 Solution

The scale separation makes correlations between distinct observation times negligible, so $K_{\theta }\simeq AI$. Put $v_i=A+{\sigma }_i^2$. With a flat prior,The next latent value is likewise approximately independent of the past conditional on $\mu$, with $f_{\ast }\mid \mu \sim N(\mu ,A)$. Marginalizing $\mu$ givesWhen every ${\sigma }_i=0$, $\bar{\mu }=\bar{y}$, $V_{\mu }=A/N$, and the predictive variance is $A(1+1/N)$.