Scientific, Medical, and Social Worlds › Prediction, Simulation, and Counterfactual Inference

Prediction, intervention, and counterfactual queries require different information. The causal hierarchy discussed in Part IV has three levels, not four. Simulation is a computational procedure that can implement several kinds of query; decision support adds a choice criterion rather than another causal rung. Bareinboim and colleagues formalize why lower-level information generally underdetermines higher-level queries without additional assumptions.

Prediction asks for an outcome conditional on available information. A ten-day weather forecast is one example. A sequence-to-structure protein prediction is another, although it does not predict a time trajectory. A function fitted to historical inputs and outputs can support useful predictions without identifying their causal mechanism. That does not make prediction easy: noise, limited data, computation, and distribution shift can each limit accuracy.

Simulation rolls a modeled transition forward. A simulator can be autonomous, stochastic, or action-conditioned; it need not encode a valid causal intervention. For interventional use, the action must enter a justified mechanism. In our compartment simulator, reducing contacts multiplies the incidence term, changing the next susceptible and infectious states and therefore later incidence. A static outcome lookup does not implement that recursion, although a lookup table of state transitions could. What matters is the transition semantics and their evidence, not whether the implementation is an equation, a table, or a neural network.

Counterfactual inference can ask what would have happened to a unit whose factual history is already observed. For example, after observing $Y$ under $A=a$, we might ask about the unreceived alternative $Y(a')$ for that same unit and episode. This retrospective query conditions on factual evidence and may require assumptions about how unobserved disturbances are shared across alternatives. A structural simulator can implement such inference, but its structure alone does not guarantee identification. Population potential-outcome contrasts, including the average treatment effect introduced below, are also counterfactual quantities; the term is not reserved for individuals. Pearl's discussion of causal queries separates the needed information from the mechanics of simulation.

Decision support combines evidence about action-dependent outcomes with a loss, feasible actions, and constraints. Where intervention effects matter, an observational conditional distribution is not automatically the required action-dependent distribution. Calibration alone also cannot select an action without specifying what counts as a costly error. Prediction losses themselves can be asymmetric; decision support makes the relevant consequences and choices explicit rather than introducing asymmetry for the first time.

Observational equivalence means equality of the distributions of observable variables under a specified observation regime. Write this as $P_{M_1}(O\in B)=P_{M_2}(O\in B)$ for every observable event $B$ in that regime. It is stronger than giving the same likelihood to one realized dataset or obtaining similar residual errors. Interventional equivalence for a specified query family means equality of the outcome distributions under each intervention in that family, for example $P_{M_1}(Y\mid\operatorname{do}(A=a))=P_{M_2}(Y\mid\operatorname{do}(A=a))$ for all actions of interest. Observational equivalence alone does not generally imply this second equality. Identification assumptions or suitable experimental evidence can connect them.

A small structural counterexample makes the non-implication exact. Let $U$ be a Bernoulli variable with probability one half, and let $A=U$ in both models. In model 1, $Y=U$; in model 2, $Y=A$. Both produce only $(A,Y)=(0,0)$ and $(1,1)$, each with probability one half. Setting $A=0$ by intervention leaves $Y=U$ in model 1 but makes $Y=0$ in model 2. The interventional mean is therefore one half in the first and zero in the second. More passive observations of these same variables cannot distinguish these particular models. That is not a claim that more data never helps: additional variables, experiments, or defensible restrictions can change identification.

A separate issue is structure versus fit. Observational predictive accuracy is useful but does not by itself validate an intervention forecast. A mechanism-based model can transfer when its relevant mechanisms and parameters remain valid under the specified action. It can also fail because its parameters, boundary conditions, or omitted processes are wrong. A highly accurate associational fit can fail on intervention because an observation-only objective need not penalize the unseen action error. Our example illustrates both failures: the two-coefficient AR mean model lacks the contact input, and the fitted SIR model responds to contacts but misestimates the response.

For treatment questions, keep outcome prediction separate from effect estimation. Predicting outcomes among patients who received a drug is not the same estimand as comparing their outcomes under alternative assignments. If treatment selection depends on severity, a group difference can mix treatment response with baseline risk. The sign and size of that bias depend on the data-generating process; the synthetic cohort below stipulates one sign-reversal example.