I have a larger problem but have presented what I believe is a minimal example. Imagine that you are trying to determine the true probability of a potentially-biased coin landing on heads, and want to take a bayesian perspective. Our prior is hence that the probability of heads is beta(1,1) distributed. Say that we flip the coin once and we get a heads. Our posterior is now beta(2,1).
Then we flip once more, but the coin lands crooked against an object on the table. It looks like it would have landed tails, but say that we are only 70% sure that it would have landed tails (so 30% sure that it would have been heads).
Obviously the 'best' solution is to ignore and retest, but if these coin flips are limited/expensive that might not be ideal. So is there anyway to include this result even with the uncertainty? Possibilities I've considered are
- Ignore result, p ~ beta(2,1)
- Include and pretend we are certain, p ~ beta(2,2)
- Include with uncertainty, p ~ beta(2, 1.7)
- Include with uncertainty for both, p ~ beta(2.3 1.7)
Option 4 seems reasonable, but I'm worried this is a statistical golem and I'm missing something obvious. I'm trying to stay in the bayesian setting so the answer here Distribution of partially observable binominal parameter isn't sufficient. Cheers!