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Acyclic directed mixed graph
Definition
An acyclic directed mixed graph, or ADMG, contains directed edges such as X → Y and bidirected edges such as X ↔ Y, with no cycle made entirely of consistently oriented directed edges. In causal applications, it can summarize relationships among observed variables when some common causes are unobserved. Its meaning depends on the statistical or causal model attached to the graph.
What it means
A dataset rarely contains every factor relevant to an outcome. Drawing an ordinary directed acyclic graph only over recorded variables can hide this limitation. An ADMG provides a way to retain information about certain omitted causes without drawing all of them explicitly.
A bidirected edge has arrowheads at both ends, but it does not mean that each variable directly causes the other. Under the common latent-variable interpretation, it represents a connection through unobserved common causes. A directed edge and a bidirected edge may coexist between the same pair; this arrangement is often called a bow. Additional restrictions apply to special graph classes, not to every ADMG.
How it works
One route to an ADMG starts with a DAG containing observed and hidden variables. Latent projection removes the hidden nodes while preserving relevant paths: hidden directed chains can induce directed edges, and suitable hidden common-cause paths can induce bidirected edges.
For independence questions, m-separation examines paths and arrowheads. A path is blocked by a conditioned non-collider, or by a collider with neither itself nor a descendant conditioned on. When every path is blocked, the graph implies the corresponding conditional independence under its global Markov property.
A graph's independence interpretation should be distinguished from stronger causal interpretations and additional constraints of particular latent-variable models. Merely drawing an ADMG neither proves the assumed structure nor identifies every causal effect.
Three practical examples
- A retailer studies promotional emails and purchases. Unrecorded customer interest may influence both email targeting and buying, motivating a bidirected connection alongside a possible directed email effect.
- A manufacturing analyst studies a machine setting and defect rates. A hidden material property may influence the chosen setting and the eventual defects, requiring more than a simple regression arrow.
- A learning-platform team studies practice activity and test results. Unmeasured motivation can affect both. An ADMG helps the team document that limitation before interpreting the association as the effect of additional practice.
Think of it this way
Think of a simplified road map that preserves important connections through areas omitted from the drawing. A bidirected connection says something relevant may pass through that unseen region; it does not identify its exact contents. The simplified map can answer some route questions, but it cannot supply missing measurements or certify that the original map was correct.
Worked example
Construct an original toy model with hidden U and independent, mean-zero noises e_X and e_Y. Give all three variance one. Let X = U + e_X and Y = 2X + U + e_Y. The full causal DAG has U → X, U → Y, and X → Y.
Removing U gives an observed ADMG with X → Y and X ↔ Y. This is allowed: the bidirected edge is not a directed return route from Y to X, so it creates no directed cycle.
Now Var(X) = Var(U) + Var(e_X) = 2. Also Cov(X,U) = 1 and Cov(X,e_Y) = 0. Therefore Cov(X,Y) = 2 Var(X) + Cov(X,U) = 5. The population least-squares slope for predicting Y from X is 5/2 = 2.5.
Under the stipulated structural equations, externally increasing X by one unit while leaving U's distribution unchanged increases mean Y by two units. Thus the observational slope 2.5 differs from the causal effect 2. The bidirected edge records the confounding structure that makes this discrepancy possible.
These numbers follow from the specified equations, not from the graph alone. The graph does not supply coefficients or prove that adjustment using the available variables can recover the effect.
Illustrative case study
Fictional retailer BasketLine finds that customers receiving a premium recommendation buy more expensive products. Its team first draws recommendation → purchase value. Interviews with staff reveal that an unrecorded assessment of customer interest also affects which recommendations are sent.
The analyst adds a bidirected connection to represent the proposed hidden common cause. The team stops presenting the predictive coefficient as an established causal effect and considers a controlled experiment within its operational constraints. This fictional scenario illustrates how a graph can expose an assumption requiring evidence; it does not report a real deployment or claim that adding an edge fixes the underlying data problem.
Common misconceptions and mistakes
Acyclic does not forbid every closed shape when arrow direction is ignored. It forbids directed cycles. Likewise, a bidirected edge is not shorthand for two directed causal arrows.
Another mistake is assuming every missing edge means unconditional independence. The relevant separation criterion depends on all paths and the conditioning set. Conditioning can open a collider path rather than block it. Finally, different hidden-variable systems can produce the same observed graph. An ADMG generally does not reveal the number, identities, or detailed mechanisms of all hidden causes.
Business use
Use an ADMG to make assumptions about unmeasured influences visible during causal analysis. Agree on the decision, outcome, observed variables, and plausible omissions with domain experts. Then assess identification before choosing an estimator. Keep a record of the evidence supporting each relationship and test sensitivity to plausible alternatives. The graph is especially useful for explaining why accurate prediction and reliable intervention advice may require different analyses.
People also ask
Is every DAG an ADMG?
Yes, a DAG is a special case with no bidirected edges.
Does a bidirected edge show observed correlation?
It represents model structure; its presence does not itself measure the size of correlation.
Can an ADMG identify hidden causes by name?
No. More evidence or assumptions are needed.
Check your understanding
Why is the worked graph acyclic?
Answer: Its directed portion contains only X → Y.
Why is the regression slope 2.5 rather than 2?
Answer: The hidden common cause contributes an additional covariance term.
Does drawing the graph identify the intervention effect?
Answer: Not automatically; identification requires further analysis of the model and available evidence.
Related terms
- Directed acyclic graph
- M-separation
- Collider
- Structural causal model
- latent projection
- unobserved confounding
- causal identification
- district
Learn next
Practice distinguishing directed paths from paths that include bidirected edges. Then study m-separation and how causal identification differs from estimating an association in observed data.
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Sources
Evans and Richardson, Markovian Acyclic Directed Mixed Graphs for Discrete Data, 2014: https://arxiv.org/pdf/1301.6624 Evans, Causal Inference, Chapter 15, latent projection: https://www.stats.ox.ac.uk/~evans/APTS/fp.html
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