The average log likelihood $$L(W,X) = \frac{1}{N}\sum_{1}^{N} log(p(x_n;W))$$ as defined by the authors in http://www.gatsby.ucl.ac.uk/aistats/fullpapers/217.pdf (first equation, first page, right column).
So far, so good.
Then the author(s) go on to say that $$L(W,X) = \sum_{1}^{N} \delta(x-x_n)log(p(x_n;W))$$.
I don't see why this is true.