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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?

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