Causal Models and Intervention

The preceding pages establish a negative result: probabilistic dependence underdetermines causal structure. Probability raising cannot supply asymmetry, screening off presupposes causal knowledge, and Simpson's paradox shows that identical statistics can require opposite analyses.

The causal modelling framework — developed by Pearl, and by Spirtes, Glymour, and Scheines — responds not by finding a better probabilistic definition but by adding structure. A causal model is a probability distribution plus a graph, and the graph carries information no distribution contains. The framework's central conceptual contribution is the distinction between observing and intervening.

The apparatus

A causal Bayesian network consists of a directed acyclic graph whose nodes are variables and whose edges represent direct causal influence, together with a probability distribution factorising according to the graph:

$$P(x_1,\dots ,x_n)=\prod _{i}P(x_i\mid \mathrm{pa}(x_i))$$

where $\mathrm{pa}(x_i)$ are the parents of $x_i$. The factorisation is exactly the Causal Markov Condition: each variable is independent of its non-descendants given its parents.

A structural equation model says more. Each variable is written as a function of its parents and an error term,

$$x_i=f_i(\mathrm{pa}(x_i),u_i)$$

and the equations are read asymmetrically: the left-hand side is determined by the right, not merely equal to it. This is the formal locus of causal asymmetry. An ordinary equation can be rearranged; a structural equation cannot, because it represents a mechanism. That $y=2x$ as a structural equation says setting $x$ changes $y$, and not that setting $y$ changes $x$.

Intervention and the do-operator

The framework's key move. Consider two questions:

• Observation. Given that a patient took the drug, how likely is recovery? This is $P(\text{recovery}\mid \text{drug})$.
• Intervention. If we give the patient the drug, how likely is recovery? This is $P(\text{recovery}\mid do(\text{drug}))$.

These differ, and the difference is the whole of the confounding problem. Observing that someone took a drug is evidence about why they took it — perhaps they were sicker, or wealthier, or better informed — and that evidence bears on recovery independently of the drug's effect. Intervening severs those pathways.

Formally, $do(X=x)$ is defined by graph surgery: delete all edges into $X$, set $X=x$, and compute in the mutilated graph. Deleting the incoming edges represents the fact that when we set $X$ by intervention, its usual causes no longer determine it — the intervention replaces the mechanism.

This is a genuine conceptual advance rather than notation. It gives a precise sense to the difference between seeing and doing, explains why randomised trials work — randomisation is an intervention, severing the edges from confounders into treatment — and provides the vocabulary in which causal claims can be stated without ambiguity. It also makes clear why the observational and interventional quantities coincide exactly when there is no confounding.

Identification

The practical question: when can an interventional quantity be computed from observational data plus a graph?

The back-door criterion gives a sufficient condition. A set $Z$ of variables suffices for adjustment if it blocks every path from $X$ to $Y$ that begins with an arrow into $X$, and contains no descendant of $X$. Then

$$P(y\mid do(x))=\sum _{z}P(y\mid x,z)P(z)$$

The criterion formalises the informal advice to "control for confounders" while explaining the two ways that advice goes wrong: conditioning on a descendant of $X$ blocks part of the effect (the mediator case), and conditioning on a collider opens a path that was closed (creating spurious association). The three-way distinction that Simpson's paradox showed to be undetermined by the data is read directly off the graph.

The front-door criterion handles cases with unmeasured confounding but a measured mediator, and the do-calculus — three rules, proved complete — determines for any graph whether a given interventional quantity is identifiable from observational data. Completeness is a strong result: if the do-calculus cannot derive the quantity, no method can, from that graph and that data.

The ladder of causation summarises the hierarchy: association ($P(y\mid x)$), intervention ($P(y\mid do(x))$), and counterfactuals ($P(y_x\mid x^{\prime },y^{\prime })$ — what would have happened to this individual had things been otherwise). Each level requires strictly more than the one below, and no amount of data at a lower level suffices for a claim at a higher one without assumptions.

Causal discovery

The converse problem: inferring the graph from data. Algorithms such as PC and FCI exploit the fact that different causal structures imply different patterns of conditional independence.

The key asymmetry is the collider or v-structure. In the chain $X\to Z\to Y$ and the fork $X\leftarrow Z\to Y$, conditioning on $Z$ makes $X$ and $Y$ independent. In the collider $X\to Z\leftarrow Y$, $X$ and $Y$ are unconditionally independent and become dependent when $Z$ is conditioned on. That signature is detectable in data, so colliders can be identified — which is what makes any orientation of edges possible at all.

Discovery requires faithfulness as well as the Causal Markov Condition, and typically causal sufficiency (no unmeasured common causes). Even then the output is generally an equivalence class of graphs rather than a unique one, since chains and forks are indistinguishable observationally. The framework is honest about this, which is a virtue: it delivers exactly what the assumptions support.

Philosophical status

Is it reductive? No, and the framework does not claim to be. Causal notions appear in the semantics of the arrows, in the interpretation of structural equations as mechanisms, and in the Causal Markov Condition. What the framework provides is a representation and calculus for causal structure, not an analysis of causation in non-causal terms. Given the failures catalogued in this folder, that is the right ambition, and the earlier programmes' insistence on reduction now looks like the error.

What are interventions? The definition of $do(X)$ presupposes the notion of setting a variable from outside the system. Characterising this without circularity is delicate — an intervention must not affect the outcome except through $X$, which is a causal condition. Woodward's interventionism embraces the circularity as non-vicious: the account is illuminating without being reductive, in the way that a good account of a family of interrelated concepts can be.

Do the arrows correspond to anything? A realist reads them as representing mechanisms in the world; an instrumentalist reads them as encoding policies for prediction under intervention. The formalism is neutral, and the choice connects to the treatment of laws of nature and dispositions.

Limitations. Acyclicity excludes feedback, which is ubiquitous in biology and economics. Variables must be well defined, and the choice of variables is itself a substantive modelling decision with no algorithm — the same model-selection problem this section keeps meeting. And the framework says nothing about level: whether to model at the level of neurons or of beliefs is not a question the graph can answer.

Where this sits

Causal models complete this folder's argument. Probability alone cannot deliver causal structure; adding a graph and a notion of intervention makes causal inference rigorous, with precise criteria for when observational data suffice and a proof of when they do not.

The philosophical moral is that the correct response to the failure of reductive probabilistic analyses was not a better definition but a richer formalism. Causation is not definable in probabilistic terms, and probability plus causal structure supports a powerful and largely complete theory of inference.

The last page of the folder draws out the practical consequences for what statistical evidence can and cannot show: correlation, prediction, and explanation.