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gradient wrt observation probabilities

Using the chain rule,


Differentiating (a time shifted version of) the eqn.1.2 wrt tex2html_wrap_inline2908


Finally we get the required probability, by substituting for tex2html_wrap_inline2914 in eqn.1.22 (keeping in mind that tex2html_wrap_inline2916 in this case), which is obtained by substituting eqns.1.28 and 1.24 in eqn.1.27.


Usually this is given the following form, by first substituting for tex2html_wrap_inline2808 from eqn.1.21 and then substituting from eqn.1.14.


If the continuous densities are used then tex2html_wrap_inline2928 can be found by further propagating the derivative tex2html_wrap_inline2930 using the chain rule.
The same method can be used to propagate the derivative (if necessary) to a front end processor of the HMM. This will be discussed in detail later.

Narada Warakagoda
Fri May 10 20:35:10 MET DST 1996

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