Posterior distribution when the domain of the likelihood depends on the parameter

Question

I am trying to calculate a posterior density given distribution and a prior. And I am a bit confused about how I should act as the domain of the distribution depends on the parameter. I am talking about the distribution with density:

$g(x) = \frac{\alpha p^{a}}{x^{\alpha+1}}\mathbb{1}_{[x\geq p]}$

and prior

$\pi(p)\propto p^{\gamma-1}e^{-\beta p}$, where $\alpha,\beta,\gamma>0$.

I am trying to use the basic approach for $iid$ $x_{i}$ with density $g$. Posterior is likelihood times prior. Then i get:

\begin{align}\pi(p|x_{1},...,x_{n})&\propto f(p|x_{1},...,x_{n}) \pi(p)\\ &\propto \prod_{i=1}^{n} g(x_{i}) p^{\gamma-1}e^{-\beta p}\\ &= \prod_{i=1}^{n} \frac{\alpha p^{a}}{x_{i}^{\alpha+1}}\mathbb{1}_{[x_{i}\geq p]} p^{\gamma-1}e^{-\beta p}\\ &\propto \alpha^{n} p^{\alpha n}\prod_{i=1}^{n}\mathbb{1}_{[x_{i}\geq p]} p^{\gamma-1}e^{-\beta p}\\ &\propto\mathbb{1}_{[\min_{i} x_{i}\geq p]} p^{n\alpha + \gamma -1}e^{-\beta p}\end{align}

Is this prior conjugate to the data model? It is if we ignore the indicator but I don't see how I can easily drop it. Or should I think of the domains in a different way? The prior and the model are conjugate in the support of the posterior? I am a little stuck and don't know how to proceed.

Answer 1

If$$\pi(p)\propto p^{\gamma-1}e^{-\beta p}\mathbb I_{(0,p^+)}(p)\qquad\gamma,\beta,p^+>0\tag{1}$$ then $$\pi(p|x_{1},...,x_{n})\propto p^{n\alpha+\gamma-1}e^{-\beta p}\mathbb I_{(0,p^+\wedge\min\{x_i\})}(p)$$ which is from the same family as (1) with \begin{align} \gamma &\mapsto n\alpha+\gamma\\ \beta &\mapsto \beta\\ p^+ &\mapsto p^+\wedge\min\{x_i\} \end{align} hence conjugate.

Note that the fact that $\beta$ is not actualised after observing the sample means that the conjugate family is over-parameterised. One could then argue that there is one conjugate family for each value of $\beta$, which can also be seen as part of the dominating measure. Or, else, that this part is altogether superfluous and that $\beta=0$ also leads to a conjugate (Beta) family.