Causal inference §

The DAG encodes causal assumptions: an edge $X \to Y$ means $X$ is a direct cause of $Y$. The model generates a joint distribution $P(X_1, \ldots, X_n)$ that is Markov with respect to the graph: each variable is independent of its non-descendants given its parents.

Example. A simple SCM for a drug trial:

Here, Age is a confounder: it affects both the treatment assignment and the outcome. The associational effect of Drug on Recovery is confounded by Age.

The SCM framework makes explicit the distinction between associational quantities (read off the observed distribution) and interventional quantities (require modifying the model). This is the key insight that separates causal inference from Estimation and Hypothesis testing.

The do-operator modifies the SCM: the structural equation for $X$ is replaced by $X = x$ (a constant), and all incoming edges to $X$ are removed. The resulting distribution is the interventional distribution.

Key distinction:

• $P(Y \mid X = x)$ (associational): the distribution of $Y$ among units that happen to have $X = x$.
• $P(Y \mid \text{do}(X = x))$ (interventional): the distribution of $Y$ if we force $X = x$.

These are equal if and only if there is no confounding (i.e., $X$ is independent of the potential outcomes). In general, $P(Y \mid X = x) \neq P(Y \mid \text{do}(X = x))$.

Example. In the drug trial SCM, $P(\text{Recovery} \mid \text{Drug} = 1)$ is confounded by Age (older patients are more likely to receive the drug and less likely to recover). But $P(\text{Recovery} \mid \text{do}(\text{Drug} = 1))$ gives the true causal effect, free of confounding.

The ladder explains why causal inference needs extra assumptions: the observed distribution $P(V)$ is a rung-1 object, and the map from rung-1 objects to rung-2 or rung-3 objects is not unique without causal assumptions. Two different SCMs can induce the same observational distribution but different interventional distributions; only the causal structure (the arrows) breaks the tie. This is why identifying $P(Y \mid \text{do}(X))$ from data requires either an experiment (which intervenes directly) or untestable structural assumptions (which the graphical criteria below encode).

Example. Observing that barometer readings predict rain (rung 1) does not tell you whether twisting the dial will cause rain (rung 2). The DAG $\text{Pressure} \to \text{Barometer}$ and $\text{Pressure} \to \text{Rain}$ explains both: the association is real, but there is no causal path from the barometer to the rain.

Why the front-door works. The effect of $X$ on $M$ is identifiable by the back-door criterion with $Z = \emptyset$ (condition (i) rules out confounding of $X$–$M$). The effect of $M$ on $Y$ is identifiable by adjusting for $X$ (condition (ii)). Condition (iii) ensures the effect of $X$ on $Y$ is fully transmitted through $M$, so the two identifiable pieces compose into the front-door formula. The derivation is a short chain of do-calculus rules — this is exactly what the calculus is for.

Example. Smoking ($X$), tar deposits in the lungs ($M$), and cancer ($Y$), with an unobserved confounder $U$ (e.g., a genetic predisposition) affecting both smoking and cancer. No set of observed variables blocks the back-door path $X \leftarrow U \to Y$. But tar deposits mediate the effect, and the front-door formula recovers the effect of smoking on cancer from observational data alone.

The back-door and front-door criteria are the two workhorse graphical identification results; the do-calculus is the general machine that generates them (and many others, such as instrumental-variable-type identifications).

The two main frameworks for causal inference are:

In the potential outcomes framework, the key assumption is unconfoundedness (also called ignorability or selection on observables):

i.e., potential outcomes are independent of treatment assignment. Unconfoundedness with no covariates corresponds exactly to the back-door criterion with the empty set $Z = \emptyset$ (no confounders). Its conditional version, $(Y(0), Y(1)) \perp\!\!\!\perp X \mid Z$ for observed covariates $Z$, corresponds exactly to the back-door criterion with adjustment set $Z$.

The two frameworks are mathematically equivalent: any SCM induces a set of potential outcomes, and any set of potential outcomes satisfying consistency and unconfoundedness can be represented by an SCM.

Connection to the do-operator. In the potential outcomes framework, $E[Y \mid \text{do}(X = 1)] = E[Y(1)]$ and $E[Y \mid \text{do}(X = 0)] = E[Y(0)]$. The ATE is the difference of interventional means.

as functions of the full assignment vector $\mathbf{x}$.

Example. In a vaccine trial, if vaccinated individuals reduce transmission, unvaccinated individuals in the same villages are partially protected too. Estimating the direct effect requires care (e.g., cluster randomization); the total effect includes these spillovers and violates the plain SUTVA setup.

