= Gaussian innovation likelihood
{c}
{title2=$\ell=-\frac12\sum_t[\log\det\Sigma_t+\epsilon_t^\top\Sigma_t^{-1}\epsilon_t+q\log(2\pi)]$}
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> $\Sigma_t=AP_t^-A^\top+R$, with innovation $\epsilon_t=y_t-Am_t^-$. 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.
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