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How do we go about calculating a posterior with a prior given N data points of the form greater than X or less than Y, when those data points are noisy (that is, greater than X can be incorrect)?

Example:

We start with the posterior normal distribution of variable X with mean 1 and variance 1. We do series of tests with the following results:

test x=1: less than X
test x=2: less than X
test x=5: greater than X
test x=6: less than X
test x=4: greater than X
test x=10: greater than X
test x=12: greater than X

Given such results, I'd think that that mean of X variable is not 1 - probably something around 5. But how can I derive this mean and variance?

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    $\begingroup$ The probabilistic bisection algorithm is closely related to your question. See e.g. Waeber, R. (2013). Probabilistic bisection search for stochastic root-finding (PhD thesis). Cornell University. Retrieved from people.orie.cornell.edu/shane/theses/ThesisRolfWaeber.pdf $\endgroup$ – Kodiologist Aug 13 '17 at 18:48
  • $\begingroup$ @Kodiologist - Yes, that's relevant. But it seems that this method confines the variable to some pre-defined range ([0,1] in the example can be scaled - but it is still some arbitrary scale). $\endgroup$ – Rogach Aug 13 '17 at 20:50
  • $\begingroup$ @Kodiologist - and not only it is scaled, the probability function appears to be discrete, while I am interested in obtaining real-valued unbounded estimate. $\endgroup$ – Rogach Aug 13 '17 at 20:59
  • $\begingroup$ The posterior distribution p(X|data) is proportion to the prior p(X) times the likelihood p(data|X). You must specify the likelihood. And if the likelihood involves unknowns other than X, you much specify priors for those unknowns as well. $\endgroup$ – mef Aug 14 '17 at 9:49
  • $\begingroup$ @mef - Prior p(X) can be taken as N~(0,1), and p(data|X) is for a case of one data point simply norm cdf(data_point, X) (can be easily extended to multiple data points). $\endgroup$ – Rogach Aug 14 '17 at 21:18

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