how to find a posterior with min of observation and constant

Question

There is a variable Y with exponential distribution with parameter theta. The prior distribution is gamma distribution with parameters alpha and beta. If we don't have an actual observation of Y, but only observations that are equal to the minimum between the value of Y and some constant a. How in this case can we find the posterior distribution?

Answer 1

The tail probability of $Z$ is $P(Z > s) = P(Y > a) = e^{-\lambda a}$ if $s < a$ and $= P(Y > s) = e^{-\lambda s}$ if $s >=a$.

The pdf of $Z$ is $f(z) = \lambda e^{-\lambda z} / e^{-\lambda a} = \lambda e^{-\lambda (z-a)}$ for $z \ge a$, and $0$ otherwise.

The posterior $p(\lambda) \propto \lambda^{\alpha - 1} e^{-\beta \lambda} * \prod_{z_i > a} \lambda e^{-\lambda(z_i - a)} = \lambda^\alpha e^{-\lambda \big (\beta + \sum_{z_i \ge a} (z_i -a) \big) } $, which implies the posterior follows a $Gamma\big(\alpha + 1, \beta + \sum_{z_i \ge a} (z_i -a)\big)$ distribution.