 As an illustration of the role of prior probabilities, consider the HIV test example described in the previous section. What the physician and the patient want to know is the value of the posterior probability, \(P_{\alpha}[h \pmid b\cdot c\cdot e]\), that the patient has HIV, \(h\), given the evidence of the positive test, \(c\cdot e\), and given the error rates of the test, described within \(b\). The value of this posterior probability depends on the likelihood (due to the error rates) of this patient obtaining a true-positive result, \(P[e \pmid h\cdot b\cdot c] = .99\), and of obtaining a false-positive result, \(P[e \pmid {\nsim}h\cdot b\cdot c] = .05\). In addition, the value of the of the posterior probability depends on how plausible it is that the patient has HIV prior to taking the test results into account, \(P_{\alpha}[h \pmid b]\). In the context of medical diagnosis, this prior probability is usually assessed on the basis of the base rate for HIV in the patient’s risk group (i.e., whether the patient is an IV drug user, has unprotected sex with multiple partners, etc.). On a rigorous approach to the logic, such information and its risk-relevance should be explicitly stated within the background information \(b\). To see the importance of this information, consider the following numerical results (which may be calculated using the formula called Bayes’ Theorem, presented in the next section). If the base rate for the patient’s risk group is relatively high, say \(P_{\alpha}[h \pmid b] = .10\), then the positive test result yields a posterior probability value for his having HIV of \(P_{\alpha}[h \pmid b\cdot c\cdot e] = .69\). However, if the patient is in a very low risk group, say \(P_{\alpha}[h \pmid b] = .001\), then a positive test result only raises the posterior probability of his having an HIV infection to \(P_{\alpha}[h \pmid b\cdot c\cdot e] = .02\). This posterior probability is much higher than the prior probability of .001, but should not worry the patient too much. This positive test result may well be due to the comparatively high false-positive rate for the test, rather than to the presence of HIV. This sort of test, with a false-positive rate as large as .05, is best used as a screening test; a positive result warrants conducting a second, more rigorous, less error-prone test.