  It has been observed by Maldacena that one can extract asymptotically anti-de
Sitter Einstein $4$-metrics from Bach-flat spacetimes by imposing simple
principles and data choices. We cast this problem in a conformally compact
Riemannian setting. Following an approach pioneered by Fefferman and Graham for
the Einstein equation, we find formal power series for conformally
compactifiable, asymptotically hyperbolic Bach-flat 4-metrics expanded about
conformal infinity. We also consider Bach-flat metrics in the special case of
constant scalar curvature and in the special case of constant $Q$-curvature.
This allows us to determine the free data at conformal infinity, and to select
those choices that lead to Einstein metrics. Interestingly, the mass is part of
that free data, in contrast to the pure Einstein case. We then choose a
convenient generalization of the Bach tensor to (bulk) dimensions $n>4$ and
consider the higher dimensional problem. We find that the free data for the
expansions split into low-order and high-order pairs. The former pair consists
of the metric on the conformal boundary and its first radial derivative, while
the latter pair consists of the radial derivatives of order $n-2$ and $n-1$.
Higher dimensional generalizations of the Bach tensor lack some of the
geometrical meaning of the 4-dimensional case. This is reflected in the
relative complexity of the higher dimensional problem, but we are able to
obtain a relatively complete result if conformal infinity is not scalar flat.
