  The IceCube report of a $\sim 3.5\sigma$ excess of $13\pm5$ neutrino events
in the direction of the blazar TXS 05056+056 in 2014-2015 and the 2017
detection of a high-energy neutrino, IceCube-170922A, during a gamma-ray flare
from the same blazar, have revived the interest in scenarios for neutrino
production in blazars. We perform comprehensive analyses on the long-term
electromagnetic emission of TXS 05056+056 using optical, X-ray, and gamma-ray
data from the All-Sky Automated Survey for Supernovae (ASAS-SN), the Neil
Gehrels Swift Observatory (Swift), the Monitor of All-sky X-ray Image (MAXI),
and the Fermi Large Area Telescope (Fermi-LAT). We also perform numerical
modeling of the spectral energy distributions (SEDs) in four epochs prior to
2017 with contemporaneous gamma-ray and lower energy (optical and/or X-ray)
data. We find that the multi-epoch SEDs are consistent with a hybrid leptonic
scenario, where the gamma-rays are produced in the blazar zone via external
inverse Compton scattering of accelerated electrons, and high-energy neutrinos
are produced via the photomeson production process of co-accelerated protons.
The multi-epoch SEDs can be satisfactorily explained with the same jet
parameters and variable external photon density and electron luminosity. Using
the maximal neutrino flux derived for each epoch, we put an upper limit of
$\sim0.4-2$ on the muon neutrino number in ten years of IceCube observations.
Our results are consistent with the IceCube-170922A detection, which can be
explained as an upper fluctuation from the average neutrino rate expected from
the source, but in strong tension with the 2014-2015 neutrino flare.
