 Gaussian innovation likelihood

In a linear Gaussian state-space model, let the predicted state mean and covariance be $m_t^{-},P_t^{-}$. The observation innovation process has conditional mean zero and covariance ${\Sigma }_t=A{P}_t^{-}{A}^{\top }+R$, with innovation ${ϵ}_t=y_t-A{m}_t^{-}$. The conditional multivariate normal distribution gives the predictive observation density. Multiplying these conditional densities by the chain rule gives the displayed likelihood function; equivalently the innovations are independent Gaussian vectors. The Kalman filter supplies $m_t^{-},P_t^{-}$ recursively, so one evaluates the exact observed-data likelihood function without integrating all latent states jointly.