Suppose that we have a bayesian linear regression problem with a gaussian framework and flat isotropic gaussian prior. We train a model with 4 features. Such that the posterior is 4 dimensional gaussian.
Suppose I want to predict for a single test instance, but one of the features is missing. I have two options, use the conditional posterior, setting the parameter of the missing value to 0. Another option is use the observed features to create another linear regression model to predict the missing value and populate the missing value with the predictive mean.
It turns out these are equivalent, but im not sure how to prove this theoretically or mathematically? Any suggestions would be appreciated.