Are you thinking about a more hierarchical model? Like, there’s a common effect between studies that is some GP, and then the param for each study is drawn where that effect is the mean?
If so, you might want to try a hierarchical GP. In the ICM, the outputs have varying amounts of correlation, but there isn’t some global average effect. You’re learning a structure that’s more like, the outputs of study 1 and 2 are closely correlated, but not correlated with study 3. That’s what the matrix B is representing.