  We present a Bayesian framework based on a new exponential likelihood
function driven by the quadratic Wasserstien metric. Compared to conventional
Bayesian models based on Gaussian likelihood functions driven by the
least-squares norm ($L_2$ norm), the new framework features several advantages.
First, the new framework does not rely on the likelihood of the measurement
noise and hence can treat complicated noise structures such as combined
additive and multiplicative noise. Secondly, unlike the normal likelihood
function, the Wasserstein-based exponential likelihood function does not
usually generate multiple local extrema. As a result, the new framework
features better convergence to correct posteriors when a Markov Chain Monte
Carlo sampling algorithm is employed. Thirdly, in the particular case of signal
processing problems, while a normal likelihood function measures only the
amplitude differences between the observed and simulated signals, the new
likelihood function can capture both the amplitude and the phase differences.
We apply the new framework to a class of signal processing problems, that is,
the inverse uncertainty quantification of waveforms, and demonstrate its
advantages compared to Bayesian models with normal likelihood functions.
