  We introduce a Bayesian framework for inference with a supervised version of
the Gaussian process latent variable model. The framework overcomes the high
correlations between latent variables and hyperparameters by using an unbiased
pseudo estimate for the marginal likelihood that approximately integrates over
the latent variables. This is used to construct a Markov Chain to explore the
posterior of the hyperparameters. We demonstrate the procedure on simulated and
real examples, showing its ability to capture uncertainty and multimodality of
the hyperparameters and improved uncertainty quantification in predictions when
compared with variational inference.
