Right, it’s (very roughly speaking) an approximation to an algorithm which maximises the probability of the entire response by factorising the distribution as a

Right, it’s (very roughly speaking) an approximation to an algorithm which maximises the probability of the entire response by factorising the distribution as a sequence of conditional distributions. If the approximation works, and the training data is coherent, it will produce coherent responses. The surprising thing to me is that this does work so much of the time.