A Logical Approach to Context-Specific Independence

Abstract

Bayesian networks constitute a qualitative representation for conditional independence (CI) properties of a probability distribution. It is known that every CI statement implied by the topology of a Bayesian network G is witnessed over G under a graph-theoretic criterion called d-separation. Alternatively, all such implied CI statements have been shown to be derivable using the so-called semi-graphoid axioms. In this article we consider Labeled Directed Acyclic Graphs (LDAG) the purpose of which is to graphically model situations exhibiting context-specific independence (CSI). We define an analogue of dependence logic suitable to express context-specific independence and study its basic properties. We also consider the problem of finding inference rules for deriving non-local CSI and CI statements that logically follow from the structure of a LDAG but are not explicitly encoded by it.

Appendix

Fig. 5.

LDAG for the proof of Theorem 3.

Proof of Theorem 3 . We apply the proof suggested by Koller et al. (p. 196) to LDAG structures [14]. We will reduce a 3-SAT problem instance into deciding whether a CI statement is implied by an LDAG structure.

Define the corresponding LDAG to the 3-SAT instance as follows (see Fig. 5). Let binary nodes $Z_1,\cdots ,Z_l$ correspond to variables in the 3-SAT instance. Let $Y_0,Y_1,\cdots ,Y_k$ denote additional binary nodes of which $Y_1,\cdots ,Y_k$ represent the clauses of the 3-SAT instance. Let the parents of node $Y_i$ ($i \ge 1$) be the node $Y_{i-1}$, and the Z-nodes appearing in the clause i, let us call them $Z_a,Z_b,Z_c$. The labels on the edge $Y_{i-1}\rightarrow Y_i$ consist of assignments to the nodes $Z_a,Z_b,Z_c$. Let the label $\mathcal {L}_i$ on the arc $Y_{i-1} \rightarrow Y_i$ be exactly the set of assignments to $Z_a,Z_b,Z_c$ that do not satisfy the ith clause of the 3-SAT problem.

Consider different contexts $e_z$ over variables $Z_1,\cdots ,Z_l$. If $e_z$ does not satisfy the 3-SAT instance, there is a clause i which is unsatisfied, and thus the corresponding edge $Y_{i-1}\rightarrow Y_i$ does not appear in $G(e_z)$. Thus, $Y_0$ and $Y_k$ are d-separated in $G(e_z)$ and according to Theorem 2: $Y_0 \perp Y_k | Z_1,\ldots ,Z_l=e_z$.

If $e_z$ satisfies the 3-SAT instance, all clauses are satisfied and thus all edges $Y_{i-1}\rightarrow Y_i$ appear in $G(e_z)$. Thus, $Y_0$ and $Y_k$ are not d-separated in $G(e_z)$. We can define a parameterization for the LDAG under which there is a dependence. Let $Y_0,Z_1,\ldots ,Z_l$ be distributed uniformly. Let $Y_i=Y_{i-1}$ if $Z_a,Z_b,Z_c$ satisfy the clause i and 0 otherwise. Now under a satisfying context $e_z$: $Y_k = Y_{k-1} =\cdots = Y_0$ hence $Y_0 \not \perp Y_k | Z_1,\ldots ,Z_k = e_z$. Thus, $Y_0 \perp Y_k | Z_1,\ldots ,Z_k = e_z$ cannot follow from the LDAG structure.

If the 3-SAT problem is satisfiable there is a context $e_z$ such that $Y_0 \perp Y_k | Z_1,\ldots ,Z_k = e_z$ does not follow from the LDAG structure, hence $Y_0 \perp Y_k | Z_1,\ldots ,Z_k$ does not follow from the structure either. If the 3-SAT problem is unsatisfiable we have that for all contexts $e_z$: $Y_0 \perp Y_k | Z_1,\ldots ,Z_k = e_z$, from which it directly follows that $Y_0 \perp Y_k | Z_1,\ldots ,Z_k$. Thus, the defined LDAG structure implies independence $Y_0 \perp Y_k | Z_1,\ldots ,Z_k$ if and only if the 3-SAT problem is unsatisfiable. If we could decide whether an independence is implied by an LDAG in polynomial time, we could also solve 3-SAT in polynomial time.