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I know that one can approximate this density ($p(v) \propto 1/v$) using its truncated version and implement it this way:

   B~dunif(log(BInf),log(BSup))
   v<-exp(B)

but I would like to use the exact form (I checked that under this prior my posterior is proper). Is there any solution to achieve this ?

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I believe JAGS does not allow improper priors. So your suggested approach is as close as you can get.

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