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The PC algorithm learns conditional-independence structure from a collection of observed random variables. Its standard initialization is the complete undirected graph, which represents every pair as a candidate adjacency. It then tests conditional independence over increasing conditioning-set sizes, using subsets of current neighboring vertices. Whenever $X\perp Y\mid S$ is established, it removes $X-Y$ and records $S$ as a separating set.^[ref]

For an unshielded triple $X-Z-Y$ with $X,Y$ nonadjacent, a recorded separating set that excludes $Z$ supports the collider orientation $X\to Z\leftarrow Y$. Further orientation rules propagate directions while avoiding directed cycles and unsupported new collider patterns. For example, a chain $X\to Z\to Y$ typically has $X\perp Y\mid Z$, so its separating set contains $Z$; a collider $X\to Z\leftarrow Y$ instead has marginal independence in the simple three-variable graph, so the empty separating set identifies the collider.

The oracle guarantee assumes an acyclic causal model over the observed variables, causal sufficiency, correct conditional independence decisions and faithfulness of a directed acyclic graph. The output represents a class under Markov equivalence of directed acyclic graphs; it need not identify every arrow of one unique causal graph.

An arbitrary incomplete initial graph does not satisfy that guarantee. With only two genuinely adjacent dependent variables, starting with their edge absent cannot recover it: the skeleton stage only removes edges. PC initialization from a conditional independence graph is a qualified screening construction that retains the true skeleton and records the required separating information for absent pairs. In finite samples, incorrect early independence decisions can propagate through both edge removal and orientation.

Peter Spirtes develops mathematical and computational methods for causal discovery. His work asks which causal conclusions can be recovered from statistical data and background knowledge when fully controlled experiments are unavailable, and how to turn the corresponding guarantees into practical algorithms.^[ref]

With Clark Glymour and Richard Scheines, he coauthored Causation, Prediction, and Search by Peter Spirtes, Clark Glymour and Richard Scheines. One important example is the PC algorithm, which uses conditional independence information to remove candidate edges and orient some of the remaining ones. Under its assumptions, the resulting graph represents an equivalence class of possible causal explanations; it need not recover every arrow in a unique graph. Making these limitations explicit is part of the method's contribution, rather than an exception to it.^[ref]

Richard Scheines studies causal discovery: learning causal structure from statistical evidence under explicit assumptions. With Peter Spirtes, Clark Glymour and collaborators, he developed the Tetrad research programme, connecting causal inference, algorithms and machine learning.^[ref]

In Causation, Prediction, and Search by Peter Spirtes, Clark Glymour and Richard Scheines, the collaborators examine how patterns of conditional independence constrain a causal directed acyclic graph. The connection is informative but conditional: faithfulness of a directed acyclic graph, restrictions on unobserved causes and the correctness of the independence information affect what can be identified. Different graphs can remain indistinguishable under Markov equivalence of directed acyclic graphs, so observational data need not select one unique causal story.^[ref]