you want all the individual humans to share approximately no mutual information conditioned on the "human" abstraction.
Obviously, there are subclusters of humans (e.g. women, children, ashkenazi jews, ...) where members share more properties (which I'd say is the relevant sense of "mutual information" here) than properties that are universally shared among humans.
So given what I intuitively want the "human" abstraction to predict, there would be lots of mutual information between many humans.
However, (IIUC,) your definition of natural latents permits there to be waaayyy more information encoded in the "human" abstraction, s.t. it can predict all the subclusters of humans that exist on earth, since it only needs to be insensitive to removing one particular human from the dataset. This complex human abstraction does render all individual humans approximately independent, but I would say this abstraction seems very ugly and not what I actually want.

I don't think we need this conditional independence condition, but rather something else that finds clusters of thingies which share unusually much (relevant) mutual information.
I like to think of abstractions as similarity clusters. I think it would be nice if we find a formalization of what a cluster of thingies is without needing to postulate an underlying thingspace / space of possible properties, and instead find a natural definition of "similarity cluster" based on (relevant) mutual information. But not sure, haven't really thought about it.

(But possibly I misunderstood sth. If it already exists, feel free to invite me to a draft of the conceptual story behind natural latents.)

[-]JessRiedel2y*32

Further, assume that mediates between $Λ$ and $Λ^{'}$ (third diagram below).

I can't tell if X is supposed to be another variable, distinct from X_1 and X_2, or if it's suppose to be X=(X_1,X_2), or what. EDIT: From reading further it looks like X=(X_1,X_2). This should be clarified where the variables are first introduced. Just to make it clear that this is not obvious even just within the field of Bayes nets, I open up Pearl's "Causality" to page 17 and see "In Figure 1.2, X={X_2} and Y={X_3} are d-separated by Z={X_1}", i.e. X is not assumed to be a vector (X_1, X_2, ...). And obviously there is more variability in other fields.

[-]johnswentworth2y30

Edited, thanks.

[-]Review Bot2y*30

The LessWrong Review runs every year to select the posts that have most stood the test of time. This post is not yet eligible for review, but will be at the end of 2024. The top fifty or so posts are featured prominently on the site throughout the year.

Hopefully, the review is better than karma at judging enduring value. If we have accurate prediction markets on the review results, maybe we can have better incentives on LessWrong today. Will this post make the top fifty?

[-]Gurkenglas2y30

Fix some atom of information. It's contained in some of Lambda, X1, X2, and Lambda'. Call the corresponding four statements a,b,c,d. Then you assume "b&c implies a, c&d implies b, b&d implies c, a&d implies b or c.".

These compress into "b&c implies a, d implies a=b=c."; after concluding that, I read that you conclude "d&(b or c) implies a", which seems to be a special case. My approach feels too gainfully simpler, so I'm checking in to ask whether it fails.

[-]Morpheus2y10

Just pasting this into a calculator your expressions don't seem to be equivalent:

first expression e1: (((b∧c)=>a) ∧ ((c ∧ d) => b) ∧ (d ∧ b => c) ∧ ((a ∧ d) =>(b or c)))
second expression e2: ((b∧c)=>a)∧(d =>((a=b)=c))

e1 = e2: (a ∧ b) ∨ (a ∧ c) ∨ (b ∧ c) ∨ ¬ d

e1 simplifies to (a ∧ b ∧ c) ∨ (¬ a ∧ ¬ b ∧ ¬ c) ∨ (¬ b ∧ ¬ d) ∨ (¬ c ∧ ¬ d)

[This comment is no longer endorsed by its author]Reply

[-]kave2y10

For e2 I think you want ((b∧c)=>a)∧(d =>((a=b)∧(a=c)))

[-]Morpheus2y10

Oops!

LESSWRONG
LW

LESSWRONG
LW

131

Natural Latents: The Math

131

Ω 49

131

Ω 49

Resampling