Poisson-gamma posterior mean expectation

Question

Let's have a gamma prior $\lambda\sim \operatorname{Gamma}(a,b)$ (mean: $\frac{a}{b}$)

With Poisson data $Y\mid \lambda\sim \operatorname{Pois}(N\lambda)$ (mean: $N\lambda$)

The posterior is $\lambda\mid Y\sim \operatorname{Gamma}(a+Y,b+N)$

The posterior mean is $E(\lambda\mid Y)=\frac{Y+a}{N+b}$.

What is its expectation? $E(E(\lambda\mid Y))=\frac{\tfrac{a}{b}+a}{N+b}$? How is this estimator called?

(I would be interested in, when I repeatedly sample from a Poisson, given samples from a gamma distribution, what will be the mean of this samples.)

Thank you very much.

Answer 1

From your first two lines, your prior implies $E[Y] = N\frac{a}{b}$

So with $E(\lambda\mid Y)=\frac{Y+a}{N+b}$, you have $E\left[E(\lambda\mid Y)\right] =E\left[\frac{Y+a}{N+b}\right]=\frac{E\left[Y\right] +a}{N+b}= \frac{N\frac{a}{b}+a}{N+b} = \frac{a}{b}$

which is what you might have thought from your first line and the law of total expectation