The Price of a Causal Variable
Prediction is cheap. Intervention is not. The identifiability theorems set out exactly what a causal world model costs — and no quantity of video pays the bill.
Thesis. A generative model trained on sequences learns to answer “what usually follows this?” An agent that changes its environment needs to answer “what happens if I do this?” These are different quantities, and the second is not recoverable from the first by any amount of data or scale. The mathematics is settled: a latent representation fitted by reconstruction is not identified, because infinitely many encoder–decoder pairs reconstruct equally well, and the true causal variables are not distinguished among them. What identifies them is intervention — specific, structured, per-variable intervention, in numbers that scale with the size of the latent space. The current world-model literature trains against an evidence lower bound consisting of a reconstruction term and a KL term. That objective contains no intervention operator, and conditioning on a logged action is not one. Until a system can name the interventional environments it learned from, its “world model” is a very good conditional density and nothing more.
Abstract. I set the world-model training objective beside the identifiability results for causal representation learning, and read off the gap. The standard world model is a partially observed Markov process whose parameters are fitted by maximising a bound consisting of observation reconstruction plus a KL alignment between the filtered posterior and the dynamics prior — a reconstruction-regularisation paradigm with no interventional content. Action-conditioning does not repair this: fitting p(next state | action) on logged trajectories recovers a conditional under the behaviour policy, which equals the interventional quantity only when the action was independent of the unobserved state, and the largest available datasets are precisely those where it was not. I then set out what recovery actually costs, using the score-based identifiability results: one hard intervention per latent node under linear mixing buys the exact graph and the variables up to scale; two hard interventions per node under arbitrary mixing buys the exact graph and the variables up to componentwise reparameterisation, which is the ceiling. I give the measured gap between those guarantees and what the same algorithms achieve when the score functions are estimated rather than supplied — for five latent variables and fifty thousand samples per environment, mean correlation falls from 0.87 to 0.60 in the hardest of the reported settings. I close with the nine world-model desiderata sorted by which are identifiability problems, which are representation problems, and which are neither. Two figures.
I. Two questions that look alike
The training objective of a modern generative system is a conditional density over what comes next given what came before. It answers: given this history, what usually follows?
The question an agent needs answered is different: if I do this, what happens? Not “in situations where the umbrella went up, what did the sky look like”, but “if I put the umbrella up, does the sky change”.
The distinction is not philosophical fastidiousness. It is the exact point at which correlations stop working, and it is not a rare edge case — it is the defining condition of agency. Correlations are summaries of a distribution that some process generated. An agent that acts is a modification of that process. The moment it intervenes, it is drawing from a distribution its training data did not contain, and the relationships it learned may hold, may weaken, or may reverse, with nothing in the model to say which.
The wet-street example is worn but precise. Wet streets predict umbrellas beautifully. From that association you cannot tell whether rain causes both, whether the streets cause the umbrellas, or whether raising an umbrella changes the weather. All three hypotheses fit the observed joint distribution. Only one survives contact with an agent that raises an umbrella on a dry day, and the model has no way to know that before it tries.
So: what would it take to learn the version that survives?
II. What a world model is actually trained to do
Write the objective out, because it is rarely written out in the discussion.
The standard formulation is a partially observed Markov decision process. The agent receives an observation and takes an action; the true state is unobserved; the model maintains a learned latent state inferred by a one-step filtering posterior, on the assumption that the previous latent summarises the relevant history. Three distributions are parameterised: a dynamics prior over the next latent given the previous latent and the previous action; a filtered posterior over the latent given the previous latent, the previous action and the current observation; and a reconstruction of the observation from the latent.
The likelihood is intractable, so training maximises an evidence lower bound. That bound decomposes into two terms: the expected log-likelihood of the observation given the latent, and a KL divergence aligning the filtered posterior with the dynamics prior. The current survey literature describes this, accurately, as a reconstruction-regularisation paradigm.
Now look at what is in that objective and what is not.
What is in it: a pressure to compress observations into a latent from which they can be reconstructed, and a pressure for the inferred latent to be predictable from the previous latent and action.
