# Does it make sense to infer a rate (as a probability distribution or upper limits) for a Poisson process if there are "no events"

I have an inhomogeneous Poisson process with a rate $$\lambda (\mathbf{t})$$ defined on some parameters $$\mathbf{t}$$. I am trying to infer $$\lambda (\mathbf{t})$$ from some data, which are events (really $$\lambda (\mathbf{t})$$ is a function of the parameter I am interested in but it's the same idea). Would it make sense to estimate $$\lambda (\mathbf{t})$$ in an area of the parameter space where there are no events? As poorly constraining as that may be.

For a simpler case, if we take a normal Poisson process (removing $$\mathbf{t}$$), here's what I would think:

1. The likelihood distibution for k events is $$p(k|\lambda) = e^{-\lambda} \frac{\lambda^k}{k!}$$

2. The "uninformative prior" on $$\lambda$$, if we're taking a Bayesian perspective, is $$p(\lambda) = \sqrt \lambda$$ (correct?)

3. If we have no events and $$k=0$$, the likelihood is $$e^{-\lambda}$$.

From a frequentist perspective, as far as I can see, estimating a rate doesn't make much sense. Or rather, rate $$\lambda = 0$$ has the highest likelihood, and the higher the rate, the less likely it is, but that's all we can say.

From a Bayesian perspective, the posterior on the rate is $$p(\lambda) = e^{-\lambda} \sqrt \lambda$$. This would actually have a maximum, and I'm wondering if it makes sense then to make inferences on $$\lambda$$. Quantities like upper limits (i.e $$\lambda$$ below which 95% of the pdf lies etc...) could be computed, the posterior is proper, and so on...

I've seen papers where authors decide to limit themselves to cases where there are non-zero events, and I was wondering why that is.