  Streamlined qBOLD acquisitions enable experimentally straightforward
observations of brain oxygen metabolism. $R_2^\prime$ maps are easily inferred;
however, the Oxygen extraction fraction (OEF) and deoxygenated blood volume
(DBV) are more ambiguously determined from the data. As such, existing
inference methods tend to yield very noisy and underestimated OEF maps, while
overestimating DBV.
  This work describes a novel probabilistic machine learning approach that can
infer plausible distributions of OEF and DBV. Initially, we create a model that
produces informative voxelwise prior distribution based on synthetic training
data. Contrary to prior work, we model the joint distribution of OEF and DBV
through a scaled multivariate logit-Normal distribution, which enables the
values to be constrained within a plausible range. The prior distribution model
is used to train an efficient amortized variational Bayesian inference model.
This model learns to infer OEF and DBV by predicting real image data, with few
training data required, using the signal equations as a forward model.
  We demonstrate that our approach enables the inference of smooth OEF and DBV
maps, with a physiologically plausible distribution that can be adapted through
specification of an informative prior distribution. Other benefits include
model comparison (via the evidence lower bound) and uncertainty quantification
for identifying image artefacts. Results are demonstrated on a small study
comparing subjects undergoing hyperventilation and at rest. We illustrate that
the proposed approach allows measurement of gray matter differences in OEF and
DBV and enables voxelwise comparison between conditions, where we observe
significant increases in OEF and $R_2^\prime$ during hyperventilation.
