  We develop efficient ways to consider and correct for the effects of hidden
units for the paradigmatic case of the inverse kinetic Ising model with fully
asymmetric couplings. We identify two sources of error in reconstructing the
connectivity among the observed units while ignoring part of the network. One
leads to a systematic bias in the inferred parameters, whereas the other
involves correlations between the visible and hidden populations and has a
magnitude that depends on the coupling strength. We estimate these two terms
using a mean field approach and derive self-consistent equations for the
couplings accounting for the systematic bias. Through application of these
methods on simple networks of varying relative population size and connectivity
strength, we assess how and under what conditions the hidden portion can
influence inference and to what degree it can be crudely estimated. We find
that for weak to moderately coupled systems, the effects of the hidden units is
a simple rotation that can be easily corrected for. For strongly coupled
systems, the non-systematic term becomes large and can no longer be safely
ignored, further highlighting the importance of understanding the average
strength of couplings for a given system of interest.
