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The following question speaks about binomial distribution with known probability $p$, but unknown number of trials $n$.

Binomial confidence interval over the number of trials

Trying to think of how a Bayesian interval would be constructed for such a case I passed first at the stage to thinking about the Jeffreys prior. However, for a discrete parameter space this is not defined because the derivative does not exist.

Are there approaches to find a prior according to the same ideas? Of course, the property of invariance of the distribution under coordinate transformations is obsolete since probability mass functions don't transform like probability density functions. Is that the only property/motivation for Jeffreys prior, or are there other properties that can be applied to probability mass functions as well?

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  • $\begingroup$ Possibly we could try some continuous analogue of a binomial distribution? That might lead to different options, but I am sure there are some more simple/desired cases among them. $\endgroup$
    – Sextus Empiricus
    Commented yesterday
  • $\begingroup$ The binomial distribution has a relationship between variance and expectation like $Var[X] = (1-p) E[X]$. Are there known distributions that match that? Or otherwise we could maybe model it as a normal distribution $X \sim N(np,np(1-p))$ and for that case a Jeffreys prior should exist and might be computed. $\endgroup$
    – Sextus Empiricus
    Commented yesterday
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    $\begingroup$ Relevant: stats.stackexchange.com/questions/500781/…, stats.stackexchange.com/questions/275600/…, stats.stackexchange.com/questions/113851/…, stats.stackexchange.com/questions/588863/…, stats.stackexchange.com/questions/502124/… and search for more ... $\endgroup$
    – kjetil b halvorsen ♦
    Commented yesterday
  • $\begingroup$ Possibly another approach could be to have a multistage model with a continuous parameter that models a discrete distribution from which the parameter $n$ is obtained, and we let the variance of that distribution approach zero. Should different of such multistage models lead to the same result (same prior). $\endgroup$
    – Sextus Empiricus
    Commented yesterday
  • $\begingroup$ Number of trials to get a known count of successes (the observed data) would be negative binomial. $\endgroup$
    – Glen_b
    Commented 19 hours ago

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