# #StackBounty: #probability #mathematical-statistics #maximum-likelihood #inference #birthday-paradox Birthday puzzle

### Bounty: 50

I need some help with Bayesian statistics likelihoods. Consider the following question: Given a number of persons. Each person $$p$$ knows $$n(p)$$ other persons – $$p$$‘s neighbourhood $$N(p)$$. Knowing a person $$p’ in N(p)$$ may imply (as an assumption) that $$p$$ knows if $$p’$$ was born on the same day as $$p$$ (neglecting the year). The probability that there are $$k$$ persons $$p’ in N(p)$$ born on the same day as $$p$$ is

$$P_1(X = k | n(p) = N) = binom{N}{k}alpha^k(1- alpha)^{N-k}$$

with $$alpha = 1/365$$.

Now consider a survey where each person $$p$$ was asked how many of the persons $$p$$ knows were born on the same day as $$p$$. Let the result of the survey (the evidence) be a distribution $$P_2(X = a)$$ giving the frequency that the answer was $$a$$.

By which (Bayesian) argument can be told which distribution $$P_3(X = N)$$ of the size of neighbourhoods $$N(p)$$ is the most probable one that would yield $$P_2$$? $$P_3(X = N)$$ gives the probability that a person has a neighbourhood of size $$N$$.

Before that: Is this question even well-posed and correctly worded? Which tacit assumptions have to be made explicit, possibly?

Edit: Maybe it’s easier to ask and answer what the most probable mean neighbourhood size $$overline{N}$$ is, assuming that it is distributed a) normally, b) Poisson, c) scale-free.

Get this bounty!!!

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