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The binomial distribution gives me a distribution for the number of successes in several Bernoulli trials, k, given parameters N the total number of trials and q, the success probability for one trial. This probability can be thought of as P(k|N,q).

Assuming that q is known I would be interested in P(N|k,q), which should follows from Bayes rule, and I am not sure which prior should be used for N. I also wonder if a classical distribution exists in the literature that is what I am looking for.

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From a foundational perspective, there is no "good" or "classical" prior. Whatever the sampling distribution, any prior that does not clash with the observation(s) is acceptable. The corresponding posterior is simply conditional or relative to this prior (and never reflects an overall reality or meta-reality!).

When building one's prior, consider the likelihood function $$\ell(N|k,q)={N \choose k} q^k (1-q)^{N-k}\propto \dfrac{N!}{(N-k)!} (1-q)^{N}\mathbb{I}_{N\ge k}$$ which is not from an exponential family wrt to $N$. Hence does not allow for the so-called conjugate priors which are often used because of their rather simple features, rather than for theoretical reasons. This means that the prior on $N$ has to be constructed from prior information on the possible size of the population, $N$. If this information is found missing, an alternative is to consider the marginal distribution on $k$ $$m(k)=\sum_{N=k}^{\infty} \pi(N) \ell(N|k,q)$$ as the shape of this posterior may help comparing different choices of priors.

If nothing else works, a default (but not the default) prior is the non-informative $$\pi(N)\propto 1/N \mathbb{I}_{\mathbb{N}^+}(N)$$ which stems from the notion that $N$ is a form of scale parameter, $k$ being of order $qN$.

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