  We present algorithms for estimating the forward reachable set of a dynamical
system using only a finite collection of independent and identically
distributed samples. The produced estimate is the sublevel set of a function
called an empirical inverse Christoffel function: empirical inverse Christoffel
functions are known to provide good approximations to the support of
probability distributions. In addition to reachability analysis, the same
approach can be applied to general problems of estimating the support of a
random variable, which has applications in data science towards detection of
novelties and outliers in data sets. In applications where safety is a concern,
having a guarantee of accuracy that holds on finite data sets is critical. In
this paper, we prove such bounds for our algorithms under the Probably
Approximately Correct (PAC) framework. In addition to applying classical
Vapnik-Chervonenkis (VC) dimension bound arguments, we apply the PAC-Bayes
theorem by leveraging a formal connection between kernelized empirical inverse
Christoffel functions and Gaussian process regression models. The bound based
on PAC-Bayes applies to a more general class of Christoffel functions than the
VC dimension argument, and achieves greater sample efficiency in experiments.
