We define a copula process which describes the dependencies between
arbitrarily many random variables independently of their marginal
distributions. As an example, we develop a stochastic volatility model,
Gaussian Copula Process Volatility (GCPV), to predict the latent standard
deviations of a sequence of random variables. To learn the parameters of GCPV
we use Bayesian inference, with the Laplace approximation, and with Markov
chain Monte Carlo as an alternative. We find both methods comparable. We also
find our model can outperform GARCH, on simulated and financial data. And
unlike GARCH, GCPV can easily handle missing data, incorporate covariates other
than time, and model a rich class of covariance structures.