# Bayes theorem in a clinical setting

Say I have a test with the following characteristics:

$P(B|A)$ = positive test in disease population = 0.8

$P(A)$ = incidence of disease = $\frac{1}{5,000}$

$P(B)$ = positive test in general population = 0.3

Thus I end up with the following probability of disease given a positive test: $P (A|B)=\frac{P(B|A) P(A)}{P(B)} =\frac{0.8*\frac{1}{5,000}}{0.3}$ = 0.0005

Now, this only reflects the probability of disease given a positive test, neglecting any other clinical signs.

How can I update this probability given clinical signs as well? In other words, is it possible to extend Bayes into $$P (A|(B,x1,...x_n)$$ somehow, and how can I provide a measure of the added information gained by the test (expensive)?

Say for instance that x1 is ubiquitous for the diagnosis, but that x2 occurs in 75% of patients with the disease. On the other hand, x1 and x2 have incidences of 0.1 and 0.01 in the general population.

• – Tim Nov 23 '16 at 21:36
• Are you interested in $P(A|x_1,\ldots,x_n)$ or $P(A|B,x_1,\ldots,x_n)$? – Thoth Nov 24 '16 at 2:02
• See edit. I'm interested in both, but my main interest is as per the edit. – Misha Nov 24 '16 at 11:20