  The chaos expansion of a general non-linear function of a Gaussian stationary
increment process conditioned on its past realizations is derived. This work
combines Wiener chaos expansion approach to study the dynamics of a stochastic
system with the classical problem of the prediction of a Gaussian process based
on a realization of its past. This is done by constructing a special basis for
the Fock space of the Gaussian space generated by the process, such that each
basis element is either measurable or independent with respect to the given
samples. This property of the basis allows us to derive the chaos expansion of
a random variable conditioned on part of the sample path. We provide a general
method for the construction of such basis when the underlying process is
Gaussian with stationary increment. We evaluate the basis elements in the case
of the fractional Brownian motion, which leads to a prediction formula for this
process.
