The QMR-DT network (Shwe et al., 1991) is a two-level or bi-partite graphical model (see Figure 1). The top level of the graph contains nodes for the diseases, and the bottom level contains nodes for the findings.
There are a number of conditional independence assumptions reflected in the bi-partite graphical structure. In particular, the diseases are assumed to be marginally independent. (I.e., they are independent in the absence of findings. Note that diseases are not assumed to be mutually exclusive; a patient can have multiple diseases). Also, given the states of the disease nodes, the findings are assumed to be conditionally independent. (For a discussion regarding the medical validity and the diagnostic consequences of these and other assumptions embedded into the QMR-DT belief network, see Shwe et al., 1991).
Figure 1: The QMR belief network is a two-level graph where the dependencies between the diseases and their associated findings have been modeled via noisy-OR gates.
To state more precisely the probability model implied by the QMR-DT model, we write the joint probability of diseases and findings as:
where d and f are binary (1/0) vectors referring to presence/absence states of the diseases and the positive/negative states or outcomes of the findings, respectively. The conditional probabilities are represented by the ``noisy-OR model'' (Pearl, 1988):
where is the set of diseases that are parents of the finding in the QMR graph, is the probability that the disease j, if present, could alone cause the finding to have a positive outcome, and is the ``leak'' probability, i.e., the probability that the finding is caused by means other than the diseases included in the QMR model. In the final line, we reparameterize the noisy-OR probability model using an exponentiated notation. In this notation, the model parameters are given by .