  We present a next-to-next-to-leading order (NNLO) QCD calculation of the
bottom-induced contributions to the production of a Higgs boson plus a jet,
i.e. the process $p p \rightarrow H +j$ to $\mathcal{O}(y_b^2 \alpha_s^3)$. We
work in the five-flavor scheme (5FS) in which the bottom quark mass is retained
only in the coupling to the Higgs boson. Our calculation uses $N$-jettiness
slicing to regulate infrared divergences, allowing for fully-differential
predictions for collider observables. After extensively validating the
methodology, we present results for the 13 TeV LHC. Our NNLO predictions show a
marked improvement in the overall renormalization and factorization scale
dependence, the latter of which proves to be particularly troublesome for 5FS
calculations at lower orders. In addition, using the same methodology we
present a NNLO computation of $b\overline{b} \rightarrow H$. Our results are
implemented into MCFM.
