Bayesian updates with cumulative gamma distribution?

Question

Say I have this situation with an exponential distribution and it's gamma conjugates:

$y\mid\lambda \sim exp(\lambda)$

$\lambda \sim gamma(\theta,\beta)$

$\lambda \mid y,\theta,\beta \sim gamma(\theta + 1, \beta + y)$

A trial shows that $y>x$, (where $x$ is just a constant) and we'd like to update $\lambda$. Am I correct to think that the posterior density would be given by the following equation?

$p(\lambda\mid \theta,\beta,y>x)= \int_{y=x}^\infty 1- gamcdf(x|\theta+1,\beta+y)dy$

Is there are better way to do this?

Answer 1

When $\lambda\sim\mathcal G(\theta,\beta)$ and it is known that $Y>c$ for $Y\sim\mathcal E(\lambda)$, this amouns to observing a Bernoulli random variable$$Z\sim\mathcal{Be}(\mathbb P_\lambda(Y>c))$$to be equal to one. The posterior is therefore $$\pi(\lambda) \propto \lambda^{\theta-1}e^{-\beta\lambda}\mathbb P_\lambda(Y>c)=\lambda^{\theta-1}e^{-\beta\lambda}e^{-\lambda c}=\lambda^{\theta-1}e^{-(\beta+c)\lambda}$$