  We review the information geometry of linear systems and its application to
Bayesian inference, and the simplification available in the K\"ahler manifold
case. We find conditions for the information geometry of linear systems to be
K\"ahler, and the relation of the K\"ahler potential to information geometric
quantities such as $\alpha $-divergence, information distance and the dual
$\alpha $-connection structure. The K\"ahler structure simplifies the
calculation of the metric tensor, connection, Ricci tensor and scalar
curvature, and the $\alpha $-generalization of the geometric objects. The
Laplace--Beltrami operator is also simplified in the K\"ahler geometry. One of
the goals in information geometry is the construction of Bayesian priors
outperforming the Jeffreys prior, which we use to demonstrate the utility of
the K\"ahler structure.
