  Measurements of $t\bar{t} H$ production in the $H\to b\bar{b}$ channel depend
in a critical way on the theoretical uncertainty associated with the
irreducible $t\bar{t}+b$-jet background. In this paper, analysing the various
topologies that account for $b$-jet production in association with a $t\bar{t}$
pair, we demonstrate that the process at hand is largely driven by final-state
$g\to b\bar{b}$ splittings. We also show that in five-flavour simulations based
on $t\bar{t}+$multi-jet merging $b$-jet production is mostly driven by the
parton shower, while matrix elements play only a marginal role in the
description of $g\to b\bar{b}$ splittings. Based on these observations we
advocate the use of NLOPS simulations of $pp\to t\bar{t}b\bar{b}$ in the
four-flavour scheme, and we present a new POWHEG generator of this kind.
Predictions and uncertainties for $t\bar{t}+b$-jet observables at the 13 TeV
LHC are presented both for the case of stable top quarks and with
spin-correlated top decays. Besides QCD scale variations we consider also
theoretical uncertainties related to the POWHEG matching method and to the
parton shower modelling, with emphasis on $g\to b\bar{b}$ splittings. In
general, matching and shower uncertainties turn out to be remarkably small.
This is confirmed also by a tuned comparison against Sherpa+OpenLoops.