What is not in it: any operator corresponding to intervention. No do. No distinction between a variable that was set and a variable that was observed. Nothing that would penalise a latent representation which reconstructs perfectly, predicts perfectly, and encodes the causal structure backwards.
This is not a criticism of the formulation. It is an accurate objective for the thing it names — a simulator of the distribution the data came from. It is a criticism of the inference people draw from it, which is that a model which reconstructs the future well must have understood how the world works.
III. Action-conditioning is not intervention
Here is the objection that has to be answered first, because it is the good one: the dynamics prior conditions on the action. The model is not learning a passive sequence density; it is learning how the state evolves given what the agent did. Is that not the interventional quantity?
No, and the reason is exact.
Conditioning on an action recorded in a log gives you the distribution of the next state among trajectories in which that action was taken. The interventional quantity is the distribution of the next state when the action is imposed. These coincide only when the action was independent of everything unobserved that also influences the outcome. Randomisation delivers that independence. A behaviour policy does not.
Consider where world-model training data comes from. Expert demonstrations: the expert chose the action because of features of the situation, many of them not in the observation. Human driving logs: the driver braked because they saw something, and what they saw is the confounder. Internet video: no action labels at all, and where latent actions are inferred between adjacent frames, they are inferred from the very footage whose generating process is in question. In each case the action is a function of the state, including the parts the model cannot see, and the fitted conditional inherits that dependence.
The consequence is specific rather than vague. A model trained on expert data learns that a particular manoeuvre is followed by a good outcome. What produced the good outcome was the expert’s unobserved assessment, of which the manoeuvre was a symptom. Deploy a policy that reproduces the manoeuvre without the assessment and you get the symptom without the cause. This is the standard confounding failure, and it appears in world models exactly where you would predict: at deployment, off the behaviour distribution, when the agent’s action is no longer being selected by the process that made the correlation hold.
Simulated environments with random exploration are the honourable exception, and it is worth being clear about why they work: in simulation you can randomise, so the action really is independent of the hidden state, and the fitted conditional really is interventional. Which is also why sim-to-real transfer is the recurring difficulty in that literature. The interventional structure was learned in the place where interventions were free.
IV. Why reconstruction cannot identify the variables
Now the deeper result, and the one that ends the argument that scale will handle it.
Set up the problem as the causal-representation literature does. There are latent causal variables related by a directed acyclic graph. An unknown transformation maps them to the high-dimensional observations we actually see — pixels, sensor readings, tokens. The task is to recover both the variables and the graph.
Recovering the variables means finding the inverse of that transformation. And here is the difficulty, stated plainly in the literature: infinitely many valid encoder–decoder pairs satisfy the reconstruction property. Any invertible remixing of the latent space, followed by the compensating remixing of the decoder, reconstructs the observations exactly as well. Reconstruction cannot choose among them, because reconstruction is blind to the choice.
It is worse than “the objective is under-determined”. Identifiability is known to be impossible without additional supervision or sufficient statistical diversity in the data — and this holds even in the simplest case, where the latent variables are statistically independent and there is no causal structure to recover at all. If you cannot identify independent latents from observational data, you certainly cannot identify a causal graph over them.
So the position is this. Whatever a world model’s latent state is, it is one member of an equivalence class of representations that fit the data equally well. Nothing in the training procedure selects the member that corresponds to the causal variables, and the model has no way to report which member it landed on. A latent that has entangled two physical quantities into a single coordinate will reconstruct beautifully, predict beautifully, and answer interventional questions wrongly — and every metric in the training loop will be satisfied.
What breaks the tie is intervention. An intervention alters the mechanism generating one variable, leaving the others alone, and thereby produces a change in the data that is sparse in the right coordinate system — the true one. That sparsity is a signature of the true representation, and it is a signature the wrong representations do not have. That, in one sentence, is the whole idea behind the identifiability results.
V. What recovery costs
The results are precise about the price. Here is the ladder.
