I have been thinking about the following problem that is often given in introductory bayesian text books. Suppose we have a medical procedure that tests for the presence of some disease. We want to find the posterior probability of $$P(D+|T+) = \frac{P(T+|D+)P(D+)}{P(T+)}$$ where $D+=$ positive for disease and $T+=$ positive for test.

How can we incorporate another variable into our model, say age? $$P(D+|T+,A=65) = \frac{P(T+|D+,A=65)P(D+|A=65)}{P(T+| A=65)}$$

Furthermore, how could we learn this posterior from a bunch of data of the counts of $D+,T+,$ and $A$?


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