  We prove semi-empirical concentration inequalities for random variables which
are given as possibly nonlinear functions of independent random variables.
These inequalities describe concentration of random variable in terms of the
data/distribution-dependent Efron-Stein (ES) estimate of its variance and they
do not require any additional assumptions on the moments. In particular, this
allows us to state semi-empirical Bernstein type inequalities for general
functions of unbounded random variables, which gives user-friendly
concentration bounds for cases where related methods (e.g. bounded differences)
might be more challenging to apply. We extend these results to Efron-Stein
PAC-Bayesian inequalities which hold for arbitrary probability kernels that
define a random, data-dependent choice of the function of interest. Finally, we
demonstrate a number of applications, including PAC-Bayesian generalization
bounds for unbounded loss functions, empirical Bernstein type generalization
bounds, new truncation-free bounds for off-policy evaluation with Weighted
Importance Sampling (WIS), and off-policy PAC-Bayesian learning with WIS.
