[neilkakkar]

> In real life usually we have no idea about priors

Priors are your previous knowledge on the topic.

> One textbook example of Bayes theorem is how doctors overestimate the probability of being positive for a disease. But what are the priors?

In this example, doctors overestimate precisely because they don't take the priors into account.

Doing something risky the day before / feeling funny is extra evidence that is assimilated (or should be) into the likelihood ratio P(D|H) / P(D). This is information the patient should share with the doctor.

Of course, if they don't, then the Bayes estimate is the best guess given all the information the doctor has.

Edit: Your criticism about how we choose priors is fair. The better you are at this, the more accurate your answers become. I mention more about this in the "putting it to practice" section.

[new2628]

I agree with you, but my point is more broadly that in reality we often don't go through the steps "1. estimate probability" -> "2. make a decision based on the probability distribution", because step 1. is so error-prone and intractable, that we typically jump directly to step 2. and try to limit our downside.

Of course you could look back and say, given the fact that I took some decision, what would have been my prior if I had used Bayes theorem, but my point is that we don't actually use it for taking the decision.