Medical diagnosis:

Imagine that a doctor has to deal with a patient who presents with sneezing (call this event S). The underlying disease might be pneumonic plague (P, and dangerous) or a cold (C, and not a major worry). Obviously, the doctor needs to form an opinion about what is likely to be going on. In our terms he needs to estimate both P(P|S) and P(C|S). These are the probabilities of the diseases given the symptoms. The answer is not intuitively obvious. It obviously isn't enough to know the probabilities P(S|C) and P(S|P) which are the probabilities of sneezing if you have the relevant diseases (most doctors would assume P(S|C)=1.0 and P(S|P)=1.0 ; you are pretty well certain to sneeze if you have either of the diseases).

Fortunately, it is general knowledge among doctors that, all other things being equal, the common cold is more common than pneumonic plague. In Scotland you can assume that P(C) = 0.25 (at least a quarter of the patients visiting the doctor have a cold) and P(P) = 10^-6 (about 1 in a million visitors to a doctor's surgery have plague). It might also be reasonable to assume (from experience) that P(S) = 0.35, about 1 in 3 of the patients are sneezing at a given time (this is called the prior on sneezing). Putting all this together, you get

$$P(P\vert S) = \frac{P(S\vert P) P(P)}{P(S)} = \frac{1.0 \times 10^{-6}}{0.30} = 3 \times 10^{-6}$$

and

$$P(C\vert S) = \frac{P(S\vert C) P(C)}{P(S)} = \frac{1.0 \times 0.25}{0.30} = 0.83$$

That is, sneezing patients are much more likely to be cold victims than harbingers of doom. Colds are 250,000 times more likely than plague.

Note the following:

• The dominant factor in the calculation is the assessment of the base rates of the two illnesses (i.e. P(P) and P(S). Small changes in the other factors do not affect the conclusion much, since Understanding the derivation allows us to see which changes matter.
• It would be bad if the doctor had simply learnt in medical school that the P(P|S) is low and P(C|S) is high, without deriving it from more robust prior information. If P(P) changes markedly, as would be the case in an epidemic of plague,the doctor might not realize that this will have a proportionate effect on P(P|S). A colleague who understands Bayes' rule would be much better placed.
• The causal information P(S|C) and P(S|P) is unaffected by the prevalence of either disease. Medical schools can and do provide this sort of reliable knowledge to doctors.

What happens if the common cold is eradicated? Notice that we assumed 5 in 100 patients were sneezing for reasons which were neither colds nor plague.

The details change, since P(S) drops to 0.05. The effect on the estimate of the probability of plague is to increase it sixfold to $1.2 \times 10^{-4}$. Whether this matters much depends on what the consequences of missing a patient with pneumonic plague actually are.