Been watching this video by Tom Minka on Expectation Propagation (http://videolectures.net/mlss09uk_minka_ai/). At about 19:12, he says that the reason the moment matching technique works is that when you multiply a lot of likelihood terms together, the resulting posterior distribution is compact and its variance is very small and it will look more Gaussian (assuming IID). At some point in the video, he says this is because of the central limit theorem. However, However, I am unable to see how the CLT applies to multiplying such likelihoods and why this should approximate a compact Gaussian.
Would appreciate any thoughts/suggestions on understanding this.