# Truncated Gamma Distribution

The Gamma distribution is the conjugate prior of Poisson distribution. What about the Truncated Gamma distribution? Is it still the conjugate prior of Poisson distribution?

• Thanks for your answer. I thought it as still the conjugate prior but my lecturer told me it wouldn't be the conjugate prior. I am really confused. – Suki Hao Mar 8 at 14:53

A generic if rarely mentioned result about conjugate families is that they are defined in terms of an arbitrary dominating measure $$\lambda$$. This means that their density wrt this dominating measure is provided by the corresponding exponential family shape $$\exp\{A(\theta)\cdot S_0 -\lambda \psi(\theta)\}$$ but that the dominating measure $$\lambda$$ may include an arbitrary function $$C(\theta)$$ of $$\theta$$ wrt a standard dominating measure like the Lebesgue measure (or any other), in particular the indicator of a particular subset of the parameter space.
In the case of the Poisson example, with likelihood$$\theta^S\exp\{-n\theta\}$$the prior with density against the Lebesgue measure$$\pi(\theta)\propto \theta^{S_0}\exp\{-\lambda\theta\}\mathbb{I}_{(a,b)}(\theta)$$is associated with the posterior$$\pi(\theta)\propto \theta^{S+S_0}\exp\{-(n+\lambda)\theta\}\mathbb{I}_{(a,b)}(\theta)$$which is indeed of the same shape, $$S_0$$ being replaced with $$S+S_0$$ and $$\lambda$$ with $$n+\lambda$$.