  We extend the Landauer-B\"uttiker probe formalism for conductances to the
high bias regime, and study the effects of environmentally-induced elastic and
inelastic scattering on charge current in single molecule junctions, focusing
on high-bias effects. The probe technique phenomenologically incorporates
incoherent elastic and inelastic effects to the fully coherent case, mimicking
a rich physical environment at trivial cost. We further identify
environmentally-induced mechanisms which generate an asymmetry in the current,
manifested as a weak diode behavior. This rectifying behavior, found in two
types of molecular junction models, is absent in the coherent-elastic limit,
and is only active in the case with incoherent-inelastic scattering. Our work
illustrates that in the low bias - linear response regime, the commonly used
"dephasing probe" (mimicking only elastic decoherence effects) operates nearly
indistinguishably from a "voltage probe" (admitting inelastic-dissipative
effects). However, these probes realize fundamentally distinct $I$-$V$
characteristics at high biases, reflecting the central roles of dissipation and
inelastic scattering processes on molecular electronic transport
far-from-equilibrium.
