    \begin{align}
    \label{equ_post_mean}
    m_t(\phi(x)) &= \mathbf{k_{t-1}}(\phi(x))^{\top} \left(K_{t-1}+ \lambda I \right)^{-1} \tilde F_{t-1},\\
    \nonumber k_t \left(\phi(x), \phi(x^{\prime})\right) &= k\left(\phi(x), \phi(x^{\prime})\right) \\
    \label{equ_post_cov}& -\mathbf{k_{t-1}}(\phi(x))^{\top} \left(K_{t-1}+ \lambda I \right)^{-1}\mathbf{k_{t-1}}(\phi(x^{\prime})),
    \end{align}
