"Conditional Inter-Causally Independent" Node Distributions, a Property of "Noisy-OR" Models

This paper examines the interdependence generated between two parent nodes with a common instantiated child node, such as two hypotheses sharing common evidence. The relation so generated has been termed "inter­ causal." It is shown by construction that inter-causal independence is possible for bi­ nary distributions at one state of evidence. For such "CICI" distributions, the two mea­ sures of inter-causal effect, "multiplicative synergy" and "additive synergy" are equal. The well known "noisy-or" model is an ex­ ample of such a distribution. This introduces novel semantics for the noisy-or, as a model of the degree of conflict among competing hy­ potheses of a common observation. In a general Bayesian network, the relation between a pair of nodes can be predictive, meaning we are inter­ ested in the effect of a node upon its successors, or, oppositely, diagnostic, where we infer the state of a node from knowledge of its successors. \\1e can define yet a third relation between nodes that are neither suc­ cessors of each other, but share a common successor. Such a relation has been termed inter-causal. (Henrion and Druzel 1990, p.10] For example, in the simplest di­ agram with this property, nodes A and B in Figure one are inter-causally related to each other by their com­ mon evidence at node e. This relation is a property of the clique formed by "marrying the parents" of e, not by the individual effects of the arcs into e. In this paper I derive the quantitative inter-causal properties due to evidence nodes constructed from the noisy-or" model. The interest in inter-causal relations occurs in the pro­ cess of abduction, that is, reasoning from evidence back to the hypotheses that explain the evidence. This arises in problems of interpretation, where more than one hypothesis may be suggested by a piece of evi­ dence. (Goldman and Charniak 1990] Having multiple explanations denotes the ambiguity due to not having enough information to entirely resolve which hypothe­ sis offers the true explanation. This paper shows how to construct an evidence node that expresses this am­ biguity by the degree of conflict between hypotheses. We apply this elsewhere (Agosta 1991] as a compo­ nent in building a "recognition network" where rele­ vant hypotheses are created "on the fly" as possible interpretations of the evidence. The implicit relation between A and B due to shared evidence has been extensively explored as the prop­ erty of one hypothesis to "explain away" another. These are cases where, given evidence and the asser­ tion of one hypothesis, the other hypothesis can be disqualified as a cause of the evidence. This paper ex­ plores how this dependency induced between hypothe­ ses changes with the evidence. Interestingly, with bi­ nary variables, the induced dependency may vary, and as shown by the noisy-or, disappear for certain states of evidence. Figure 1: The relationship between hypotheses is de­ termined by their common evidence 1 EVIDENCE NODES THAT ARE