Why Factorization and Separation Belong Together
An undirected graphical model represents a joint distribution through local factors attached to groups of connected variables. Its edges indicate which variables may participate together in a factor; missing edges restrict how the joint distribution can depend on those variables. The key consequence is that a set of nodes separating two other sets in the graph also separates their random variables probabilistically, under the graph factorization.
In a multi-omics network, nodes might represent measured molecular features from different assay layers. A graph separation statement concerns conditional association in the modeled distribution. It does not establish a causal pathway, and the absence of an edge is meaningful only relative to the model, its assumptions, and the data used to estimate it.
The empty set may be a clique, corresponding to a constant factor. A factor attached to a smaller clique is also allowed when that clique is contained in a larger one. For a density, the factorization is understood relative to the product measure for the variables. The argument below assumes the relevant integrals or sums exist and that conditioning is considered only at values of \(X_S\) with positive probability or density.
Three Markov Properties
Each Markov property translates a different graph statement into a conditional independence statement. In the expressions below, \(X_A\) denotes the collection of variables at nodes in \(A\).
| Property | Graph condition | Conditional independence |
|---|---|---|
| Pairwise Markov | Nodes \(u\) and \(v\) are not adjacent | \(X_u\perp X_v\mid X_{V\setminus\{u,v\}}\) |
| Local Markov | Node \(v\) and its neighbors \(N(v)\) | \(X_v\perp X_{V\setminus(\{v\}\cup N(v))}\mid X_{N(v)}\) |
| Global Markov | Disjoint sets \(A,B\) are separated by \(S\) | \(X_A\perp X_B\mid X_S\) |
The pairwise property conditions on every other node, not merely on the endpoints' common neighbors. The local property conditions on the node's entire neighborhood. The global property allows arbitrary separated sets and conditions on the separating set. Thus the properties ask related but distinct questions; their equivalence requires assumptions.
Worked Example: Conditional Independence Along a Three-Node Chain
Consider the chain \(X_1-X_2-X_3\), with binary variables and the factorization \(p(x_1,x_2,x_3)=p(x_2)p(x_1\mid x_2)p(x_3\mid x_2)\). Suppose \(P(X_2=1)=0.4\), \(P(X_1=1\mid X_2=0)=0.2\), \(P(X_1=1\mid X_2=1)=0.8\), \(P(X_3=1\mid X_2=0)=0.1\), and \(P(X_3=1\mid X_2=1)=0.7\). The remaining conditional probabilities are the complements, so these are valid binary distributions.
Given \(X_2=1\), the factorization gives \(P(X_1=1,X_3=1\mid X_2=1)=0.8(0.7)=0.56\). This equals \(P(X_1=1\mid X_2=1)P(X_3=1\mid X_2=1)\), so the joint event has the product probability. More generally, the conditional joint distribution factors into the two conditional distributions, which establishes \(X_1\perp X_3\mid X_2\).
For a check on the arithmetic in the joint distribution, \(P(X_1=1,X_2=1,X_3=1)=0.4(0.8)(0.7)=0.224\). For \(X_2=0\), the corresponding joint probability is \(0.6(0.2)(0.1)=0.012\). The node \(X_2\) separates nodes 1 and 3, and the numerical factorization agrees with the global Markov statement.
Why Factorization Implies the Global Markov Property
The essential observation is that after removing a separator, no clique can contain nodes from two different connected components of the remaining graph. If it did, two of its nodes would be joined by an edge, placing them in the same component. This lets the factors split into groups that cannot couple variables in components separated by \(S\).
Let \(C_1,\ldots,C_m\) be the connected components of the graph induced by \(V\setminus S\). No edge joins distinct components.
Every clique lies within \(S\) and at most one component \(C_j\). Collect factors involving nodes of \(C_j\) into \(g_j(x_{C_j},x_S)\); factors involving only \(S\) form \(h(x_S)\).
The factors depending only on \(x_S\) cancel in the conditional distribution. The remaining unnormalized conditional density is \(\prod_{j=1}^m g_j(x_{C_j},x_S)\).
Summing or integrating over \(x_{V\setminus S}\) factors into a product of component-specific normalizers. Thus, conditional on \(x_S\), the component variables have a product distribution.
In equation form, when the component normalizers are finite and positive,
For continuous variables, the sum is replaced by an integral. A set \(A\) and a set \(B\) separated by \(S\) cannot have nodes in the same component \(C_j\). The componentwise product distribution therefore implies \(X_A\perp X_B\mid X_S\), including when other components contain variables in neither set. This proves the factorization-to-global-Markov implication, subject to the stated existence and conditioning conditions. Strict positivity is not needed for this direction.
Worked Example: Separating Sets in a Four-Node Path
Suppose four variables have graph \(1-2-3-4\), with a distribution factorizing as \(p(x_V)\propto\psi_{12}(x_1,x_2)\psi_{23}(x_2,x_3)\psi_{34}(x_3,x_4)\). Assume the potentials are nonnegative and the product has a finite, positive normalizing constant.
Removing node 2 leaves component \(\{1\}\) and component \(\{3,4\}\). Thus node 2 separates node 1 from node 4, and the factorization result gives \(X_1\perp X_4\mid X_2\). Notice that node 3 is not conditioned on in this global statement.
