[10/10/23] Aim 3: Posterior Sampling and Uncertainty

y \sim \text{Poission}(N_0~e^{-Ax})

y \sim \text{Poission}(N_0~e^{-Ax})

$x$

$y$

$\text{FBP}(y)$

$x$

$y$

$\text{FBP}(y)$

Poisson model

Gaussian approximation

$\lambda = N_0~e^{-Ax}$

$y \sim \text{Poisson}(\lambda)$

$y \sim \text{Normal}(\lambda, \lambda)$

DDPM Schedule

$p(x_t \mid x_0) = \mathcal{N}(\sqrt{\alpha_t}x_0, (1 - \alpha_t)\mathbb{I})$

Tweedie's formula

If $x_i \sim \mathcal{N}(a_i~x_0, b_i^2 \mathbb{I})$ , then

$\hat{x}_0(x_t) = \mathbb{E}[x_0 \mid x_t] = \frac{x_t + b_i^2~\nabla_{x_t} \log p_t(x_t)}{a_i}$

$x_0$

$\hat{x}_0$

$t=0$

$t=1$

DPS:

$\nabla_{x_t} \log p(y \mid x_t) \approx \nabla_{x_t} \log p(y \mid \hat{x}_0(x_t))$

Poisson

Gaussian

DPS:

$\nabla_{x_t} \log p(y \mid x_t) \approx \nabla_{x_t} \log p(y \mid \hat{x}_0(x_t))$

Poisson

Gaussian

DPS:

$\nabla_{x_t} \log p(y \mid x_t) \approx \nabla_{x_t} \log p(y \mid \hat{x}_0(x_t))$

Poisson

Gaussian

$\hat{l}_{\alpha}$

$\hat{u}_{\alpha}$

$\hat{u}_{\alpha} - \hat{l}_{\alpha}$