Figure 1. What a causal variable costs. Each rung is a theorem with its own conditions. With observational data alone, nothing is recovered. One soft intervention per node under a linear mixing recovers the ancestral relations and leaves the variables entangled with their parents. One hard intervention per node recovers the graph exactly and the variables up to permutation and scale. Under an arbitrary unknown mixing, two hard interventions per node recover the graph exactly and the variables up to permutation and componentwise reparameterisation — which is the ceiling: no interventional method does better without further assumptions.
Three things about that ladder deserve emphasis.
The unit of cost is the node. Not the dataset, not the compute budget — one interventional environment per latent variable for the linear case, two per latent variable for the general case. If your environment has fifty causal variables, you need fifty or a hundred interventional regimes, each of which involves actually setting one variable and letting the rest run.
The ceiling is lower than people assume. Even at the top of the ladder, you do not recover the variables. You recover them up to permutation and an unknown monotone reparameterisation of each coordinate. You learn that there is a quantity here that behaves causally like temperature; you do not learn that it is measured in kelvin, or that it is temperature. For control this is often enough. For any claim that the system has discovered the true variables of the domain, it is not, and this is not a limitation of current algorithms — it is proved to be the best possible under interventions without additional structure.
Softer interventions buy less, and the drop is sharp. Hard interventions sever a variable from its parents; soft ones merely change the mechanism. The difference in what you recover is the difference between the exact graph and its transitive closure — between knowing which variable causes which, and knowing only which variables are upstream of which. Since most interventions available in a real environment are soft, this matters more than the theory’s tidy presentation suggests.
VI. The rest of the bill
The ladder gives the headline price. The conditions attached to each rung are the rest of it, and they are not incidental.
You must know how many latent variables there are. The theorems are stated for a fixed n. Nothing tells you what n is for a kitchen.
The mixing must be a diffeomorphism onto its image, with the observed dimension at least the latent dimension. Information must not be destroyed on the way from causes to observations. Occlusion destroys information. So does saturation, motion blur, and a camera pointed the other way. The formal condition is a real one and real sensors violate it routinely.
Interventions must act on the latent variables. This is the condition that determines whether the whole programme is applicable to your domain, and it is domain-specific in a way that should be stated loudly. For a robot arm, the causal variables can be taken to be joint angles, the observations are camera images, and an intervention is setting a joint to a value — the framework fits perfectly, because the latent variables are exactly the things you can grasp and move. For weather, or an economy, or a social environment, the latent causal variables are not the things you have handles on, and no amount of acting on the observations substitutes for acting on the causes.
The interventions must be single-node and atomic, one variable per environment, in the standard formulation. Interventions that touch several variables at once are a harder and less complete case.
The two interventions on a node must be sufficiently distinct. The formal statement — interventional discrepancy — requires that the ratio of the two intervention densities has non-vanishing derivative almost everywhere. Intervene twice in nearly the same way and you learn nearly nothing.
And you must be able to estimate score functions — the gradients of log densities — in the observed space, well enough to detect which coordinates changed. The algorithms are agnostic to how, which is elegant, and it also means the entire empirical burden lands there.
Notice what is absent from that list: more data of the kind you already have. None of these conditions is relaxed by scale. They are conditions on the kind of data, not the amount.
VII. Why interventions work, and when they stop working
It is worth seeing the mechanism, because it explains both why the price is what it is and where the whole approach breaks.
The score of a distribution is the gradient of its log density — a vector with one coordinate per variable. Because the joint density of the latent variables factorises into one conditional mechanism per node, the score decomposes into a sum of terms, one per mechanism.
Now intervene on a single node. An intervention changes that node’s mechanism and leaves every other mechanism untouched. So when you subtract the observational score from the interventional score, everything cancels except the term belonging to the intervened node — and that term is a function only of the node and its parents. The difference is zero at every other coordinate. The intervention leaves a sparse signature, and the signature is sparse only in the true coordinate system.
Sharpen it further. Take two different hard interventions on the same node. A hard intervention severs the node from its parents, so in both environments the parents’ contribution is identical. Subtract one from the other and the difference is nonzero at exactly one coordinate: the intervened node itself. One-sparse.
