Aim 3: Posterior Sampling and Uncertainty

April 25, 2023

Nonlinear measurement process

\[y = Be^{-A\mu} + v,~v \sim \mathcal{N}(0, K)\]

forward diffusion on signal

\[\mu_t = \alpha(t)\mu + \beta(t)z,~z \sim \mathcal{N}(0, K)\]

forward diffusion on measurement

\[Be^{-A\mu_t} = Be^{-A\alpha(t)\mu}e^{-A\beta(t)z}\]

\[y_t = Be^{-A\mu_t} + e^{-A\beta(t)z} v = e^{-A\beta(t)z}\left[Be^{-A\alpha(t)\mu} + v\right]\]

Nonlinear measurement process

\[y = Be^{-A\mu} + v,~v \sim \mathcal{N}(0, K)\]

forward diffusion on signal

\[\mu_t = \alpha(t)\mu + \beta(t)z,~z \sim \mathcal{N}(0, K)\]

forward diffusion on measurement

\[Be^{-A\mu_t} = Be^{-A\alpha(t)\mu}e^{-A\beta(t)z}\]

\[y_t = Be^{-A\mu_t} + e^{-A\beta(t)z} v = e^{-A\beta(t)z}\left[Be^{-A\alpha(t)\mu} + v\right]\]

set \(\alpha(t) = 1\)

\(= e^{-A\beta(t)z}y\)