Why consistency matters. Consistency links the observational data to the potential-outcome notation: without it, “treated” in the data may not correspond to any single well-defined intervention, and $E[Y \mid X = 1]$ cannot be connected to $E[Y(1)]$. This is a practical concern when treatments are heterogeneous (e.g., “smoking” covers many intensities).

When a valid instrument exists, the causal effect can be estimated by the Wald estimator (ratio of two differences):

Example. In a drug trial, suppose a doctor’s prescribing habit (a binary variable) affects whether a patient receives the drug, but the habit itself does not affect recovery except through the drug. The doctor’s habit is an instrument. The Wald estimator gives the causal effect of the drug on recovery, even though unobserved patient severity confounds the drug–recovery association.

The IV structure ($Z \to X \to Y$ with no $Z \to Y$ edge and no confounding on $Z$) is a distinct graphical pattern from the front-door criterion ($X \to M \to Y$ with an unconfounded mediator). Both are identifiable via the do-calculus, but they correspond to different graphical configurations and different identifying formulas. Note that the Wald estimator identifies the local average treatment effect (LATE): the effect for the compliers whose treatment was actually changed by the instrument, not for the whole population.

Counterfactuals are the most specific level of causal reasoning. They require the full SCM, not just the graph: two SCMs with the same graph can induce different counterfactuals.

Example. A patient took a drug ($X = 1$) and recovered ($Y = 1$). The counterfactual question is: would this patient have recovered without the drug ($Y_{x=0} = ?$)? The answer requires the structural equations and the unit’s exogenous variables.

Counterfactuals are computed from the SCM by abduction–action–prediction:

1. Abduction: use the evidence $(X = x, Y = y)$ to infer the posterior over the exogenous variables $U$ for this unit.
2. Action: modify the SCM, replacing the equation for $X$ by $X = x'$.
3. Prediction: propagate the inferred $U$ through the modified equations to obtain $Y_{x'}$.

The resulting quantity $P(Y_{x'} \mid X = x, Y = y)$ mixes levels: it conditions on observations (rung 1) but answers a counterfactual question (rung 3). This is why it cannot in general be written as a function of $P(V)$ alone — the structural equations, not just the graph, are needed.

Why the propensity score suffices. By Rosenbaum and Rubin (1983), $e(Z)$ is a balancing score: $X \perp\!\!\!\perp Z \mid e(Z)$. Hence, under conditional unconfoundedness $(Y(0), Y(1)) \perp\!\!\!\perp X \mid Z$, adjusting for the scalar $e(Z)$ removes all confounding — this is the back-door adjustment with the score replacing the full covariate vector. In practice one estimates $e(Z)$ (e.g., by logistic regression) and either matches treated to control units with similar scores, or weights units by $1/e(Z)$ (treated) and $1/(1 - e(Z))$ (controls) to build a pseudo-population with no confounding.

Why parallel trends is a causal assumption. The DiD estimator cancels the pre/post trend common to both groups. It is identified only if the counterfactual trend of the treated group equals the observed trend of the control group — an untestable claim about rung 2 of the ladder, not a statistical association. It is the time-varying analogue of unconfoundedness: time-invariant unobserved confounders are allowed, but their effect must evolve identically in both groups.

Why RD is “local randomization”. Units cannot precisely manipulate the running variable at the cutoff, so just above and just below $c$ the populations are comparable in expectation: the discontinuity in $E[Y \mid S = s]$ at $c$ is the causal effect at the cutoff (a local average treatment effect). The price is external validity: the estimate says nothing about units far from $c$.

Each method also requires SUTVA (see SUTVA) and correct functional-form assumptions; violations of any of these produce biased estimates even when the underlying identification argument is valid.

Key causal ML methods:

• Double/debiased machine learning (DML): use any ML method to fit the nuisance functions $e(Z)$ and $m(Z) = E[Y \mid Z]$, then average the residuals:

Cross-fitting (fitting the nuisances on one fold, evaluating on another) plus the Neyman orthogonality of the score make the estimator root-n consistent even when the nuisances are learned at slower rates. This is the potential-outcomes/back-door machinery with ML plug-ins.

