Let a statistical model of a random variable $X$ with parameter $\theta \in R^m$ be represented by a density function $p(X=x|\theta)$. Assume that the prior, $q(\cdot)$, is on a lower dimensional function of $\theta$, $f(\theta): R^m\to R^n$ where $n<m$. Therefore we cannot translate the prior on $f(\theta)$ to $\theta$ since the Jacobian is non-square. How do we usually deal with this situation?

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Here is one approach, Optimal low-rank approximations of Bayesian linear inverse problems, see also reference 56. Typically, you construct a reduced basis for the parameter space.