Latent Variable Gaussian Process - Computing Posteriors on New Data points

Dear All,
I am having troubles finding code to compute posterior estimates for multiple quantities for a latent variable GP.
I have x as input data that is used to train the model and a latent variable z that is inferred. Then, I have a new data point x* that is not used to train the model.
How do I compute quantities like P(z*|x*, x) or P(x*|x) ?

Best !