I came across the following toy example and am lacking a final answer/step to finish the analysis.
Imagine a surgery or medical procedure where we don't know the success-probability. It can be thought of as a Bernoulli processes (hit or miss) with unknown theta.
Now for the first patient the first try of the surgery is a success. I know how I can calculate the posterior when I start with a uniform prior.
But know I want to include a 2nd patient. For him the surgery only works at the tenth trial. So he has 9-misses before the first hit. I also know how I would get a likelihood function from that and with a uniform prior that would be my posterior.
But what if my prior is not uniform, but I have strong belief that for parts of the population the operation has a high success-probability but for another part of the population it only works in one out of ten or even twenty attempts.
I know how to produce a prior distribution (or grid) that represents that success-probability beliefe (and bimodal distribution, as a mix of the population of easy and hard patients).
What I don't know is how to calculate the likelihood function that strengthens that prior. I have two independent observations here (1,1) for instant hit and (9,1). Both have equal weight.
If I update my bimodal prior with the first patient the bump of the pdf to the left (low-success-theta) would decrease and the posterior would increase in the middle and to the right. If I then use that posterior as a basis for the second patient the right-hand side of the distribution (high success) would decrease and the middle again would grow.
So I was wondering if I actually need a bimodal likelihood function to update my belief.
Am I making a mistake in my thinking here or do I miss some Bayesian fundamentals?