  The energy levels of light hypernuclei are experimentally accessible
observables that contain valuable information about the interaction between
hyperons and nucleons. In this work we study strangeness $S = -1$ systems
$^{3,4}_\Lambda$H and $^{4,5}_\Lambda$He using the ab initio no-core shell
model (NCSM) with realistic interactions obtained from chiral effective field
theory ($\chi$EFT). In particular, we quantify the finite precision of
theoretical predictions that can be attributed to nuclear physics
uncertainties. We study both the convergence of the solution of the many-body
problem (method uncertainty) and the regulator- and calibration data-dependence
of the nuclear $\chi$EFT Hamiltonian (model uncertainty). For the former, we
implement infrared correction formulas and extrapolate finite-space NCSM
results to infinite model space. We then use Bayesian parameter estimation to
quantify the resulting method uncertainties. For the latter, we employ a family
of 42 realistic Hamiltonians and measure the standard deviation of predictions
while keeping the leading-order hyperon-nucleon interaction fixed. Following
this procedure we find that model uncertainties of ground-state $\Lambda$
separation energies amount to $\sim 20(100)$ keV in
$^3_\Lambda$H($^4_\Lambda$H,He) and $\sim 400$ keV in $^5_\Lambda$He. Method
uncertainties are comparable in magnitude for the $^4_\Lambda$H,He $1^+$
excited states and $^5_\Lambda$He, which are computed in limited model spaces,
but otherwise much smaller. This knowledge of expected theoretical precision is
crucial for the use of binding energies of light hypernuclei to infer the
elusive hyperon-nucleon interaction.
