Parameter Estimation

Abstract

The overview of the Kalman filter estimation procedure in figure 4.1 shows the likelihood function and the maximization procedure as the last relevant steps. We start to examine the appropriate log-likelihood function within our framework. Let ψ ∈ ψ ⊂ ℝⁿ be the vector of the n unknown parameters, which various system matrices of the state space model formulated in equations (3.1) and (3.2), or (4.10) and (4.11), as well as the variances of the measurement errors depend on. The likelihood function of the state space model is given by the joint density of the observational data y=(y _T ,y _T-1,...,y₁)

$$ l\left( {y;\Psi } \right) = p\left( {{y_T},{y_{T - 1}}, \ldots ,{y_1};\Psi } \right) $$

which reflects how likely it would have been to have observed the data if ψ were the true values for the parameters. Using the definition of conditional probability, we can split the likelihood up into conditional densities using Bayes’s theorem recursively and write the joint density as the product of conditional densities

$$l\left( {y;\Psi } \right) = p\left( {{y_T}|{y_{T - 1}}, \ldots ,{y_1};\Psi } \right) \ldots p\left( {{y_t}|{y_{t - 1}}, \ldots ,{y_1};\Psi } \right) \ldots p\left( {{y_1};\Psi } \right) $$

((5.1))

, where we approximate the initial density function p (y₁; ψ) by p (y₁|y₀; ψ).