Bayesian modelling of two sequential tests

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In this case it helps to go back to the basics of probability. Let X be the medical disease variable of interest and A, B the outcome of the two medical tests.

P(X|A,B) = P(A,B|X)P(X) = P(B|A,X) P(A|X)P(X)

Where = means proportional to (ignoring the normalization constant P(A,B))

This tells you you need 3 pieces for your analysis:

  1. Prior probability of medical disease X, P(X)
  2. Conditional probability of A test results given all possible X, P(A|X)
  3. Conditional probability of B test results given all possible combinations of A and X, P(B|A,X)

Only then can you compute an answer for a given set of observations.

The analysis would be simpler if you could assume that P(B|A,X) = P(B|X) but that’s not generally the case in medical testing.

If you only have A, you compute P(X|A) = P(A|X)P(X), again ignoring the normalization constant P(A)

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