I have a variable that is Poisson distributed. Let's say I have a number of boxes each with a number of balls inside according to a Poisson distribution, with $\lambda=0.4$, (the average number of balls per box is 0.4):
$X\sim Poiss(\lambda=0.4)$
A random sample of 20 boxes would give us these values:
X=[0, 2, 1, 0, 0, 2, 1, 0, 2, 1, 1, 0, 0, 2, 0, 0, 1, 1, 0, 0]
An average of these values as an estimate of $\lambda$ gives us $0.7$, which is off due to the low sample count.
However, I can't actually see the number of balls in each box, just whether the box is empty or not:
X_1=[0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0]
I can estimate $\lambda$ from this via $\lambda=-ln(Empty Box\%)$ since the probability of a box having $k$ balls is:
$p(k) = \displaystyle\frac{\lambda^ke^{-\lambda}}{k!}$
and therefore:
$Empty Box\%=p(0) = \displaystyle\frac{\lambda^0e^{-\lambda}}{0!}=e^{-\lambda}$
In this case I get an estimate of $\lambda$ as $0.69$, again because of the low sample size for this demonstration. I'm aware that this approach works for sufficiently large sample size, and appropriate range of $\lambda$s.
In my case, I have several thousand boxes, and I am much closer to the true value when running simulations. But I am wondering if there is any Bayesian inference I can do in determining $\lambda$ for an unknown population. If I had the original counts per box instead of the binarized states, I'm aware that I could create a Gamma conjugate prior and perform Bayesian inference with my data to get a credible range of $\lambda$ predictions. Is there any way for me to do this with the data I have?