That is the identifying signal, and it is the thing reconstruction is blind to. Remix the latent space with any invertible transformation and reconstruction is unaffected — but the one-sparse difference is smeared across several coordinates of the new basis. So the search is: find the encoder under which the score differences come out sparse. The right one does. The wrong ones do not. That is the whole of it, and it is why the algorithms need two interventions per node in the general case: one pair per variable to produce one one-sparse vector per variable, which together pin down the transformation.
Now the failure condition falls out of the same argument. The signature is sparse because the intervention was surgical — it changed one mechanism. An intervention that changes several mechanisms at once produces a difference that is not sparse, and the identifying signal degrades toward nothing.
Which is exactly what physical interventions in unstructured environments do. Move the arm and you change the joint angle, the shadow, the contact forces, the occlusion pattern and the reflection off the countertop. Some of those are downstream effects, which is fine — the framework handles descendants. But some are separate mechanisms perturbed by the same physical act, and each one you disturb costs you sparsity. The experiment that identifies the structure is not “do something and watch”; it is “change exactly one thing”, and the whole difficulty of experimental science is that changing exactly one thing is hard.
VIII. The theorem holds. The estimator is what you have.
Identifiability results are statements about the ideal case: what could be recovered given perfect information about the distributions. They are necessary and they are not sufficient, and the same papers that prove them show what happens when the perfect information is replaced by estimates.
Figure 2. Measured results: five latent variables, one hundred observed dimensions, fifty thousand samples per environment. Blue is the algorithm supplied with exact score functions — the identifiability guarantee realised. Red is the same algorithm, same data-generating process, with the scores estimated from samples. Where the causal mechanisms are linear, the estimate is as good as the oracle. As the mechanisms become nonlinear and the interventions become soft, the two diverge: mean correlation falls from 0.87 to 0.60, and structural error rises from roughly half an edge to nearly three, against an expected edge count of five.
Read the two panels together, because they say different things.
On the left, latent variable recovery survives noisy estimation reasonably well in most settings — around 0.93 to 0.94 correlation for the hard-intervention cases with nonlinear mechanisms. That is a real result and it should be acknowledged as such.
On the right, graph recovery degrades much further and much faster. Structural error rises to 2.62 and 4.27 edges in the nonlinear hard-intervention settings, on graphs expected to have five edges. The reason given is instructive: the true score differences at some coordinates are small, and thresholding a small quantity in the presence of estimation noise is exactly the operation that fails first. You can recover approximately what the variables are and still get their causal ordering substantially wrong.
And the worst cell in the table is the one closest to reality. Soft interventions — the only kind available in most environments — on a nonlinear causal mechanism — the only kind most environments have — with estimated rather than exact scores. Mean correlation 0.60. That is the honest state of the art on a synthetic problem with five variables, a hundred observed dimensions, a known latent dimension, a smooth invertible mixing, complete interventional coverage, and fifty thousand clean samples per environment.
Now count what a room contains.
IX. The arithmetic of a real environment
Put the numbers together and the position becomes concrete rather than rhetorical.
For five latent variables in the general setting, you need two interventional environments per node plus observational data: eleven regimes. At the sample sizes used, that is over half a million samples to identify five variables — up to permutation and reparameterisation — in a problem where somebody has already guaranteed the mixing is smooth and invertible, told you the latent dimension, and let you intervene on the latents directly.
A kitchen has more than five causal variables. So does a road. So does a conversation.
The scaling is linear in the number of variables, which sounds benign until you notice what each unit of it requires: a physical capability to hold one causal quantity fixed while everything else runs, exercised once or twice per quantity, with the intervention distinct enough from its partner to be informative. In a robotics laboratory this is a Tuesday. In an open environment it is not obviously possible at all, and the difficulty is not cost — it is that you cannot intervene on a variable you have not yet identified, and identifying it is what the interventions were for.
