Discussions (about the course or NLP in general)

It does! This is what I mentioned several times and very precisely when emphasizing the backward part of the Viterbi algorithm, but I know it's tricky (carefully look what the question is about): do not confuse what is computed with the way it is parametrized: it's not because we parametrize $P(T_1,...,T_n)$ with $P(T_i|T_{i-1})$ that we are not maximizing $P(T_1,...,T_n)$.

More precisely w.r.t. that very question: if you want the formal proof in this very case: $P(T_3|T_1 T_2 T_4) = P(T_1 T_2 T_3 T_4) / P(T_1 T_2 T_4) = P(T_1) P(T_2 | T_1) P(T_3 | T_2) P(T_4|T_3) / \sum_t P(T_1) P(T_2 | T_1) P(t | T_2) P(T_4|t) = P(T_1) P(T_2 | T_1) P(T_3 | T_2) P(T_4|T_3) / (P(T_1) P(T_2 | T_1) \sum_t P(t | T_2) P(T_4|t)) = P(T_3 | T_2) P(T_4|T_3) / \sum_t P(t | T_2) P(T_4|t)) = P(T_3 | T_2) P(T_4|T_3 T_2) / \sum_t P(t | T_2) P(T_4|t T_2)) = P(T_3 T_4| T_2) / \sum_t P(t T_4| T_2) = P(T_3 T_4| T_2) / P(T_4| T_2) = P(T_3| T_2 T_4)$. QED 

But pay attention to what is referred to: parameters, not considered altogether, I mean, not when talking about the global proba $P(T_1,...,T_n)$ are indeed only left to right : $P(T_i|T_{i-1})$; but as soon as it's about the whole (global) proba, there is no "left-to-right biais" anymore, only the global proba is under consideration.
I hope it makes sense....