  We explore the constraints on the three-nucleon force (3NF) of chiral
effective field theory ($\chi$EFT) that are provided by bound-state observables
in the $A=3$ and $A=4$ sectors. Our statistically rigorous analysis
incorporates experimental error, computational method uncertainty, and the
uncertainty due to truncation of the $\chi$EFT expansion at
next-to-next-to-leading order. A consistent solution for the ${}^3$H binding
energy, the ${}^4$He binding energy and radius, and the ${}^3$H $\beta$-decay
rate can only be obtained if $\chi$EFT truncation errors are included in the
analysis. All of these except the $\beta$-decay rate give essentially
degenerate constraints on the 3NF low-energy constants, so it is crucial for
estimating these parameters. We use eigenvector continuation for fast and
accurate emulation of No-Core Shell Model calculations of the considered
few-nucleon observables. This facilitates sampling of the posterior probability
distribution, allowing us to also determine the distributions of the
hyperparameters that quantify the truncation error. We find a $\chi$EFT
expansion parameter of $Q=0.33 \pm 0.06$ for these observables.
