I want to find the Bayesian solution to a game where one has to infer the probability of a latent variable being within a certain range conditional on the outcomes of a Bernoulli experiment. I first provide a story intended to frame the problem and then try to express it (somewhat) mathematically. Please feel free to suggest how I can improve my post - this is my first one here.
An art dealer is trying to sell a painting which is one-of-a-kind, but does not know how much to charge for it. He would like to sell it for the highest price anyone would want to pay, but does not know people's willingness to pay for the painting. He sets a price and guesses that if there is someone in the market who wants to pay that price then the painting should sell within a week with a certain probability, but it might also not sell within a week even if there is someone who would want to pay that price. If no-one wants to pay that price he will definitely not sell the painting in the coming week.
After a week, he observes if the painting was sold or not. If the painting is sold the game is up. If it is not sold, he updates his subjective probability of there being someone who would want to pay the price he set for the painting. He compares this to the expected values of a selection of lower price tags before deciding whether to lower the price to one of those or wait another week. This process is repeated each week.
My attempt to phrase the story mathematically
A person is trying to infer the probability of the true value of a latent variable with the range $$(0, ∞)$$ being higher than each of a selection of values (e.g. 0, 10, 20, 30). The information they acquire is the outcome of a Bernoulli experiment which occurs on every trial. The experiment has two outcomes: 1 and 0. If outcome 1 is realized the game stops, and no more experiments occur. Updating thus only happens when outcome 0 is realized.
Before each trial, the person sets a criterion equal to one of the selected values (e.g. 30) If the true value of the latent variable is higher than the criterion, they observe outcome 1 with some probability $$0 < p_1 < 1$$ and outcome 0 with the probability $$1 - p_1$$ If the true value is lower than the criterion they will never observe outcome 1: $$(p_1|value < criterion = 0)$$.
I want to calculate the Bayesian posterior of each selected value after each trial. However, I don't know how to do this in this case. What makes it difficult for me is mainly two things:
1. How are the values which were lower than the criterion on the last trial affected by the outcome of the experiment? The experiment provides no information about the probability of the true value being smaller than the criterion but the probability of it being greater than the criterion decreases with every failure. The probability density has to go somewhere, so how should it be distributed in the range between the criterion and 0?
2. Are there any other restrictions on the priors in excess of transitivity? 30 must have a lower prior than 20, 20 than 10 and 10 than 0. This is because any value which is greater than 30 necessarily also is greater than 20, and so on. The probability density "east" of 20 must thus include both the density from 20 to 30 and the density from 30 to infinity.
Due to these two points, I don't see how I can apply the regular Bayes formula. Basically, how should I deal with the fact the outcome of the experiment is informative about only a section of the CDF of the true value?