The prior distribution for $\theta$ is Gamma($\alpha,\beta$) and $(y|\theta)$ is exponentially with rate $\theta$.
- For the first case A, we observe that $y\geq 10$, but we don't observe the exact number.
- For the second case B, we observe that $y=10$.
I know the posterior distributions for these two case.
- For case A, posterior distribution is Gamma($\alpha,\beta+10$)
- For case B, posterior distribution is Gamma($\alpha+1,\beta+10$)
According to the formula for the Gamma distribution we know that the posterior variance of A is smaller.
However, is there an intuitive or simple reason for the case A will have a smaller posterior variance without calculating their variance by formula?