Bayesian inversion: can I force a Jeffreys prior on the discrepancy model?

Hi UQ people,

I am doing a model parameters calibration with the Bayes module in UQlab.
I assume that I have no information about the discrepancy variance in the discrepancy model. However, I know that the term is a variance of a normal distribution. Therefore, I assume that this little information is more than no information at all.

Before, I considered the variance as uniformly distributed as in \epsilon \sim \mathcal{U} [a,b]. Now, I assumed I could impose a Jeffreys distribution on the parameter \epsilon. I noticed that the Jeffreys distribution is not implemented in UQlab. Therefore I created a Gamma distribution with very low values in parameter.

A Gamma distribution is defined by the parameters k and \theta as

f(x) = \frac{1}{\Gamma(k)\theta^k} x^{k-1}e^{-x/\theta} ,

while the Jeffreys is defined as

f(x)=\frac{1}{x} .

So for k,\theta = 0, I obtain a Jeffreys distribution. However, UQlab gives some instability warnings for this value, and the internet is skeptical in assuming that the Jeffreys distribution is a special case of the Gamma distribution, i.e., Gamma(0,0).

What do you think? And would you implement a Jeffreys distribution in UQlab?