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Let $X,Y$ be independent random variables, each with a Bernoulli distribution of parameter $1/2$, and put $S=X+Y$. Unconditionally, the four ordered pairs $(0,0),(0,1),(1,0),(1,1)$ each have probability $1/4$. On the event $S=1$, only $(0,1)$ and $(1,0)$ remain, each with conditional probability $1/2$. ThusKnowing that exactly one trial succeeded makes observing one outcome informative about the other. This violates conditional independence given $S$, even though the original random variables are independent. Given $S=0$ or $S=2$, both variables are constant random variables and are conditionally independent there; that does not repair the failure at $S=1$, an event of positive probability.

The conditional covariance here is $\mathbb{E}[XY\mid S=1]-\mathbb{E}[X\mid S=1]\mathbb{E}[Y\mid S=1]=0-1/4=-1/4$. Both conditional variances are $1/4$, so the conditional correlation coefficient is $-1$: within the selected group, $Y=1-X$. These conclusions follow from restricting a finite sample space, without positing any physical interaction between the trials.

This example and the shared-environment example establish both logical directions: independent random variables need not be conditionally independent, and conditionally independent variables need not be independent. When $S$ is represented as the joint result of two inputs in a causal directed acyclic graph, conditioning on it has the collider structure associated with collider bias. A probability calculation by itself establishes a dependence relation, not a physical causal mechanism or an intervention.

Under the displayed causal directed acyclic graphs, with nonzero effects along the relevant arrows and no exact cancellation of the effects on $B$, the valid instrumental variables are scenarios (i), (ii), and (iv). The path arguments are:

An upstream intermediate variable need not invalidate an instrumental variable if its routes to the outcome all pass through the target exposure; an outcome intermediate reached without that exposure does invalidate it.

In the second causal directed acyclic graph, $Y(a_1,a_2)$ depends on $U_2$ but not on $U_1$, whereas $A_1$ depends on $U_1$ and the two latent roots are independent. HenceAlthough conditioning on $X$ conveys information about $U_2$, the assignment $A_2$ uses only $(A_1,X)$ and fresh randomization, soThese are the two sequential exchangeability conditions. Using them successively, together with consistency of potential outcomes, giveswhich is the formula from part a.

Adding $X\to Y$ invalidates the argument in general. The latent variable $U_1$ confounds $A_1$ and $X$, so $\mathbb{P}(X\mid A_1=a_1)$ need not equal the distribution of $X(a_1)$. When $X$ directly affects $Y$, that discrepancy no longer cancels after summing over $x$. The same observed distribution can then correspond to different intervention means, so the displayed formula need not identify the effect.

Let $G$ be the genetic instrument, $X$ vitamin-D concentration, $U$ an unmeasured cause of vitamin D and mortality, and $Y$ mortality. The causal directed acyclic graph containsAlthough $G$ and $U$ are marginally independent, $X$ is a collider. Conditioning on $X$ opens the path $G\leftrightarrow U\to Y$, violating instrumental-variable independence within the resulting strata. Residual exposure stratification instead removes the component of $X$ predicted by $G$ before stratification; under the additive first-stage model, the stratifying variable is no longer caused by $G$.

Take all components to be centered and let their marginal variances be $V_A,V_C,V_E$. A linear structural equation model iswhere $C_i,E_{i1},E_{i2}$ are mutually independent and the $E_{ij}$ are identically distributed. The causal directed acyclic graph has arrows

For monozygotic twins, set $A_{i1}=A_{i2}=G_i$ with $\mathrm{Var}(G_i)=V_A$; the genetic cause is completely shared. For dizygotic twins, one explicit construction iswhere $G_i,G_{i1},G_{i2}$ are independent with variance $V_A/2$. Then both additive genetic terms have variance $V_A$, while their covariance is $V_A/2$ and their correlation coefficient is $1/2$. In a fuller graph, the shared $G_i$ points to both genetic components and the unique $G_{ij}$ points only to its own component.

A path in a causal directed acyclic graph is active given a conditioning set $K$ when every noncollider on the path is outside $K$ and every collider has itself or a descendant in $K$. Two vertex sets are d-separated by $K$ when no path between them is active given $K$. The global Markov property of a directed acyclic graph then turns d-separation into conditional independence.

In the displayed graph, the edges areEvery observed pair except $(X_1,X_4)$ and $(X_2,X_4)$ is joined by a direct edge, which remains active under conditioning on any other observed variables. The pair $(X_2,X_4)$ is always joined by the fork $X_2\leftarrow U\to X_4$, because the unobserved noncollider $U$ cannot be conditioned on.

For $(X_1,X_4)$, if $X_3$ is not conditioned on, $X_1\to X_3\to X_4$ is active. If $X_3$ is conditioned on, the pathbecomes active because its collider $X_2$ has conditioned descendant $X_3$; conditioning on $X_2$ itself also opens it. Thus every observed pair is d-connected given every subset of the other observed variables. Any conditional independence between two nonempty observed subvectors would imply one between each selected pair, so the graph entails none.

A minimal causal directed acyclic graph has edges