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From: nemo@INRS-Telecom.UQuebec.CA (Capt. Nemo Semret)
Subject: Re: HMM-isolated word recognizer
Message-ID: <1994Jul11.191008.24306@INRS-Telecom.UQuebec.CA>
Organization: INRS Telecommunications
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References: <2va0ne$hhj@prakinf2.PrakInf.TU-Ilmenau.DE> <LUCKE.94Jul8105817@atrq28.itl.atr.co.jp>
Date: Mon, 11 Jul 1994 19:10:08 GMT
Lines: 63

Helmut Lucke (lucke@itl.atr.co.jp) wrote:

: Whether you use Viterbi decoding or forward-backward calculations, your
: likelihood should always increase. If it doesn't you certainly have
: a bug in your programme. My advice: check your code again.

: The reason why Likelihood monotonically increases for the Viterbi training
: as well, is simple:

: Consider first the problem of supervised training where we are given
: the `correct' state sequence. In this case the re-estimation 
: equations will give you a monotonically increasing likelihood for
: each iteration. (Infact since the HMM is no longer `hidden' you will
: get the optimal parameters in the first iteration and the parameters will
: remain constant from then onwards, but this is not the point here).

: Now if at each iteration you are allowed to choose the optimal
: path you can only do better: The same state path as the previous
: iteration would already result in a likelihood at least as good as in the
: previous iteration. 

I believe that's not necessarily the case. The iteration might have
improved the total likelihood while worsening the likelihood of that
particular state path.

: If the viterbi algorithm yealds an even better path, then
: so be it. The Likelihood  must be even higher in this case.

For example, suppose there are only two paths, call them a and b. Let
V(n,a) be the likelihood (Viterbi score) of path a after n
iterations. Suppose, we have

V(n,a)   > V(n,b)
V(n+1,b) > V(n+1,a)

ie after re-estimation, the optimal path has changed. The total
likelihood is guaranteed to increase so

p(n,a)V(n,a) + p(n,b)V(n,b) < p(n+1,a)V(n+1,a) + p(n+1,b)V(n+1,b)

with p(n,a)+p(n,b)=1. 

Now you could still have 

V(n,a) > V(n+1,b) > V(n+1,a)

ie the Viterbi score has not improved, neither on the old optimal
path, nor on the new one.

So in theory, to be guaranteed a monotonically increasing score, I
think you have to use the forward-backward score. In practice, you
probably almost always get an increasing Viterbi socre, because the
optimal path outscores the others by orders of magnitude, and so the
total likelihood is almost the same as the Viterbi likelihood.

Regards,

	-nemo-
-- 
C N aturally
A E xpressing
P M y
T O pinions
