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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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1 Answer 1

It is an interesting remark that I have not seen explicitly spelled out, however the parameter space for conjugate priors is often chosen in the opposite way, namely the largest possible set that keeps the sampling distribution well defined. See Brown's Fundamentals of Statistical Exponential Families (1986).

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thanks. do you know where it is mentioned explicitly in the text you sent? I can't seem to find it. –  singleton Oct 13 '12 at 15:33
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Not 100% related to the question, but my former PhD advisor told me that one of the favorities ironies of his PhD avisor, Prof. Debrabata Basu, was to point out that a point mass at an arbitrary parameter space point is a conjugate prior. –  Zen Oct 13 '12 at 21:15
    
@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

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