  We present results from the MOSFIRE Deep Evolution Field (MOSDEF) survey on
the identification, selection biases, and host galaxy properties of 55 X-ray,
IR and optically-selected active galactic nuclei (AGN) at $1.4 < z < 3.8$. We
obtain rest-frame optical spectra of galaxies and AGN and use the BPT diagram
to identify optical AGN. We examine the uniqueness and overlap of the AGN
identified at different wavelengths. There is a strong bias against identifying
AGN at any wavelength in low mass galaxies, and an additional bias against
identifying IR AGN in the most massive galaxies. AGN hosts span a wide range of
star formation rate (SFR), similar to inactive galaxies once stellar mass
selection effects are accounted for. However, we find (at $\sim 2-3\sigma$
significance) that IR AGN are in less dusty galaxies with relatively higher SFR
and optical AGN in dusty galaxies with relatively lower SFR. X-ray AGN
selection does not display a bias with host galaxy SFR. These results are
consistent with those from larger studies at lower redshifts. Within
star-forming galaxies, once selection biases are accounted for, we find AGN in
galaxies with similar physical properties as inactive galaxies, with no
evidence for AGN activity in particular types of galaxies. This is consistent
with AGN being fueled stochastically in any star-forming host galaxy. We do not
detect a significant correlation between SFR and AGN luminosity for individual
AGN hosts, which may indicate the timescale difference between the growth of
galaxies and their supermassive black holes.
