I am struggling with the following question, as illustrated as well per below. I have a prior multivariate distribution, with correlation between the variables. I have obtained additional data on one variable. I would like to use the information to update my prior distribution.
I am struggling on what the best and most efficient way is to do this, and I would like to find a method that can be scaled to higher dimensional and non-conjugate distributions. I was thinking of
- approximating each marginal,
- calculating the correlation / co-variance matrix,
- performing a standard Bayesian update on the marginal of interest, and then
- using the updated marginal and the other un-changed marginals and the co-variance matrix to draw samples and approximate the updated posterior.
However, this feels quite laborious and sub-optimal. What are better ways to tackle this?