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https://sphweb.bumc.bu.edu/otlt/mph-modules/bs/bs704_probability/bs704_probability6.html

A patient goes to see a doctor. The doctor performs a test with 99 percent reliability--that is, 99 percent of people who are sick test positive and 99 percent of the healthy people test negative. The doctor knows that only 1 percent of the people in the country are sick. Now the question is: if the patient tests positive, what are the chances the patient is sick?

The intuitive answer is 99 percent, but the correct answer is 50 percent....

Here, they consider only the reliability of the test & the prevalence of the disease in the population. Should we also consider the fact whether the patient is symptomatic or not - i.e. if the patient is asymptomatic, will it further reduce the reliability of the test?

Many people routinely test for a few lifestyle diseases once every few years as they get older - this is even if they don't exhibit any symptoms. I was wondering if this counter productive as per Bayesian probability.

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    $\begingroup$ I assume it depends on the disease. Should be possible to study the false positive/negative probabilities on the symptomatic population. $\endgroup$
    – lulu
    Commented Jul 6, 2022 at 23:41

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The point of Bayesian analysis is to take advantage of prior information to reach a probability estimate.

If all you know are the general false positive and false negative rates and the incidence in the population then that 50% estimate is about right. If the patient is symptomatic that's more evidence that they have the disease, and increases the probability that a positive test is a true positive. You would need more data to quantify that increase.

As for

Many people routinely test for a few lifestyle diseases once every few years as they get older - this is even if they don't exhibit any symptoms.

whether that is worth doing is a cost benefit analysis. What is the cost of the test, in dollars which might come from the patient or be better spent elsewhere, in anxiety? What are the treatments possible if the condition is found? What are the consequences of leaving it untreated?

Bayesian analysis does not provide answers, just a framework for a part of the problem.

See Applied Probability- Bayes theorem

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  • $\begingroup$ If the patient is symptomatic that's more evidence that they have the disease, and increases the probability that a positive test is a true positive - the opposite would also be true, right? i.e. if the testee is asymptomatic, then the probability of the test being a false positive is more than that calculated earlier (using just the 2 earlier probabilities -i.e. greater than $\frac {P(B|A) P(A)}{P(B)}$) $\endgroup$
    – user93353
    Commented Jul 8, 2022 at 23:24
  • $\begingroup$ Yes. The symptom information pushes the probability in either case. That's what a good doctor will pay attention to. In your simple example the doctor might have ordered the test precisely because the patient had symptoms. $\endgroup$
    – Ethan Bolker
    Commented Jul 9, 2022 at 1:01

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