I've hit a problem trying to sample an inverse gamma distribution, 'scaled' using a temperature variable, T. If my distribution is defined as (where the normalising constant k=(b^a)/Gamma(a) ): IG(x|a,b) = k * x^(-a-1) * exp(-b/x) then the scaled version is (IG(x|a,b))^(1/T) = k^(1/T) * x^(-(-a+1)/T) * exp(-b/(Tx)). which I want to sample as part of a simulated annealing procedure. If I am not mistaken, this is also proportional to a new IG distribution, IG(x|a',b') where a'+1 = (a+1)/T and b'=b/T. Hence a' = (a+1-T)/T. The problem is however, that a' and b' should be strictly > 0 for the distribution to be valid. Thus for temperatures T>a+1, a' becomes negative and the samples can't be drawn. (However I can still evaluate the IG pdf for a'<0... so I'm not sure why the condition is strictly necessary? Is it just a physical interpretation?) I can do the sampling for X~IG(a',b') by transformation of a Gamma variate G(x|a,b)=k * x^(a-1) exp(-bx) by drawing Y~G(a',b') and letting X=1/Y. Now the strange thing is if I apply the same temperature scaling to the Gamma distribution, I get the new transformation to a Gamma distribution with a' = (a-1+T)/T which will always be positive for all T>=1 if a>0. As whether I'm working with x or 1/x (and therefore working with the IG or Gamma) should be purely a matter of convenience, and not depend on which definition I start from, there is something at odds here... I've probably missed something obvious but I can't think what it is. Any suggestions appreciated!