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.



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