Using the chain rule for any of the likelihoods, *free* or *clamped*,

Differentiating eqns.1.39 and 1.40 wrt
, to get two results for *free* and *clamped* cases, and
using the common result in eqn.1.28, we get substitutions for
both terms on the right hand side of eqn. 1.45. This substitution
yields two separate results for *free* and *clamped* cases.

where is a Kronecker delta. And

Substitution of eqns. 1.46 and 1.47 in eqn.1.38 we get the required result,

This equation can be given a somewhat ``nice'' form by defining,

where is a Kronecker delta, and

With these variables we express the eqn.1.48 in the following form.

This equation completely defines the update of observation probabilities. If however continuous densities are used then we can further propagate this derivative using the chain rule, in exactly the same way as mentioned in the case ML. A similar comments are valid also for preprocessors.

Fri May 10 20:35:10 MET DST 1996