  We study product regular conditional probabilities under measures of two
coordinates with respect to the second coordinate that are weakly continuous on
the support of the marginal of the second coordinate. Assuming that there
exists a sequence of probability measures on the product space that satisfies a
large deviation principle, we present necessary and sufficient conditions for
the conditional probabilities under these measures to satisfy a large deviation
principle. The arguments of these conditional probabilities are assumed to
converge. A way to view regular conditional probabilities as a special case of
product regular conditional probabilities is presented. This is used to derive
conditions for large deviations of regular conditional probabilities. In
addition, we derive a Sanov-type theorem for large deviations of the empirical
distribution of the first coordinate conditioned on fixing the empirical
distribution of the second coordinate.
