  We explicitly rewrite the path integral for the free or critical $O(N)$ (or
$U(N)$) bosonic vector models in $d$ space-time dimensions as a path integral
over fields (including massless high-spin fields) living on ($d+1$)-dimensional
anti-de Sitter space. Inspired by de Mello Koch, Jevicki, Suzuki and Yoon and
earlier work, we first rewrite the vector models in terms of bi-local fields,
then expand these fields in eigenmodes of the conformal group, and finally map
these eigenmodes to those of fields on anti-de Sitter space. Our results
provide an explicit (non-local) action for a high-spin theory on anti-de Sitter
space, which is presumably equivalent in the large $N$ limit to Vasiliev's
classical high-spin gravity theory (with some specific gauge-fixing to a fixed
background), but which can be used also for loop computations. Our mapping is
explicit within the $1/N$ expansion, but in principle can be extended also to
finite $N$ theories, where extra constraints on products of bulk fields need to
be taken into account.
