  Recently quantum prediction problem was proposed in the Bayesian framework.
It is shown that Bayesian predictive density operators are the best predictive
density operators when we evaluate them by using the average relative entropy
based on a prior.As an illustrative example, we treat the Gaussian states
family adopting the Gaussian distribution as a prior and give the Bayesian
predictive density operator with the heterodyne measurement fixed. We show that
it is better than the plug-in predictive density operator based on the maximum
likelihood estimate by calculating each average relative entropy.
