  We investigate the initial conditions of inflation in a Bianchi~I universe
that is homogeneous but not isotropic. We use the Eisenhart lift to describe
such a theory geometrically as geodesics on a field space manifold. We
construct the phase-space manifold of the theory by considering the tangent
bundle of the field space and equipping it with a natural metric. We find that
the total volume of this manifold is finite for a wide class of inflationary
models. We therefore take the initial conditions to be uniformly distributed
over it in accordance with Laplace's principle of indifference. This results in
a normalisable, reparametrisation invariant measure on the set of initial
conditions of inflation in a Bianchi~I universe. We find that this measure
favours an initial state in which the inflaton field is at or near its minimum,
with a mild preference for some initial anisotropy. Since inflation requires an
initial field value with a large displacement from its minimum, we therefore
conclude that the theory of inflation requires finely tuned initial conditions.
