  A spectral approach to Bayesian inference is presented. It pursues the
emulation of the posterior probability density. The starting point is a series
expansion of the likelihood function in terms of orthogonal polynomials. From
this spectral likelihood expansion all statistical quantities of interest can
be calculated semi-analytically. The posterior is formally represented as the
product of a reference density and a linear combination of polynomial basis
functions. Both the model evidence and the posterior moments are related to the
expansion coefficients. This formulation avoids Markov chain Monte Carlo
simulation and allows one to make use of linear least squares instead. The pros
and cons of spectral Bayesian inference are discussed and demonstrated on the
basis of simple applications from classical statistics and inverse modeling.
