MMPC

MMPC (Max-Min Parents and Children) is a constraint-based algorithm for local causal discovery that identifies the direct neighbors — the parents and children — of a target variable in a causal graph without recovering the full graph structure. Unlike global algorithms such as the PC algorithm or GES, which attempt to reconstruct the entire graph topology, MMPC operates locally: given a target variable T and a dataset, it returns the set of variables that are directly connected to T in the underlying causal graph. This local focus makes MMPC computationally efficient and statistically powerful when the research question concerns a specific variable of interest rather than the full causal architecture of the system.

The algorithm proceeds in two phases. First, the forward phase iteratively adds variables to the candidate neighbor set. For each variable not yet selected, MMPC computes the maximum association with T across all possible conditioning sets drawn from the current candidate set. The variable with the highest minimum association — the max-min heuristic — is added to the neighbor set. This heuristic is designed to be conservative: a variable must show strong association with T even after conditioning on the best possible subset of current candidates. Second, the backward phase removes false positives by testing whether each candidate neighbor remains associated with T after conditioning on all subsets of the final neighbor set. If a variable becomes independent of T given some subset of the other neighbors, it is removed.

Local Discovery and the Scalability Frontier

The computational advantage of MMPC over global algorithms is dramatic. The PC algorithm requires O(n^2) conditional independence tests in the worst case, where n is the number of variables. MMPC requires O(n) tests for a single target, making it feasible for high-dimensional domains such as genomics, where the number of variables (genes) can exceed tens of thousands. But this efficiency comes at a cost: MMPC discovers only the local structure around T. It cannot orient edges — it returns an undirected neighborhood — and it cannot identify indirect causal paths. If the research goal is to understand how T influences distant variables in the system, MMPC is insufficient.

MMPC is part of a family of local causal discovery algorithms that includes HITON, IAMB, and their variants. These algorithms share the insight that global structure learning is often overkill: most scientific questions are local. A biologist studying a disease gene does not need the full gene regulatory network; they need the immediate regulators of that gene. A economist studying inflation does not need the full macroeconomic model; they need the direct drivers of inflation. Local algorithms trade the ambition of global discovery for the tractability of focused inquiry. But this trade is not always well-understood, and MMPC is sometimes applied to problems that require global structure.

The Faithfulness Trap in Local Discovery

Like all constraint-based methods, MMPC relies on the Faithfulness assumption: it assumes that conditional independencies in the data reflect structural absences of edges, not accidental parameter cancellations. In local discovery, faithfulness failures are particularly insidious because they cannot be detected from the local neighborhood alone. If two parents of T are nearly perfectly correlated, MMPC may fail to identify either of them, because conditioning on one removes the association of the other. This is the synergy problem: the joint effect of multiple parents can be hidden when they are considered individually. The max-min heuristic mitigates but does not eliminate this problem.

Furthermore, MMPC assumes Causal sufficiency — that all common causes of the target and its neighbors are measured. When hidden confounders exist, MMPC may include variables that are not true parents or children but are merely correlated with T through unmeasured common causes. The algorithm has no mechanism to detect such spurious associations. In this sense, MMPC is not merely a scaled-down version of global causal discovery; it is a method with its own distinct failure modes, and those failure modes are harder to diagnose because the algorithm provides no global context in which to evaluate them.

MMPC represents a pragmatic turn in causal discovery — the recognition that we do not always need the full graph. But pragmatism can become complacency. The local structure that MMPC recovers is only as good as the assumptions that justify looking locally in the first place. If the system contains feedback loops, hidden confounders, or non-linear interactions that propagate globally, then the local neighborhood is not a meaningful unit of analysis. MMPC succeeds when the causal graph is approximately tree-like and the target variable is well-isolated. In complex, interconnected systems — which is to say, in most systems that matter — the local neighborhood is an artifact of the algorithm's limitations, not a natural boundary of causal influence.