  We introduce the notion of a weighted $\delta$-vector of a lattice polytope.
Although the definition is motivated by motivic integration, we study weighted
$\delta$-vectors from a combinatorial perspective. We present a version of
Ehrhart Reciprocity and prove a change of variables formula. We deduce a new
geometric interpretation of the coefficients of the Ehrhart $\delta$-vector.
More specifically, they are sums of dimensions of orbifold cohomology groups of
a toric stack.
