Suppose that we want to estimate the value $p$ of a binomial experiment with only knowing that a heads has occurred without any information about the number of trials. How can we write likelihood function for this observation? How can we maximize this function?
Intuitively it seems that the likelihood for all possible $p$ should be the same but it's not possible to write down the exact equation for likelihood.
anyway the likelihood is the following
$L(p) \propto p^a(1-p)^{n-a}$
with $a$ known (number of Heads observed) and $n \geq a$
If you want to estimate $p$ in a bayesian way, supposing not to have any prior information about the value of $p$ and supposing you are wondering to find the Miminum MSE estimator,
$\hat{p}=\frac{a+1}{n+2}$
to get a number for your estimation, of course you must know the number of trials