Screening Off and Common Causes

If two events are correlated and neither causes the other, something explains the correlation. Reichenbach's Principle of the Common Cause says what:

If events $A$ and $B$ are correlated, and neither causes the other, then there is a common cause $C$ that screens off the correlation.

Screening off is the technical core:

$$P(A\wedge B\mid C)=P(A\mid C)\cdot P(B\mid C)$$

Conditional on the common cause, the correlation vanishes — $A$ and $B$ become independent. Falling barometers and storms are correlated; conditional on atmospheric pressure they are not. The pressure screens off, and the correlation between barometer and storm is fully explained by their shared dependence on it.

This principle does more work than any other in the inference from statistics to causal structure. It is also, in its unrestricted form, false — and the most interesting place it fails is in quantum mechanics.

The conjunctive fork

Reichenbach's structure: a common cause $C$ of $A$ and $B$ forms a conjunctive fork when

$$P(A\wedge B\mid C)=P(A\mid C)P(B\mid C),P(A\wedge B\mid \neg C)=P(A\mid \neg C)P(B\mid \neg C)$$

together with the requirement that $C$ raises the probability of each. From these it follows that $A$ and $B$ are unconditionally correlated: the correlation is derived from the fork structure rather than assumed. This is what makes the principle explanatory — it shows how a common cause manufactures a correlation between causally unrelated events.

The principle also supplies a temporal asymmetry that pure probabilistic dependence lacks. Conjunctive forks open toward the future: common causes screen off their joint effects, but common effects do not screen off their causes. Indeed conditioning on a common effect typically creates a correlation between independent causes — the phenomenon now called collider bias or explaining away. If a burglar alarm sounds when either a burglar or an earthquake occurs, then burglary and earthquakes are independent, but conditional on the alarm they are negatively correlated: learning there was an earthquake explains the alarm and lowers the probability of a burglar.

That asymmetry — forks open forward, colliders create dependence — is the structural fact that makes causal inference from observational data possible at all, and it reappears as the d-separation rules in causal models.

Where the principle fails

Genuine coincidence. Two unrelated quantities can be correlated by chance, particularly in small samples or in the vast space of possible comparisons. Spurious correlations found by data-dredging require no common cause.

Non-causal explanations. Correlations can arise from logical or mathematical relations, from shared definitions, or from selection effects in how the data were assembled. A correlation between two ratios sharing a denominator needs no causal story.

Common causes that do not screen off. Where the common cause is coarsely specified or heterogeneous, conditioning on it may leave residual correlation, because the true screening variable is finer-grained than the one available. This is an epistemic limitation rather than a failure of the principle, but it matters in practice: apparent violations of screening off are usually evidence that a variable is unmeasured or badly measured.

Selection. If the sample is selected on a common effect of both variables, a correlation appears with no common cause. This is collider bias again, and it is treated on the correlation and explanation page.

The quantum failure

The most significant counterexample, and the reason this material connects to physics.

In an EPR–Bell setup, two entangled particles are measured at spacelike separation. The outcomes are correlated. Neither measurement causes the other, since they are spacelike separated and any influence would have to be superluminal. So by Reichenbach's principle there must be a common cause in the shared past — the state $\lambda$ prepared at the source — screening off the correlation.

Suppose there is. Then, writing $a$ and $b$ for the measurement settings and $A$, $B$ for outcomes:

$$P(A\wedge B\mid a,b,\lambda )=P(A\mid a,\lambda )\cdot P(B\mid b,\lambda )$$

This factorisability condition — the conjunction of what are usually separated as parameter independence and outcome independence — is precisely the assumption from which Bell's inequality is derived. And the inequality is violated, by quantum mechanical prediction and by experiment.

So no common cause of the required kind exists. The correlation is not explicable by a screening-off variable in the common past, and one of the following must be given up: the principle of the common cause, locality, the assumption that measurement settings are freely chosen (statistical independence), or the single-outcome assumption. The philosophy of Bell's theorem works through the options in detail, and it is the natural companion to this page.

Two points bear emphasis here. First, the derivation of Bell's inequality is an application of Reichenbach's principle — the factorisability condition is screening off — so Bell's theorem can be read as an experimental refutation of the unrestricted principle. Second, this is a rare case where a philosophical principle about causation has been tested empirically and failed, which is a striking fact about the principle's status: it was never a conceptual truth.

Causal Markov and faithfulness

The modern generalisation of Reichenbach's principle, and the assumption on which causal discovery algorithms rest.

The Causal Markov Condition. In a causal graph, every variable is probabilistically independent of its non-descendants conditional on its direct causes. This generalises screening off to arbitrary causal structures: conditioning on a variable's parents renders it independent of everything except its effects.

Faithfulness. All and only the independences implied by the Causal Markov Condition actually hold in the distribution. This is the converse assumption, and it is what licenses inferring causal structure from observed independences: without it, a genuine causal connection could be hidden by an exactly cancelling path.

Faithfulness is the more contentious. Cancellation requires precise parameter tuning and so is, in a natural measure, improbable — the standard defence is that unfaithful distributions form a measure-zero set. But that argument has the structure of a typicality claim and inherits its difficulties: measure-zero sets can be exactly where selection or equilibrium puts a system. Biological systems with homeostatic feedback are plausibly tuned to cancellation, since that is what regulation is, so near-unfaithfulness may be common in exactly the domains where causal discovery is most wanted.

Where this sits

Screening off is the principle that converts statistical data into causal conclusions, and everything downstream — Simpson's paradox, causal models, the practice of controlling for confounders — is an application of it. Its asymmetry, forks opening forward and colliders creating dependence, supplies what pure probabilistic dependence lacks.

Its status is empirical rather than conceptual. That was not obvious before Bell, and the quantum violation is the clearest demonstration in the section that a principle connecting probability to the world can be tested and found wanting. What remains is a principle that holds in ordinary macroscopic contexts, is presupposed by all statistical causal inference, and fails in a specific and well-understood physical regime.

The next page examines what happens when a correlation reverses under conditioning, and why no purely statistical rule says which conditioning is the right one: Simpson's paradox.