FB-GDM: Fully-Bayesian Guided Diffusion Models for High-Dimensional Linear Inverse Problems via Unsupervised Variational Inference

Diffusion models are powerful priors for linear inverse problems, but the reference guidance methods, Diffusion Posterior Sampling (DPS) and Pseudoinverse-Guided Diffusion Models ($Π$GDM), rely on scalar hyperparameters tuned per task, usually against the ground truth. We introduce FB-GDM, a fully-Bayesian guided diffusion method that removes this calibration step. Starting from the Gaussian approximation of $Π$GDM, we derive a closed-form conditional score that depends on two precision parameters (inverse variances), one associated with the denoising approximation and one with the observation likelihood, and treat them as latent variables inferred by variational inference at each reverse step. A separable factorization makes each update scale linearly with the number of pixels, so the inference stays tractable at full image resolution, at a cost comparable to one $Π$GDM run. FB-GDM requires neither the noise level nor the ground truth: its only inputs are the observation and the forward operator. Experiments on CelebA-HQ inverse problems establish two results. (i) The precision parameters, inferred from the observation alone, allow FB-GDM to outperform $Π$GDM at its nominal setting, even when the latter is given the true noise level, by up to 14 dB depending on the operator, and to match the ground-truth-calibrated $Π$GDM oracle within 0.1 dB. (ii) FB-GDM is robust when the forward operator, the noise level, or the image distribution changes: it stays close to a per-problem $Π$GDM oracle throughout and does not exhibit the hallucinations observed with DPS, whereas DPS substantially degrades at a fixed scale and $Π$GDM stays competitive only if it is re-tuned against the ground truth for each new problem. When the prior is applied to images outside its training set, this re-balancing between data and prior keeps FB-GDM faithful where a fixed face-prior guidance can otherwise hallucinate.