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A Logical Approach to Context-Specific Independence

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 9803)

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.

Keywords

  • Context-specific Independence (CSI)
  • Semi-graphoid Axioms
  • Bayesian Network Framework
  • Conditional Independence (CI)
  • Directed Acyclic Graph (DAG)

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Acknowledgements

The third author was supported by grant 292767 of the Academy of Finland. The fourth author was supported by FDPSS via grant 141318 of the Academy of Finland.

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Correspondence to Juha Kontinen .

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Appendix

Appendix

Fig. 5.
figure 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.

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Corander, J., Hyttinen, A., Kontinen, J., Pensar, J., Väänänen, J. (2016). A Logical Approach to Context-Specific Independence. In: Väänänen, J., Hirvonen, Å., de Queiroz, R. (eds) Logic, Language, Information, and Computation. WoLLIC 2016. Lecture Notes in Computer Science(), vol 9803. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-52921-8_11

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