Solution
= Solution
For draw $m$, let $C_m$ be the observed covariance matrix, $k_{*m}=(k_{\theta_m}(t_*,t_i))_i$, and
$$
m_{*m}=\mu_m+k_{*m}^TC_m^{-1}(y-\mu_m\mathbf1),\qquad
v_{*m}=A_m-k_{*m}^TC_m^{-1}k_{*m}.
$$
The posterior predictive distribution is a mixture of these conditional Gaussians. Its Monte Carlo mean and variance are
$$
\widehat m_*=\frac1M\sum_mm_{*m},\qquad
\widehat v_*=\frac1M\sum_m(v_{*m}+m_{*m}^2)-\widehat m_*^2.
$$