The local property for node 1 instead states \(X_1\perp X_{\{3,4\}}\mid X_2\), since \(N(1)=\{2\}\). The pairwise property for nodes 1 and 4 states \(X_1\perp X_4\mid X_{\{2,3\}}\). Both statements use different sets of variables on the right side of the conditioning bar. Treating these three statements as interchangeable without checking their conditioning sets is an error.
When the Markov Properties Are Equivalent
For a finite set of discrete variables with a strictly positive joint mass function (every configuration has positive probability), the pairwise, local, and global Markov properties are equivalent. Under the same positivity condition, the Hammersley–Clifford theorem states that the global Markov property is equivalent to factorization over the graph's cliques. Related results hold for continuous distributions under suitable regularity conditions, but positivity and existence conditions must be assessed for the model at hand.
The equivalence is useful because different properties are easier to check in different settings. The pairwise property gives one statement per missing edge, the local property focuses on each node and its neighborhood, and the global property expresses the full implication of graph separation. A clique factorization gives a direct route from the graph to global conditional independence.
Strict positivity matters particularly for converse implications. When a distribution has structural zeros—configurations assigned probability zero—some conditional distributions may be undefined, and the equivalence between Markov properties or between a Markov property and a clique factorization need not hold in general. A factorization with valid nonnegative potentials still implies the separation result proved above, but one should not infer the converse merely from that implication.
Worked Example: Checking the Three Properties on a Path
Return to the positive four-node path in the previous example. The nonadjacent pairs are \((1,3)\), \((1,4)\), and \((2,4)\). The pairwise property requires, among other statements, \(X_1\perp X_3\mid X_{\{2,4\}}\), \(X_1\perp X_4\mid X_{\{2,3\}}\), and \(X_2\perp X_4\mid X_{\{1,3\}}\). These are pairwise statements conditioned on all nodes other than the pair being tested.
The local property for node 2, whose neighbors are 1 and 3, is \(X_2\perp X_4\mid X_{\{1,3\}}\). For node 1 it is \(X_1\perp X_{\{3,4\}}\mid X_2\). Each local statement follows from the global property because the neighborhood separates the node from its nonneighbors. In this strictly positive, clique-factorized model, the equivalence result also guarantees the pairwise statements. The example shows why their conditioning sets can differ even when the properties are equivalent as collections of statements.
Diagnostics, Alternatives, and Limitations
A graph encodes conditional independence restrictions, not a guarantee that those restrictions fit the data. In an estimated multi-omics network, inspect whether the assumed graph and factorization are plausible before interpreting a missing edge as conditional independence. Practical warning signs include unstable edge patterns across reasonable modeling choices, substantial residual dependence where the model imposes independence, and sparse or empty conditioning strata that make conditional comparisons unreliable. In finite data, these are evidence of potential model or estimation problems, not definitive tests of the population Markov property.
A common mistake is to say, “There is no path between \(A\) and \(B\), so \(X_A\) and \(X_B\) are independent.” The correction is to identify a separator \(S\), then state the conditional conclusion \(X_A\perp X_B\mid X_S\). A second mistake is to condition on an arbitrary subset of neighboring variables and invoke the pairwise property; pairwise Markov independence conditions on every other node. A third is to treat an absent edge as evidence of no marginal association. The graph's missing-edge statements concern conditional, not necessarily marginal, independence.
Factorization and separation are not the only ways to represent dependence. A directed acyclic graph uses parent sets and directed factorization, so its separation criterion is not the undirected path rule used here. A general joint distribution without graphical restrictions need not have any useful conditional independence pattern. These alternatives answer related modeling questions but are not interchangeable: choose the graph type and its separation rule to match the assumed structure.
Finally, a graph's conditional independences are properties of a specified distribution. In applications, estimated graphs are affected by measurement error, unmeasured variables, sample size, and model misspecification. Even a well-supported conditional independence is not, by itself, evidence that one molecular feature causes another. These limitations are important when translating a statistical network into biological claims.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- In a graph, \(S\) separates \(A\) from \(B\). What conditional independence follows from a valid clique factorization?
- For a four-node path \(1-2-3-4\), what does the global Markov property imply when \(A=\{1\}\), \(B=\{4\}\), and \(S=\{2\}\)?
- For nonadjacent nodes \(u,v\), which variables are conditioned on in the pairwise Markov property?
- Does strict positivity need to be assumed to derive global Markov independence from a valid factorization? What role does positivity play in the equivalence result?
- Why does a missing edge not generally imply marginal independence?
Solutions. (1) \(X_A\perp X_B\mid X_S\), because factors split across the components remaining after \(S\) is removed. (2) \(X_1\perp X_4\mid X_2\); node 2 blocks the only path. (3) All remaining variables, \(X_{V\setminus\{u,v\}}\), are conditioned on. (4) No: under the factorization and appropriate normalizing and conditioning conditions, the separation implication holds without strict positivity. Positivity is used in the standard equivalence between the Markov properties and clique factorization. (5) A missing edge represents a conditional-independence restriction in the graph, not a claim about the marginal distribution; dependence can remain when other variables are not conditioned on.