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Measurements of $t\bar{t} H$ production in the $H\to b\bar{b}$ channel dependin a critical way on the theoretical uncertainty associated with theirreducible $t\bar{t}+b$-jet background. In this paper, analysing the varioustopologies 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 basedon $t\bar{t}+$multi-jet merging $b$-jet production is mostly driven by theparton shower, while matrix elements play only a marginal role in thedescription of $g\to b\bar{b}$ splittings. Based on these observations weadvocate the use of NLOPS simulations of $pp\to t\bar{t}b\bar{b}$ in thefour-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 TeVLHC are presented both for the case of stable top quarks and withspin-correlated top decays. Besides QCD scale variations we consider alsotheoretical uncertainties related to the POWHEG matching method and to theparton shower modelling, with emphasis on $g\to b\bar{b}$ splittings. Ingeneral, matching and shower uncertainties turn out to be remarkably small.This is confirmed also by a tuned comparison against Sherpa+OpenLoops.

New NLOPS predictions for $\boldsymbol{t\bar{t}+b}$-jet production at the LHC