I am reading a paper where they talk about keeping a prior explicit as opposed to an implicit prior. To be honest, I have never came across the terms explicit/implicit in context of priors and I was wondering if these are technical terms associated with prior distributions. Does an explicit prior simply means when the prior distribution is specified? How does one get an implicit prior from the problem specification?
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4$\begingroup$ I have never heard that too and it's hard to guess without the context. However, when you choose a prior on $\theta$, then you implicitly assign a prior on $\theta^2$ (for example). Perhaps this is the what the author means ? $\endgroup$– Stéphane LaurentCommented Mar 5, 2014 at 14:03
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$\begingroup$ I would have thought working with the likelihood function as if it were a probability distribution (up to proportionalty) was the equivalent of an implicit uniform prior. $\endgroup$– HenryCommented Dec 3, 2020 at 15:59
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2 Answers
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In my understanding the probability density function is explicitly given for an explicit prior. If this is not possible, the prior can be still implicitly defined which I would then call an implicit prior. Consider estimation of a parameter $\theta$. Assume that because of some circumstances the prior distribution is not specified for $\theta$ directly but for a functional transform of $\theta$, i.e. $f(\theta)$ follows distribution $X$. In order to get back to the distribution of $\theta$ it would be necessary to find the inverse transform $f^{-1}$. Since this is sometimes computationally infeasible, it is not done. Yet, one can still use this implicitly defined distribution for $\theta$ to do i.e. parameter estimation.
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$\begingroup$ Thanks for the answer guys. Unfortunately, the writing is very vague and does not say much other than keeping the prior explicit, which is confusing and frustrating. $\endgroup$– LucaCommented Mar 5, 2014 at 15:28
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I like to think of an implicit distribution as an algorithm able to produce samples. An explicit distribution on the other hand has an analytical expression for its density.
