  For radioactive nuclear data, $\beta$ decay is one of the most important
information and is applied to various fields. However, some of the
$\beta$-decay data are not available due to experimental difficulties. From
this respect, theoretically calculated results have been embedded in the
$\beta$-decay data to compensate the missing information. To calculate the
$\beta$-decay half-lives, a proton-neutron quasi-particle random phase
approximation on top of a Skryme energy density functional is applied with an
assumption of spherical symmetry. The isoscalar pairing strength is estimated
by a Bayesian neutral network (BNN). We verify the predicted isoscalar pairing
strengths by preparing the training data and test data. It was confirmed that
the finite-range isovector pairing ensures the $\beta$-decay half-lives
insensitive to the model space, while the zero-range one was largely dependent
on it. The half-lives calculated with the BNN isoscalar pairing strengths
reproduced most of experimental data, although those of highly deformed nuclei
were underestimated. We also studied that the predictive performance on new
experimental data that were not used for the BNN training and found that they
were reproduced well. Our study demonstrates that the isoscalar pairing
strengths determined by the BNN can reproduce experimental data in the same
accuracy as other theoretical works.
