Let us end this sequence of Sophie and her cancer screening saga. We applied Bayes’ theorem and showed that the probability of having the disease is low, even with a positive test result. But the purpose was not to downplay the importance of diagnostics tests. In fact, it was not about diagnostics at all!
Screening a random person
Earlier, we have used a prior of 1.5% based on what is generally found in the population (corrected for age). And that was the main reason why the conclusion (the posterior) was so low. It was also considered a random event. Sophie had no reason to suspect a condition; she just went for screening.
Is different from Diagnostics
You can not consider a person in front of a specialist as random. She was there for a reason – maybe discomfort, symptoms, or recommendation from the GP after a positive result from a screening. In other words, the previous prior of 1.5% is not applicable in this case; it becomes higher. Based on the specialist’s database or gutfeel, imagine that the assigned value was 10%. If you substitute 0.1 as the prior in the Bayes’ formula, we get about 50% as the updated probability (for the set of screening devices).
Typically, the diagnostic test would have a better specificity. If the specificity goes up from 90 to 95%, the new posterior becomes close to 70%. It remains high, even if the sensitivity of the equipment dropped from, say, 95% to 90%.
