How to compare models with different dimensions

I have some data (~200 dim) and not that many examples (~300). I am trying to get the best generative model of the data. I am using GMMs to create the model however I am adding a PCA reduction prior to this and building the GMM over the reduced subspace, the problem arises in comparing distributions with different dimensionality. I attempted to use cross validation of the likelihood however the problem is in one model the input is 100 dim in another model the input is some other dimension size so my understanding is that i cannot compare these likelihoods directly since lower dimensional data will have more approximation error than the higher dimensional data(but less parameters and noise specific to the training data? one would hope) . Any ideas in how to compare these in order to choose the best dimension to represent the data?