  Loop quantum cosmology (LQC) is used to provide concrete evidence in support
of the general paradigm underlying spin foam models (SFMs). Specifically, it is
shown that: i) the physical inner product in the timeless framework equals the
transition amplitude in the deparameterized theory; ii) this quantity admits a
%convergent vertex expansion a la SFMs in which the $M$-th term refers just to
$M$ volume transitions, without any reference to the time at which the
transition takes place; iii) the exact physical inner product is obtained by
summing over just the discrete geometries; no `continuum limit' is involved;
and, iv) the vertex expansion can be interpreted as a perturbative expansion in
the spirit of group field theory. This sum over histories reformulation of LQC
also addresses certain other issues which are briefly summarized.
