The likelihood for a state space model

This paper derives an expression for the likelihood for a state space model. The expression can be evaluated with the Kalman filter initialized at a starting state estimate of zero and associated estimation error covariance matrix of zero. Adjustment for initial conditions can be made after filtering. Accordingly, initial conditions can be modelled without filtering implications. In particular initial conditions can be modelled as 'diffuse'. The connection between the 'diffuse' and concentrated likelihood is also displayed.