where P(f+[d,ξ) is a product of upper-bound transformed conditional probabilities and exact (untransformed) conditionals. Analogously we can compute a lower bound P(f+, djlq)
by applying the lower bound transformations:

$$P(f^{+},d_{j})=\begin{array}{c}P(f^{+}|d)P(d)\geq\\ \end{array}\begin{array}{c}P(f^{+}|d,\,q)P(d)\equiv P(f^{+},d_{j}|q)\end{array}\tag{32}$$

Combining these bounds we can obtain interval bounds on the posterior marginal probabilities for the diseases (cf. Draper & Hanks 1994):

$$\frac{P(f^{+},\,d_{j}|q)}{P(f^{+},\,d_{j}|\xi)+P(f^{+},\,d_{j}|q)}\leq\,P(d_{j}|f^{+})\leq\frac{P(f^{+},\,d_{j}|\xi)}{P(f^{+},\,d_{j}|\xi)+P(f^{+},\,d_{j}|q)},\tag{33}$$

w here d is the binary complement of d i .

## 5. Experimental Evaluation

The diagnostic cases that we used in evaluating the performance of the variational techniques were cases abstracted from clinocopathologic conference ("CPC" ) cases. These cases generally involve multiple diseases and are considered to be clinically difficult cases. They are the cases in which Middleton et al. (1990) did not find their importance sampling method to work satisfactorily.

Our evaluation of the variational methodology consists of three parts. In the first part we exploit the fact that for a subset of the CPC cases (4 of the 48 cases) there are a sufficiently small number of positive findings that we can calculate exact values of the postenor marginals using the Quickscore algorithm. That is, for these four cases we were able to obtain a "gold standard" for comparison. We provide an assessment of the accuracy and efficiency of variational methods on those four CPC cases. We present variational upper and lower bounds on the likelihood as well as scatterplots that compare variational approximations of the postenor marginals to the exact values. We also present compansons with the likelihood-weighted sampler of Shwe and Cooper (1991).

In the second section we present results for the remaining, intractable CPC cases. We use lengthy runs of the Shwe and Cooper sampling algorithm to provide asurrogate for the gold standard in these cases.

Finally, in the third section we consider the problem of obtaining interval bounds on the posterior marginals.

## 5.1 Comparison To Exact Marginals

Four of the CPC cases have 20 or fewer positive findings (see Table 1), and for these cases it is possible to calculate the exact values of the likelihood and the posterior marginals in a reasonable amount of time. We used Heckerman's "Quickscore" algorithm (Heckerman 1989)  an algorithm tailored to the QMR-DT architecture  to perform these exact clculations.

Figure 3shows the log-likelihood for the four tractable CPC cases. The figure also shows the variational lower and upper bounds. We calculated the variational bounds twice, with differing numbers of positive findings treated exactly in the two cases ("treated exactly"