Causal discovery: Definition & Examples

Causal discovery is the attempt to learn causal structure from data under explicit assumptions about how the data were generated. For a quant deciding whether changing an execution policy would change costs, its useful output may be a set of compatible mechanisms and a clear statement of which directions remain unresolved. A graph fitted to observed co-movement does not automatically identify an intervention's effect.

The distinction from broader causal inference is the unknown structure: discovery asks which arrows the evidence supports; effect estimation asks what a specified intervention would do within an identified model or design.

One distribution, two incompatible policy answers

Consider three synthetic, dimensionless, jointly Gaussian variables

X,M,Y

, with mean zero and covariance matrix, in that order,

\Sigma=\begin{pmatrix} 2&1&2\\ 1&1&2\\ 2&2&5 \end{pmatrix}.

Their conditional covariance is

\text{Cov}(X,Y\mid M) =2-\frac{1\times2}{1}=0.

Because the variables are jointly Gaussian, this establishes conditional independence,

X\perp Y\mid M

. A zero conditional covariance alone would not establish that conclusion for an arbitrary distribution.

Now construct two different structural causal models. Every disturbance is centred Gaussian, and the disturbances within each model are mutually independent. The second parameter of

\mathcal N

below is a variance.

Fork: $X\leftarrow M\to Y$ . Let

M\sim\mathcal N(0,1),\qquad X=M+\varepsilon_X,\qquad Y=2M+\varepsilon_Y,

with $\text{Var}(\varepsilon_X)=\text{Var}(\varepsilon_Y)=1$ . Then $\text{Var}(X)=2$ , $\text{Var}(Y)=5$ and $\text{Cov}(X,Y)=2$ . Setting $X=x$ externally replaces the equation for $X$ ; it does not change $M$ or the equation for $Y$ . Consequently,

\mathbb E[Y\mid\operatorname{do}(X=x)]=0.

Chain: $X\to M\to Y$ . Instead let

X\sim\mathcal N(0,2),\qquad M=\tfrac12 X+\eta_M,\qquad Y=2M+\eta_Y,

with

\text{Var}(\eta_M)=\tfrac12

and

\text{Var}(\eta_Y)=1

. This gives the same entire covariance matrix, hence the same centred Gaussian observational distribution. But setting

X=x

now changes the mean of

M

x/2

and the mean of

Y

x

\mathbb E[Y\mid\operatorname{do}(X=x)]=x.

The response per unit change in

X

is therefore 0 in the fork and 1 in the chain. In both models, the observational regression gives

\mathbb E[Y\mid X=x]=x

. That common fitted slope cannot tell us which intervention response applies. These equations are stipulated mechanisms, not estimates from financial data. The intervention interpretation uses the equation-replacement semantics described by Peters and co-authors, §§1.2–1.4.

Synthetic dimensionless Gaussian fork M to both X and Y and chain X to M to Y share variances 2, 1, 5 and covariances 1, 2, 2. Conditional covariance of X and Y given M is zero. Setting X=x gives mean Y zero in the fork and x in the chain. The equivalence class has CPDAG X–M–Y, with neither edge oriented.

Original exact Gaussian construction. Identical observations leave different intervention responses possible; undirected CPDAG lines record unresolved directions.

What conditional independences can recover

For a directed acyclic graph, or DAG, its skeleton keeps the adjacencies but removes arrowheads. An unshielded collider is

A\to B\leftarrow C

with no edge between

A

and

C

. DAGs with the same skeleton and unshielded colliders are Markov equivalent: they impose the same graphical independence restrictions. Meek, §1.1.

For the example, the fork and both chain directions form one three-DAG equivalence class. A completed partially directed acyclic graph (CPDAG) represents the class: an arrow is shared by every member; an undirected edge can take either direction across members. Here the CPDAG is

X-M-Y

. The lines do not mean reciprocal causation: orienting both into

M

would introduce a collider and leave this class.

