Posterior covariance matrix (${\Sigma }_y=\mathbb{E}[(\theta -m_y)(\theta -m_y{)}^T\mid y]$)

For a vector parameter with finite second moments and posterior mean $m_y$, the posterior covariance matrix records its uncertainty and linear dependence after conditioning on the observations. Its diagonal entries are posterior variances. In a regular locally flat-prior normal distribution approximation it is the inverse observed information. A quadratic form has posterior expectation equal to the form at the posterior mean plus the trace of its matrix times this covariance.