• Causal forests (Wager & Athey): random forests modified so that estimates of the conditional average treatment effect $\tau(z) = E[Y(1) - Y(0) \mid Z = z]$ are asymptotically normal and honest (estimation and splitting use separate samples).
• Meta-learners for heterogeneous effects with binary treatment: the S-learner fits one model $E[Y \mid X, Z]$ and reads off $\hat{\tau}(z)$ as the difference of predictions at $X = 1$ and $X = 0$; the T-learner fits separate models per arm and takes their difference; the X-learner imputes each unit’s missing potential outcome, models the imputed effects per arm, and combines them weighted by the propensity score — often best when one arm is much larger than the other.

All of these assume an identification result already holds (typically conditional unconfoundedness given $Z$, i.e., the back-door criterion); the machine learning only makes the estimation flexible, not the identification.

(a) No. The correlation between $X$ and $Y$ is spurious: both are caused by $T$. There is no causal path from $X$ to $Y$ in the graph $T \to X$, $T \to Y$. The causal effect of $X$ on $Y$ is zero.

(b) The back-door paths from $X$ to $Y$ are: $X \leftarrow T \to Y$. The set $Z = \{T\}$ blocks this path (it is a common cause, not a descendant of $X$). By the back-door criterion:

Since $Y \perp\!\!\!\perp X \mid T$ (no direct edge $X \to Y$), this simplifies to $P(Y \mid T = t)$ integrated over $T$, which does not depend on $x$. Hence the causal effect is zero.

(c) From the structural equations, $Y = 0.5\,T + U_Y$ does not contain $X$. Setting $X = x$ (do-operator) changes the equation for $X$ but not for $Y$. Therefore $E[Y \mid \text{do}(X = x)] = E[0.5\,T + U_Y] = 0.5\,E[T]$ regardless of $x$. The causal effect is $0.5\,E[T] - 0.5\,E[T] = 0$, confirming part (a).

Question 1. In the graph $Z \to X \to Y$ with an additional edge $Z \to Y$, is $Z$ a valid instrument for the effect of $X$ on $Y$? Which assumption fails?

Question 2. A study measures the effect of a minimum-wage increase by comparing employment just above and just below the wage threshold. Which estimation method is this, and what is its key identifying assumption?

Question 3. At which rung of the ladder of causation is the quantity $P(Y_{x'} \mid X = x, Y = y)$? Why can it not be computed from the observational distribution alone, even with the DAG known?

Question 1. No. The exclusion restriction fails: $Z$ has a direct edge to $Y$, so $Z$ affects $Y$ through paths other than $X$. The Wald estimator would then mix the direct effect of $Z$ into the estimated effect of $X$. In the IV graph, $Z$ must affect $Y$ only through $X$.

Question 2. Regression discontinuity. Treatment is assigned by a threshold rule on a running variable (wage level relative to the threshold). The key assumption is continuity of the potential-outcome functions at the cutoff — equivalently, that units just above and below the threshold are comparable (local randomization), so the discontinuity in $E[Y \mid S = s]$ at the cutoff identifies the local causal effect.

Question 3. Rung 3 (counterfactuals). It conditions on observed evidence ($X = x, Y = y$) and asks about a different intervention ($x'$). The observational distribution $P(V)$ is a rung-1 object: many different SCMs — with the same DAG and the same $P(V)$ but different structural equations — induce different values of $P(Y_{x'} \mid X = x, Y = y)$. Computing it requires the full structural equations (via abduction–action–prediction), not just the graph.

The DAG encodes causal assumptions: an edge $X \to Y$ means $X$ is a direct cause of $Y$. The model generates a joint distribution $P(X_1, \ldots, X_n)$ that is Markov with respect to the graph: each variable is independent of its non-descendants given its parents.

Example. A simple SCM for a drug trial:

Here, Age is a confounder: it affects both the treatment assignment and the outcome. The associational effect of Drug on Recovery is confounded by Age.

The SCM framework makes explicit the distinction between associational quantities (read off the observed distribution) and interventional quantities (require modifying the model). This is the key insight that separates causal inference from Estimation and Hypothesis testing.

The do-operator modifies the SCM: the structural equation for $X$ is replaced by $X = x$ (a constant), and all incoming edges to $X$ are removed. The resulting distribution is the interventional distribution.

Key distinction:

• $P(Y \mid X = x)$ (associational): the distribution of $Y$ among units that happen to have $X = x$.
• $P(Y \mid \text{do}(X = x))$ (interventional): the distribution of $Y$ if we force $X = x$.

These are equal if and only if there is no confounding (i.e., $X$ is independent of the potential outcomes). In general, $P(Y \mid X = x) \neq P(Y \mid \text{do}(X = x))$.

