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From: lucke@itl.atr.co.jp (Helmut Lucke)
Subject: Re: Training HMM's
In-Reply-To: deisher@trcsun3.eas.asu.edu's message of Tue, 18 Jan 1994 15:54:09 GMT
Message-ID: <LUCKE.94Jan19103430@atrq28.itl.atr.co.jp>
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References: <1994Jan17.231950.9326@ccu1.aukuni.ac.nz> <CJu1IB.DJF@ennews.eas.asu.edu>
Date: Wed, 19 Jan 1994 01:34:30 GMT
Lines: 42


In article <CJu1IB.DJF@ennews.eas.asu.edu> deisher@trcsun3.eas.asu.edu (Michael E. Deisher) writes:

   In article <1994Jan17.231950.9326@ccu1.aukuni.ac.nz>, mmt@ccu1.aukuni.ac.nz (Mark Thomson) writes:

   > However, by my reckoning, the rhs of the last line is equal to
   > 
   > p(O_1...O_t occur and in state i at t) * 
   >               p(O_t+1...O_T occur | (O_1...O_t occur _and_ in state i at t))
   > 
   > The first term is clearly alpha_t(i), but it seems to me that the second
   > term is _not_ beta_t(i) as required, because it is conditioned on _both_
   > the state at t and the observation sequence up to t.

   The second term is equal to beta_t(i) if the observations are assumed
   to be statistically independent.  That is,

   p(O_t+1...O_T occur | O_1...O_t occur) = p(O_t+1...O_T occur)

   I think most authors make this assumption (e.g., see Rabiner, Feb. 89
   Proc.  IEEE, p. 262, text surrounding eqn 13).

   --Mike

The actual assumption made is that of CONDITIONAL independence, that is
knowing that we are in state i at time t everything that occured before
t is independent of what occurrs after t, or mathematically

p(O_1 ... O_T | state i at t) = 
         p(O_1 ... O_t | state i at t ) p(O_t+1 ... O_T | state i at t )

An uncoditional independence assumption would be too strong as it would 
imply that P(O_1 ... O_T) = P(O_1)P(O_2)...P(O_T) and thus
essentially remove the chronological ordering of the observation vectors.

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Helmut Lucke                                <lucke@itl.atr.co.jp>
ATR Interpreting Telecommunication Research Laboratories
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