There is one genuinely cheerful result in this area and it deserves stating. Extrapolating to combinations of interventions you have never observed can be done without doing causal representation learning at all: the score differences of single-node interventions add, so the score function of a joint intervention can be constructed from the singles and sampled from directly in observed space. The combinatorics are not the bottleneck. Getting the first n singles is the bottleneck, and it always was.
X. Video is not a world model
The survey literature on embodied world models is candid about its own open problems, and the candour is worth reproducing because it is more useful than the promotional framing that surrounds the field.
The taxonomy divides temporal modelling into two families and states the cost of each. Sequential simulation unrolls the future one step at a time: fine-grained control, natural closed-loop planning, and error accumulation over long horizons with cost linear in rollout length. Global difference prediction estimates the future in parallel: amortised computation and lower latency, at the price of weakened closed-loop interactivity and a tendency to smooth over fine local dynamics. Neither eliminates the trade; they place it in different places.
On evaluation, the surveyed conclusion is that the field needs metrics assessing physical consistency over pixel fidelity. That sentence is doing a great deal of work. It concedes that current metrics reward a rollout that looks right, and that looking right and being right are separable — which is precisely the failure mode you would predict from a reconstruction objective. A model optimised to reconstruct observations will be excellent at producing plausible observations. Whether the physics underneath them is coherent is a different question, and until recently nobody was scoring it.
Long-horizon consistency with error accumulation is named as the core modelling difficulty. Note how this interacts with everything above. An autoregressive rollout feeds its own output back as input. If the latent representation is a member of the equivalence class that is not the causal one, then each step’s error is not merely noise — it is systematic drift in a direction the model cannot detect, because detecting it would require knowing which representation was right.
There is real work in the survey that addresses parts of the problem directly, and it should be credited rather than swept into the general complaint. Some models explicitly decompose dynamics into controllable and uncontrollable components using inverse dynamics, and use the uncontrollable rollout to guide policy learning — that is the controllable-variable desideratum, addressed head on. Object-centric models represent scenes as collections of slots and model the relations between them, which is the persistent-objects desideratum. At least one line trains a causal world-model simulator explicitly to ground a language model’s causal reasoning. The field is not ignoring this. It is, however, mostly doing it as architectural inductive bias rather than as identification, and an inductive bias is a hope about which member of the equivalence class you will land on, not a guarantee.
XI. The nine desiderata, sorted
The list of what a useful world model must discover is a mixed bag, and treating it as one problem is why it looks intractable. Sorted by what kind of problem each actually is:
Genuine identifiability problems — the ones the ladder prices. Causal variables: this is the problem, and the answer is interventions per node. Counterfactual consequences: strictly harder than interventional prediction, requiring the structural functions and not merely the interventional distributions; the identifiability results give the interventional layer and stop there. Invariants that survive environmental change: this is what multiple environments buy, and it is the same currency in a different denomination — the diversity that makes identification possible is exactly the diversity that reveals what is invariant.
Representation problems, tractable with the right architecture. Persistent objects: slot-based and object-centric representations exist and work. Hidden state: this is what the filtering posterior is for, and the POMDP formulation handles it, subject to the caveat that the inferred state is only identified up to the same equivalence class as everything else. Controllable versus uncontrollable variables: decomposition via inverse dynamics is a working approach, and note that it is really a small interventional experiment run by the agent itself — the agent’s own actions are the interventions, which is why it works.
Neither, or not yet a well-posed problem. Affordances: what actions are possible is a property of the joint agent–environment pair, not of the environment, and it changes when the agent’s body changes; this is not a latent variable to be recovered but a relation to be computed. Physical and social dynamics: physics is a candidate for identification, social dynamics is not, because the other agent’s mechanism changes in response to being modelled. Other agents’ beliefs and likely responses: this is the hardest item on the list and it is misfiled with the others. Every identifiability result above assumes the mechanism generating each variable is fixed except when intervened upon. An adaptive opponent violates that assumption by construction. There is no version of the intervention story that survives an environment which learns from your interventions, and pretending otherwise by putting the other agent’s belief state in as another latent node smuggles in a modularity assumption that is exactly what a strategic agent breaks.