A constraint-based procedure such as PC searches for separating sets, removes edges when conditional independence is found, identifies unshielded colliders and propagates compelled orientations. This can orient some arrows without orienting them all. Meek, §2 explains the separation and orientation steps. Population PC uses exact independence answers; sample PC substitutes tests.

The assumptions sit behind the arrowheads

For the usual causal interpretation of PC, the important commitments include:

Acyclicity: the chosen variables and time resolution admit a DAG.
Causal sufficiency: no unmeasured common cause of the included variables creates hidden confounding, with selection also handled appropriately.
Causal Markov condition: graphical separation implies independence in the distribution.
Faithfulness: independences do not arise through parameter cancellations that the graph does not encode.

Markov and faithfulness assumptions connect probability to graph structure; neither is certified by a small correlation. Peters and co-authors show how cancellation can break faithfulness in their §1.1, Example 2.

Finite samples introduce a separate problem. Failure to reject a conditional-independence null is not proof that independence holds. Weak relationships, large conditioning sets and repeated testing can alter both the recovered skeleton and later orientations. In Gaussian PC, tests concern partial correlations; using them for a non-Gaussian nonlinear system needs additional justification. Consistency results require stated sampling and regularity conditions, not merely a large dataset. Kalisch and Bühlmann, §§2.2.2 and 3.

If hidden causes or selection are plausible, the target and algorithm must accommodate them. FCI is designed for such settings and represents remaining ambiguities differently; its output should not be read as a causally sufficient CPDAG. Allowing hidden variables broadens the compatible explanations rather than recovering every missing cause. Spirtes, Meek and Richardson, §§1–3.

Evidence that can narrow the class

A well-defined intervention can distinguish mechanisms that observations cannot. In the constructed fork and chain, manipulating $X$ and measuring the response of $Y$ supplies information absent from their common observational distribution. Real intervention data still require a credible account of what was changed and which other mechanisms remained intact.

Verified temporal ordering and substantive knowledge can forbid some directions. That is additional evidence, as in Meek's background-knowledge framework, §§1–2. A timestamp alone does not remove shared news, measurement delay or omitted variables. In financial series, a lagged predictor or Granger-causality result establishes no intervention effect by itself; the variable definitions, sampling process and time-series assumptions still matter.

Restricted functional models offer another route. Under suitable identifiability conditions, nonlinear additive-noise assumptions can distinguish directions within an observational equivalence class. Those assumptions add information beyond conditional independences; the linear Gaussian construction above remains an important non-identifiable case. Peters and co-authors, §§2–3.

For the underlying graphical mechanics, Bayesian Networks and Markov Random Fields develops chains, forks, colliders and separation. Its preview is public; the full lesson requires membership. For the financial identification problem, Endogeneity and Instrumental Variables explains why omitted influences and simultaneity can survive precise regression estimates, and what extra assumptions an instrument requires. Its preview is public; full teaching requires membership.

Sources

Markus Kalisch and Peter Bühlmann, “Estimating High-Dimensional Directed Acyclic Graphs with the PC-Algorithm”, JMLR 8 (2007), pp. 613–636, §§2–3: equivalence classes, CPDAGs, population versus sample PC and consistency conditions.
Jonas Peters, Joris M. Mooij, Dominik Janzing and Bernhard Schölkopf, “Causal Discovery with Continuous Additive Noise Models”, JMLR 15 (2014), pp. 2009–2053, §§1–3: causal semantics, faithfulness, observational ambiguity and restricted-model identification.
Christopher Meek, “Causal Inference and Causal Explanation with Background Knowledge”, UAI (1995), pp. 403–410, §§1–2: required and forbidden directions, and orientations common to compatible explanations.
Peter Spirtes, Christopher Meek and Thomas Richardson, “Causal Inference in the Presence of Latent Variables and Selection Bias”, UAI (1995), pp. 499–506, §§1–3: hidden variables, selected populations and the changed interpretation of discovery output.

One distribution, two incompatible policy answers

What conditional independences can recover

The assumptions sit behind the arrowheads

Evidence that can narrow the class

Sources

Related terms

Conditional independence

Correlation

Structural causal model