gradient wrt transition probabilities

Using the chain rule,

 

By differentiating eqn.1.21 wrt we get,

 

and differentiating (a time shifted version of) eqn 1.2 wrt

 

Eqns. 1.23,1.24 and 1.25 give, , and substituting this quantity in eqn.1.22 (keeping in mind that in this case), we get the required result,

 

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