#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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