A novel measure-theoretic framework

$$\pi^{\dagger}_\Lambda(\lambda) := \pi_\Lambda(\lambda)\frac{\pi_{\mathcal{D}}(Q(\lambda))}{\pi^{Q}_{\mathcal{D}}(Q(\lambda))} $$

Across contours, we invert $$\pi_{\mathcal{D}}(Q(\lambda)).$$

The term \(\pi^{Q}_{\mathcal{D}}(Q(\lambda))\) is called the push-forward measure of the prior.

Characterizing this measure is unique to our approach for solving stochastic inverse problems.