Page 158 - Contributed Paper Session (CPS) - Volume 4

CPS2164 Jonathan Hosking et al.

Substituting (11) into (10), and evaluating the at and Pt occurring in (11)
by iterative application of (12)–(13), yields a set of recursions from which the
derivative of the log-likelihood can be computed iteratively. Detailed
expressions have been given for example by (Zadrozny, 1989; Anderson et al.,
1995; Nagakura, 2013). However, by investigating the derivatives in more detail
we have obtained explicit expressions for the likelihood derivative in terms of
quantities routinely computed in Kalman filtering and smoothing. Details are
omitted here, but are available from the authors. The final result is

5. Practical implementation
Practical use of likelihood derivatives in model fitting requires an efficient
procedure for computing the likelihood and its derivatives. with respect to the
model’s parametrs. For each system matrix M, a practical way of computing
the contribution to the likelihood derivative is the elementwise product. of the
matrices ∂logL/∂M and M. For example, for a parameter occurring only in
matrices Tt and P1 the log-likelihood derivative with respect to θ can be
computed by the chain rule as

From (14) it is straightforward to obtain derivatives with respect to any of the

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