Sorry if the initial post was not clear. We measured T time points X_1… X_T. Instead of doing inference in the entire dataset, we want to do inference and compute the posterior with 1… (T-1) datapoints and leave one out to predict its latent value.

if you fit a guide to a model conditioned on X_{1:t-1} and that containts latent variables z_{1:t} then you can use Predictive to approximate the posterior marginal P(z_t | X_{1:t-1})

Hi Martin,
absolutely ! if I use variational inference I can trivially compute z by looking at the variational approximation to the posterior q(z | x) and compute a new value if q is amortized. I was asking, is there an easy way to do it if we are using NUTS to samples from the posterior so the only thing we have are samples but not a functional form of the posterior?