That last point is worth sitting with. The identifiability programme rests on the modularity of causal mechanisms — intervene on one and the others hold still. Social environments are the class of environment where that premise is false, and they are also the class where most of the consequential decisions get made.
XII. What a defensible claim looks like
I am not arguing that world models are useless. Model-based control works, imagination-based policy optimisation works, and a system that predicts the near future well enough to plan three seconds ahead is valuable regardless of whether its latents are the true ones.
I am arguing that “world model” has become a claim of causal understanding, and that the claim has a price list, and that nobody is showing the receipt. So:
Name the interventional environments. How many, on what, hard or soft, coupled or not. If the answer is “none, we trained on video”, then the system is a conditional density over observations and should be described as one. That is not an insult; it is what it is.
State whether the actions were randomised. If the training trajectories came from an expert, a human, or a policy, then action-conditioning is conditioning, not intervening, and the distinction should appear in the paper rather than in a reader’s inference.
Report what recovery is being claimed, on the ladder. Ancestral relations? The exact graph? The variables up to scale? Up to reparameterisation? These are different claims with different evidentiary requirements, and collapsing them into “learns the causal structure” is where the overclaiming happens.
Evaluate on interventions the model has not seen, not on held-out observations. Held-out prediction error is satisfied by every member of the equivalence class. Interventional accuracy is satisfied by one. If the evaluation is pixel error on unseen video, it does not distinguish the hypotheses in question, and its value can be high while the causal content is zero.
And state the assumptions that were required rather than inherited. Known latent dimension. Diffeomorphic mixing. Single-node atomic interventions. Modularity of mechanisms. Each is a condition on the world, each can be false, and each being false changes what the result means.
None of that is a research programme. It is a reporting standard, and it costs nothing but the ability to imply more than was shown.
XIII. What would show me wrong
If identifiability results emerge for the multi-node, uncontrolled, partially observed case — recovery from the kind of diversity that occurs naturally in unstructured environments rather than from designed atomic interventions — then the bill I have itemised is the bill for the wrong problem, and I would want to know immediately.
If a system trained purely observationally were shown to predict novel interventions accurately, across a battery of interventions genuinely outside the support of its training distribution, then either the equivalence-class argument has an exploitable structure I have not accounted for, or the architectural inductive biases are doing identification work that nobody has formalised. Either would be a substantial result and both are checkable. The test is not difficult to design: hold out interventions, not frames.
And if the modularity assumption turns out to be robust enough in practice that approximate recovery is sufficient for control — that landing on the wrong member of the equivalence class costs you little where it matters — then the theoretical gap is real and the practical gap is small, and my emphasis is misplaced. That is an empirical question about the geometry of the equivalence class near the true solution, and I have not seen it answered in either direction.
XIV. Coda
The gap between predicting and understanding is not a matter of degree, to be closed by another order of magnitude of parameters and video. It is a difference in the quantity being estimated. A conditional density over what usually follows is not an approximation to an interventional distribution; it is a different object that happens to agree with it whenever nobody is acting.
The identifiability results are not obstacles to be routed around. They are a price list. They say: here is what a causal variable costs, in interventions, per variable, under these conditions, and here is exactly how much you get for it — which is less than you would like, and is proved to be the most that is available.
An agent that can enter an unfamiliar environment, work out its causal structure through limited experiment, and predict interventions far outside anything it has seen would be a genuinely new kind of system. Nothing currently trained on video is that system, and no amount of that training will make it one, because the thing being optimised does not contain the thing being claimed.
The receipts are the point. Show which interventions were run, and the claim becomes checkable. Decline to, and “world model” is a name for a video codec with good taste.
References
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Varıcı, B., Acartürk, E., Shanmugam, K., Kumar, A., & Tajer, A. (2025). Score-based causal representation learning: linear and general transformations. Journal of Machine Learning Research, 26, 1–90.
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Li, X., He, X., Zhang, L., Wu, M., Li, X., & Liu, Y. (2026). A comprehensive survey on world models for embodied AI. arXiv:2510.16732v3.