  Kernel methods have revolutionized the fields of pattern recognition and
machine learning. Their success, however, critically depends on the choice of
kernel parameters. Using Gaussian process (GP) classification as a working
example, this paper focuses on Bayesian inference of covariance (kernel)
parameters using Markov chain Monte Carlo (MCMC) methods. The motivation is
that, compared to standard optimization of kernel parameters, they have been
systematically demonstrated to be superior in quantifying uncertainty in
predictions. Recently, the Pseudo-Marginal MCMC approach has been proposed as a
practical inference tool for GP models. In particular, it amounts in replacing
the analytically intractable marginal likelihood by an unbiased estimate
obtainable by approximate methods and importance sampling. After discussing the
potential drawbacks in employing importance sampling, this paper proposes the
application of annealed importance sampling. The results empirically
demonstrate that compared to importance sampling, annealed importance sampling
can reduce the variance of the estimate of the marginal likelihood
exponentially in the number of data at a computational cost that scales only
polynomially. The results on real data demonstrate that employing annealed
importance sampling in the Pseudo-Marginal MCMC approach represents a step
forward in the development of fully automated exact inference engines for GP
models.
