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The gradient of L(Theta) is obtained by computing, for each theta_{ij}:

pdL(Theta)/pdtheta_{ij} =
dlogp(x_{q} = a, e)/dtheta_{ij} -
dlogp(e)/dtheta_{ij}.

This expression (derivation can be found
in [10]) is:
dL(Theta)/dtheta_{ij} =
p(z'_{i} = j|x_{q} = a, e)/theta_{ij} -
p(z'_{i} = j|e)/theta_{ij},

which can be obtained through standard Bayesian network algorithms using
local computations. A conjugate gradient descent can be constructed by
selecting an initial value for Theta and, at each step, normalizing
the values of Theta to ensure they represent proper
distributions [31].

© Fabio Cozman[Send Mail?]

Fri May 30 15:55:18 EDT 1997