Of my dataset $X$ I want to calculate the likelihood using my GMM model (for BIC). Mathematically it seems to make sense that as the samples are independant $P(X) = P(X_1)P(X_2)..$, so I would get the likelihood by taking the product of the likelihoods per sample.
But it doesn't seem intuitively correct that the more samples I have the less likely my model. It also means that if just one sample that doesn't fit will crash my likelihood, it therefore seems to make more sense (to me) to average the likelihoods per sample.
What's the right thing to do here (I feel I'm messing some very basic statistics up)?