  Quantitative MR imaging is increasingly favoured for its richer information
content and standardised measures. However, extracting quantitative parameters
such as the longitudinal relaxation rate (R1), apparent transverse relaxation
rate (R2*), or magnetisation-transfer saturation (MTsat) involves inverting a
highly non-linear function. Estimations often assume noise-free measurements
and use subsets of the data to solve for different quantities in isolation,
with error propagating through each computation. Instead, a probabilistic
generative model of the entire dataset can be formulated and inverted to
jointly recover parameter estimates with a well-defined probabilistic meaning
(e.g., maximum likelihood or maximum a posteriori). In practice, iterative
methods must be used but convergence is difficult due to the non-convexity of
the log-likelihood; yet, we show that it can be achieved thanks to a novel
approximate Hessian and, with it, reliable parameter estimates obtained. Here,
we demonstrate the utility of this flexible framework in the context of the
popular multi-parameter mapping framework and further show how to incorporate a
denoising prior and predict posterior uncertainty. Our implementation uses a
PyTorch backend and benefits from GPU acceleration. It is available at
https://github.com/balbasty/nitorch.
