  We investigate the convective stability of two popular types of model of the
gas distribution in the hot Galactic halo. We first consider models in which
the halo density and temperature decrease exponentially with height above the
disk. These halo models were created to account for the fact that, on some
sight lines, the halo's X-ray emission lines and absorption lines yield
different temperatures, implying that the halo is non-isothermal. We show that
the hot gas in these exponential models is convectively unstable if
$\gamma<3/2$, where $\gamma$ is the ratio of the temperature and density scale
heights. Using published measurements of $\gamma$ and its uncertainty, we use
Bayes' Theorem to infer posterior probability distributions for $\gamma$, and
hence the probability that the halo is convectively unstable for different
sight lines. We find that, if these exponential models are good descriptions of
the hot halo gas, at least in the first few kiloparsecs from the plane, the hot
halo is reasonably likely to be convectively unstable on two of the three sight
lines for which scale height information is available. We also consider more
extended models of the halo. While isothermal halo models are convectively
stable if the density decreases with distance from the Galaxy, a model of an
extended adiabatic halo in hydrostatic equilibrium with the Galaxy's dark
matter is on the boundary between stability and instability. However, we find
that radiative cooling may perturb this model in the direction of convective
instability. If the Galactic halo is indeed convectively unstable, this would
argue in favor of supernova activity in the Galactic disk contributing to the
heating the hot halo gas.
