  The fundamental plane of black hole activity is a relation between X-ray
luminosity, radio luminosity, and black hole mass for hard state Galactic black
holes and their supermassive analogs. The fundamental plane suggests that, at
low-accretion rates, the physical processes regulating the conversion of an
accretion flow into radiative energy could be universal across the entire black
hole mass scale. However, there is still a need to further refine the
fundamental plane in order to better discern the radiative processes and their
geometry very close to the black hole, in particular the source of hard X-rays.
Further refinement is necessary because error bars on the best-fit slopes of
the fundamental plane are generally large, and also the inferred coefficients
can be sensitive to the adopted sample of black holes. In this work, we regress
the fundamental plane with a Bayesian technique. Our approach shows that
sub-Eddington black holes emit X-ray emission that is predominantly optically
thin synchrotron radiation from the jet, provided that their radio spectra are
flat or inverted. X-ray emission dominated by very radiatively inefficient
accretion flows are excluded at the >3\sigma\ level. We also show that it is
difficult to place FR I galaxies onto the fundamental plane because their X-ray
jet emission is highly affected by synchrotron cooling. On the other hand, BL
Lac objects fit onto the fundamental plane. Including a uniform subset of
high-energy peaked BL Lac objects from the SDSS, we find sub-Eddington black
holes with flat/inverted radio spectra follow log L_x=(1.45\pm0.04)log
L_R-(0.88\pm0.06)\logM_{BH}-6.07\pm1.10, with \sigma_{int}=0.07\pm0.05 dex.
Finally, we discuss how the effects of synchrotron cooling of jet emission from
the highest black hole masses can bias fundamental plane regressions, perhaps
leading to incorrect inferences on X-ray radiation mechanisms.
