  In this paper, we compute finite sample bounds for data-driven approximations
of the solution to stochastic reachability problems. Our approach uses a
nonparametric technique known as kernel distribution embeddings, and provides
probabilistic assurances of safety for stochastic systems in a model-free
manner. By implicitly embedding the stochastic kernel of a Markov control
process in a reproducing kernel Hilbert space, we can approximate the safety
probabilities for stochastic systems with arbitrary stochastic disturbances as
simple matrix operations and inner products. We present finite sample bounds
for point-based approximations of the safety probabilities through construction
of probabilistic confidence bounds that are state- and input-dependent. One
advantage of this approach is that the bounds are responsive to non-uniformly
sampled data, meaning that tighter bounds are feasible in regions of the state-
and input-space with more observations. We numerically evaluate the approach,
and demonstrate its efficacy on a neural network-controlled pendulum system.
