This morning, 2 students asked me about the proof of the recursion formula for the forward variable (slide 13/36).
Actually we were not "brutal" enough to finalize the proof: we did not introduce enough extra variables: previous tag is not enough if we want to go back as the original two hypothesis.
Back from lunch, I decided to address it "brutal force", introducing ALL intermediate variables so as to get back to HMM main formula (with all terms).
Then it's much easier as proven in the hereby attached PDF (quickly written; sorry for typos, if any).
Thanks for the question once again. I didn't practice these tricks for a while and was happy to come back to them.
