Sorry I am not sure I am expressing myself well. The sort of inverse problem is an inverse sensitivity problem. I think I have worked out the issue now, but I repeat it because it still leaves open some practical questions…

So suppose that I have some deterministic function: Q = f(lambda_1,lambda_2). From experimental data I have a probability distribution over the possible output values: Q ~ G(Q). What I would like to do is determine the corresponding distribution over input values (lambda_1,lambda_2) that would produce that output.

Note this is not a Bayesian problem as there is no likelihood here: p(Q | lambda_1,lambda_2) = 0, unless Q = f(lambda_1,lambda_2).

However I nonetheless want to compute a conditional probability distribution over the inputs, given the output distribution G(.), so I would like to compute p(lambda_1,lambda_2|G(.)). The issue is that there are many values of the inputs that correspond to a given output. In other words there are contours in Q space.

I (naively) thought that using MCMC to sample: (lambda_1i,lambda_2i) ~ G( f (lambda_1,lambda_2)) would produce the desired input distribution on (lambda_1i, lambda_2i). By ‘desired’ here I mean one such that if I were to feed the samples (lambda_1i,lambda_2i) into f(.) to obtain Q_i, then I would obtain G(Q_i) – the observed output distribution.

However this is not the case. I have worked out that I essentially need to correct the sampling for the length of contours corresponding to a particular Q value. So (using Random Walk Metropolis) the accept/reject ratio actually becomes,

r = [f(lambda_1’,lambda_2’) / f(lambda_1’,lambda_2’)] * [L( f (lambda_1,lambda_2)) / L( f (lambda_1’,lambda_2’))],

where L(Q) is the length of contours in (lambda_1,lambda_2) space, such that f(lambda_1,lambda_2) = Q for all values of the inputs.

If you’re still with me (sorry for the longwindedness), this leaves some important practical research questions. In particular, how might we determine the function L(Q) in N-dimensional space to allow us to do MCMC? Obviously this would need to be an approximate method (probably a Monte Carlo one to avoid the curse of dimensionality).

Does anyone have experience with such a problem? Or alternatively have an idea of how to sidestep it?

Best,

Ben