  $T_{1\rho}$ mapping is a promising quantitative MRI technique for the
non-invasive assessment of tissue properties. Learning-based approaches can map
$T_{1\rho}$ from a reduced number of $T_{1\rho}$ weighted images, but requires
significant amounts of high quality training data. Moreover, existing methods
do not provide the confidence level of the $T_{1\rho}$ estimation. To address
these problems, we proposed a self-supervised learning neural network that
learns a $T_{1\rho}$ mapping using the relaxation constraint in the learning
process. Epistemic uncertainty and aleatoric uncertainty are modelled for the
$T_{1\rho}$ quantification network to provide a Bayesian confidence estimation
of the $T_{1\rho}$ mapping. The uncertainty estimation can also regularize the
model to prevent it from learning imperfect data. We conducted experiments on
$T_{1\rho}$ data collected from 52 patients with non-alcoholic fatty liver
disease. The results showed that our method outperformed the existing methods
for $T_{1\rho}$ quantification of the liver using as few as two
$T_{1\rho}$-weighted images. Our uncertainty estimation provided a feasible way
of modelling the confidence of the self-supervised learning based $T_{1\rho}$
estimation, which is consistent with the reality in liver $T_{1\rho}$ imaging.
