# A re-formalization of a conjugate prior?

It is quite easy to prove that if $p(\theta)$ is a conjugate prior to some likelihood then the following:

$$q(\theta') \propto p(\theta)I(\theta \in A)$$

where $A$ is a subset of the parameter space and $I(r)$ is 1 if $r$ is true and 0 otherwise -- is also a conjugate prior.

Basically, if we take a conjugate prior and re-normalize it to a subset of the parameter space, we still get a conjugate prior family.

Is this mentioned somewhere that I can refer to?

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@singleton Sorry if my answer was unclear: I do not think this fact is mentioned in Larry Brown's monograph, I quoted him on the notion of the natural parameter space. In my own book, I also comment on the fact that an infinity of disjoint conjugate families can be defined, since the coefficient of the cumulant function, $\lambda$, gets updated as $\lambda+1$, $\lambda+2$, .... –  Xi'an Oct 16 '12 at 8:36