Median distance modulus |$\mu$| residual (MVZLTN VI |$-$| MCMC) versus PSIS |$\hat{k}$| diagnostic calculated from |$N = 5000$| parameter samples for the Foundation data set. Dotted line indicates |$\hat{k} = 0.7$|⁠, a threshold for a good variational approximation of the posterior (Vehtari et al. 2024).