If we combine independent observables and refine against all of them at once, then our uncertainty will be larger than for each individual observable. If observables are uncorrelated, then the probability distribution of the observable vector q(y) factorize into probability distributions for individual components, q(y) = M i=1 q(y i ). The same is true for p(y). The total KL divergence then becomes a sum over the KL divergences for individual components of the observable vector,