  As machine learning is increasingly used in essential systems, it is
important to reduce or eliminate the incidence of serious bugs. A growing body
of research has developed machine learning algorithms with formal guarantees
about performance, robustness, or fairness. Yet, the analysis of these
algorithms is often complex, and implementing such systems in practice
introduces room for error. Proof assistants can be used to formally verify
machine learning systems by constructing machine checked proofs of correctness
that rule out such bugs. However, reasoning about probabilistic claims inside
of a proof assistant remains challenging. We show how a probabilistic program
can be automatically represented in a theorem prover using the concept of
\emph{reparameterization}, and how some of the tedious proofs of measurability
can be generated automatically from the probabilistic program. To demonstrate
that this approach is broad enough to handle rather different types of machine
learning systems, we verify both a classic result from statistical learning
theory (PAC-learnability of decision stumps) and prove that the null model used
in a Bayesian hypothesis test satisfies a fairness criterion called demographic
parity.
