  We present a scheme by which a probabilistic forecasting system whose
predictions have poor probabilistic calibration may be recalibrated by
incorporating past performance information to produce a new forecasting system
that is demonstrably superior to the original, in that one may use it to
consistently win wagers against someone using the original system. The scheme
utilizes Gaussian process (GP) modeling to estimate a probability distribution
over the Probability Integral Transform (PIT) of a scalar predictand. The GP
density estimate gives closed-form access to information entropy measures
associated with the estimated distribution, which allows prediction of winnings
in wagers against the base forecasting system. A separate consequence of the
procedure is that the recalibrated forecast has a uniform expected PIT
distribution. A distinguishing feature of the procedure is that it is
appropriate even if the PIT values are not i.i.d. The recalibration scheme is
formulated in a framework that exploits the deep connections between
information theory, forecasting, and betting. We demonstrate the effectiveness
of the scheme in two case studies: a laboratory experiment with a nonlinear
circuit and seasonal forecasts of the intensity of the El Ni\~no-Southern
Oscillation phenomenon.
