# Intuitive reason for comparing posterior variance

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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