Example. In the drug trial SCM, $P(\text{Recovery} \mid \text{Drug} = 1)$ is confounded by Age (older patients are more likely to receive the drug and less likely to recover). But $P(\text{Recovery} \mid \text{do}(\text{Drug} = 1))$ gives the true causal effect, free of confounding.

The do-calculus is a complete decision procedure: if an interventional quantity is identifiable, the do-calculus can derive its expression in terms of observed quantities.

• Front-door criterion: if there is a confounded path $X \to Y$ but a mediator $M$ such that $X \to M \to Y$ is unconfounded, then the causal effect is identifiable via the front-door formula:

The back-door criterion is the most commonly used: it says that conditioning on the right set of covariates (confounders) removes confounding bias.

The two main frameworks for causal inference are:

In the potential outcomes framework, the key assumption is unconfoundedness (also called ignorability or selection on observables):

i.e., potential outcomes are independent of treatment assignment. This is equivalent to the back-door criterion in the SCM framework when the DAG has a single confounder.

The two frameworks are mathematically equivalent: any SCM induces a set of potential outcomes, and any set of potential outcomes satisfying consistency and unconfoundedness can be represented by an SCM.

Connection to the do-operator. In the potential outcomes framework, $E[Y \mid \text{do}(X = 1)] = E[Y(1)]$ and $E[Y \mid \text{do}(X = 0)] = E[Y(0)]$. The ATE is the difference of interventional means.

When a valid instrument exists, the causal effect can be estimated by the Wald estimator (ratio of two differences):

Example. In a drug trial, suppose a doctor’s prescribing habit (a binary variable) affects whether a patient receives the drug, but the habit itself does not affect recovery except through the drug. The doctor’s habit is an instrument. The Wald estimator gives the causal effect of the drug on recovery, even though unobserved patient severity confounds the drug–recovery association.

The IV structure ($Z \to X \to Y$ with no $Z \to Y$ edge and no confounding on $Z$) is a distinct graphical pattern from the front-door criterion ($X \to M \to Y$ with an unconfounded mediator). Both are identifiable via the do-calculus, but they correspond to different graphical configurations and different identifying formulas.

Counterfactuals are the most specific level of causal reasoning (Pearl’s “ladder of causation”: association → intervention → counterfactual). They require the full SCM, not just the graph.

Example. A patient took a drug ($X = 1$) and recovered ($Y = 1$). The counterfactual question is: would this patient have recovered without the drug ($Y_{x=0} = ?$)? The answer requires the structural equations and the unit’s exogenous variables.

Counterfactuals are identifiable from the SCM: $P(Y_{x'} \mid X = x, Y = y)$ can be computed by “abduction–action–prediction” (infer the exogenous variables from the evidence, apply the intervention, predict the outcome).

These methods are the applied counterparts of the theoretical frameworks above:

• Propensity score matching operationalizes the back-door criterion: the propensity score $e(Z) = P(X = 1 \mid Z)$ is a balancing score that, when conditioned on, makes treatment independent of the covariates.
• Difference-in-differences is a special case of the potential outcomes framework with a time-varying treatment: the “parallel trends” assumption is a dynamic version of unconfoundedness.
• Regression discontinuity exploits a local randomization: near the cutoff, treatment assignment is approximately random, so the local average treatment effect is identified without further assumptions.

Each method requires its own set of assumptions (no interference, SUTVA, correct functional form, etc.) that go beyond the basic causal model.

(a) No. The correlation between $X$ and $Y$ is spurious: both are caused by $T$. There is no causal path from $X$ to $Y$ in the graph $T \to X$, $T \to Y$. The causal effect of $X$ on $Y$ is zero.

(b) The back-door paths from $X$ to $Y$ are: $X \leftarrow T \to Y$. The set $Z = \{T\}$ blocks this path (it is a common cause, not a descendant of $X$). By the back-door criterion:

Since $Y \perp\!\!\!\perp X \mid T$ (no direct edge $X \to Y$), this simplifies to $P(Y \mid T = t)$ integrated over $T$, which does not depend on $x$. Hence the causal effect is zero.

(c) From the structural equations, $Y = 0.5\,T + U_Y$ does not contain $X$. Setting $X = x$ (do-operator) changes the equation for $X$ but not for $Y$. Therefore $E[Y \mid \text{do}(X = x)] = E[0.5\,T + U_Y] = 0.5\,E[T]$ regardless of $x$. The causal effect is $0.5\,E[T] - 0.5\,E[T] = 0$, confirming part (a).