Is it always possible to infer photon distribution from photon statistics?

Say we are detecting light in the time interval $(t,t+T)$ that is described by the intensity $I(t)$. Let the integrated intensity in this interval be given by:
$$U=int_t^{t+T}I(t’)dt’$$
Since $I(t)$ is a random variable, so is $U$, with an unknown probability distribution $p(U)$. This is the (photon) distribution I’m interested in finding out.

The probability of detecting $n$ counts in the same time interval is given by:
$$P(n,t,t+T)=int_0^{infty}frac{(alpha U)^n}{n!}e^{-alpha U}p(U)dU$$
Here $alpha$ is a parameter dependent on the sensitivity of the detector. We can experimentally measure this $P(n,t,t+T)$.

For example, in the case where it follows a Poisson distribution, we can infer that $p(U)$ is a Dirac delta centred at $bar n/alpha$ where $bar n$ is the measured mean.

So it always possible to infer $p(U)$ from $P(n,t,t+T)$?