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With the standard Binomial probability distribution we consider n trials each with a probability of success p. This can be somewhat generalized to the Beta-Binomial Distribution which is effectively the same, but this time the probability of success is a random variable with a Beta Distribution. My question is:

Could I in theory use a p which samples from some arbitrary probability distribution (continuous or discrete) so long as the support is on [0,1], or is there something special about the Beta distribution which allows for such a distribution?

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The Beta is just conjugate to the Binomial which makes life easier in a number of ways. If we are in a Bayesian setting with a Binomial likelihood, there is nothing limiting you, theoretically, from applying any prior distribution on $p$ that you like. The problem is that you may end up with an improper posterior, that is, a posterior distribution that does not integrate to one. So if you choose to use such a prior you should check that your posterior is a proper density. But perhaps most importantly, you should be prepared to defend your use of such a prior in that it should make sense for your specific application.

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  • $\begingroup$ My main concern is that my modified Binomial 'distribution' would not integrate to one with my choice of distribution for p. I guess what I was wondering is if there existed any work to prove whether or not any distribution could be used as the seed for the Binomial probability so long as this p had its support on [0,1] and was in fact a probability distribution. $\endgroup$ – mjnichol Jul 10 '14 at 19:39
  • $\begingroup$ mjnichol - perhaps you should clarify your question with this information; it sounds like the heart of your question is really just a fairly straightforward question about mixture distributions. $\endgroup$ – Glen_b -Reinstate Monica Jul 10 '14 at 23:23
  • $\begingroup$ It would help to clarify what you mean by "could be used". And perhaps clarify any specific case that you may be interested in. $\endgroup$ – Zoë Clark Jul 11 '14 at 1:36
  • $\begingroup$ Okay. I have a situation where I run n experiments and so long as a single one of those experiments succeeds I am in the clear. That led me to the Binomial Distribution as a natural way to explore the problem; however, the probability of a success is not constant and is picked from a distribution I crafted from real data at my place of work. Is it okay to simply sample from this distribution in place of a constant p? Will this preserve the integrity of the modified binomial? If anything is unclear please let me know. Also thank you for all the help! $\endgroup$ – mjnichol Jul 11 '14 at 16:27

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