Backdoor Criterion: Blocking the Confounding Trap

2020. Observational data from multiple countries: patients on ibuprofen die less from COVID. Headlines: 'ibuprofen treats COVID'. Months later, analysis controlling for disease severity found: no effect at all. Severely ill patients were shifted to heavier drugs immediately - they died more. Disease severity drove both the ibuprofen decision and the outcome. That was a backdoor path. The backdoor criterion is the algorithm that identifies such traps from the graph's arrows alone, without a single number.

• **Debiasing in ML**: algorithms trained on observational data learn spurious correlations (backdoor paths). DoWhy and CausalML implement backdoor adjustment as a pre-processing step before training - so the model learns causal effects rather than confounders
• **A/B testing**: randomization is the physical destruction of all backdoor paths. The backdoor criterion explains why randomization works: the coin flip severs the edge X <- confounder, making backdoor paths inactive
• **ML fairness audits**: Microsoft Research applies the backdoor criterion to detect protected-attribute bias. When a protected attribute (gender, race) sits on a backdoor path, controlling for it is not censorship - it is a mathematical requirement for honest effect estimation
• **Epidemiology**: G-computation (Robins, 1986) is the backdoor adjustment formula under a different name. All modern ATE estimates in clinical research are built on it

Предварительные знания

• d-separation: chains, forks, colliders

Backdoor Paths: Where Confounding Comes From

2020. Observational data from multiple countries: patients on ibuprofen die less often from COVID. Headlines read: ibuprofen treats COVID. The mechanism nobody mentioned: severely ill patients were less likely to receive ibuprofen - they were moved to heavier drugs immediately. Disease severity drove both treatment choice and death. That is a fork - a classic backdoor path.

A backdoor path from X to Y is any path that begins with an arrow pointing into X. That is: a path of the form X <- ... -> Y via a common ancestor. Such a path transmits correlation between X and Y without being causal. Not noise, not measurement error - a structural feature of the DAG that is visible without a single number.

**Definition of a backdoor path**: a path $\pi$ between $X$ and $Y$ is a backdoor path if the first edge on the path points into $X$ (that is, $\pi$ begins with $\cdot \to X$). The observed correlation $P(Y \mid X)$ mixes the causal effect with information flowing along backdoor paths.

DoWhy (Microsoft) automatically enumerates all backdoor paths when calling `model.identify_effect()`. Before issuing any effect estimate, it builds the list of active backdoor paths and checks whether the proposed adjustment set blocks all of them. If not - the effect is flagged as unidentified.

Backdoor Criterion: What Is Sufficient to Control

Pearl, 1993. One theorem - and the question of which variables to control became algorithmic rather than intuitive. Before this, econometricians argued for years about what belongs in a regression. After - it is checkable in minutes.

**Backdoor criterion (Pearl, 1993)**: a set $Z$ satisfies the backdoor criterion relative to $(X, Y)$ in DAG $G$ if: 1. no node in $Z$ is a descendant of $X$ 2. $Z$ blocks all backdoor paths between $X$ and $Y$ (i.e., d-separates $X$ from $Y$ in the subgraph $G_{\underline{X}}$ with all outgoing edges of $X$ removed).

Condition (1) - no descendants of X - is critical. Including a variable that X influences creates a collider or partially blocks a mediator. This opens new paths instead of closing old ones. This is exactly where the naive heuristic 'control everything correlated with X and Y' breaks down.

CausalML (Uber) and DoWhy (Microsoft) implement minimum adjustment set search via the O'Shaughnessy (2020) algorithm: find the smallest Z that blocks all backdoor paths. A smaller set means fewer variables to measure in an RCT and lower variance in observational estimates.

Adjustment Formula: P(Y|do(X)) from Observational Data

Once Z satisfies the backdoor criterion, the problem is solved. The do-probability is expressible in terms of the observed distribution. No experiment required - just the right set of measured variables and one formula.

**Backdoor adjustment formula**: if $Z$ satisfies the backdoor criterion for $(X, Y)$, then: $$P(Y = y \mid \mathrm{do}(X = x)) = \sum_{z} P(Y = y \mid X = x, Z = z) \cdot P(Z = z)$$ For continuous $Z$: integral over $z$ instead of sum. This is a population-weighted conditional average.

This is exactly what regression adjustment does: estimate $E[Y \mid X=x, Z=z]$ from data, then average over the marginal $P(Z)$. The method is called G-computation in epidemiology and average treatment effect (ATE) estimation in econometrics. The same result - different names across communities.

Key Ideas

• **Backdoor path**: any path from X to Y beginning with an incoming arrow into X. This is confounding - correlation without causation that naive observation mistakes for an effect
• **Backdoor criterion**: a set Z is valid if (1) it contains no descendants of X, and (2) it d-separates X from Y in the graph with all outgoing edges of X removed. Checked from the DAG structure, not from data
• **Adjustment formula**: $P(Y \mid \mathrm{do}(X)) = \sum_z P(Y \mid X, Z=z) \cdot P(Z=z)$ - population-weighted conditional average. This is G-computation, regression adjustment, and IPW under different names
• **RCT as do-operator**: randomization physically removes backdoor paths. The backdoor criterion explains why - and shows that observational analysis with the right Z is mathematically equivalent to an RCT
• **Descendants of X are forbidden**: mediators deflate the total effect; colliders open new paths. The backdoor criterion guards against both mistakes

Where to Go Next

The backdoor criterion works when all confounders are measurable. Beyond that - methods for when they are not.

• Frontdoor Criterion — Effect estimation via mediator when backdoor confounders are hidden
• do-Operator — Formalization of intervention - what the adjustment formula actually computes
• Identifiability — When do-probabilities are expressible from observational data - the general theory
• Mediation Analysis — Direct and indirect effects - extending backdoor to mediators

Вопросы для размышления

• Ibuprofen and COVID: after controlling for disease severity, the effect vanished. The backdoor criterion does not forbid observation - it specifies what to control for. What variables in current projects might be hidden confounders on such backdoor paths?
• If a team runs an A/B test but users self-select into variants (self-selection bias) - what backdoor path does this create, and what adjustment set is needed?
• The backdoor criterion requires measurable Z. When the confounder is unobservable (e.g., 'motivation' or 'management quality') - what remains? Instrumental variables, frontdoor, difference-in-differences - all address exactly this problem.

Связанные уроки

• cc-02-d-separation — d-separation is the foundation; backdoor criterion is its direct corollary
• cc-04-frontdoor — Frontdoor adjustment handles cases where backdoor sets don't exist
• cc-05-do-operator — The do-operator formalizes what the adjustment formula actually computes
• prob-03-conditional — Conditional probability is the math behind the adjustment formula
• stat-09-regression — Regression adjustment is the numerical implementation of backdoor formula
• cc-11-causal-discovery — PC algorithm finds the graph on which backdoor sets are identified
• stat-01-sampling