  The dynamics of an asymmetric kinetic Ising model is studied. Two schemes for
improving the existing mean-field description are proposed. In the first
scheme, we derive the formulas for instantaneous magnetization, equal-time
correlation, and time-delayed correlation, considering the correlation between
different local fields. To derive the time-delayed correlation, we emphasize
that the small correlation assumption adopted in previous work [M. M\'ezard and
J. Sakellariou, J. Stat. Mech., L07001 (2011)] is in fact not required. To
confirm the inference efficiency of our method, we perform extensive
simulations on single instances with either temporally constant external
driving fields or sinusoidal external fields. In the second scheme, we develop
an improved mean-field theory for instantaneous magnetization prediction
utilizing the notion of the cavity system in conjunction with a perturbative
expansion approach. Its efficiency is numerically confirmed by comparison with
the existing mean-field theory when partially asymmetric couplings are present.
