Perturbative adaptive importance sampling for Bayesian LOO cross-validation
Importance sampling (IS) is an efficient stand-in for model refitting in performing (LOO) cross-validation (CV) on a Bayesian model. IS inverts the Bayesian update for a single observation by reweighting posterior samples. The so-called importance weights have high variance -- we resolve this issue through adaptation by transformation. We observe that removing a single observation perturbs the posterior by $\mathcal{O}(1/n)$, motivating bijective transformations of the form $T(\theta)=\theta + h Q(\theta)$ for $0<h\ll 1.$ We introduce several such transformations: partial moment matching, which generalizes prior work on affine moment-matching with a tunable step size; log-likelihood descent, which partially invert the Bayesian update for an observation; and gradient flow steps that minimize the KL divergence or IS variance. The gradient flow and likelihood descent transformations require Jacobian determinants, which are available via auto-differentiation; we additionally derive closed-form expressions for logistic regression and shallow ReLU networks. We tested the methodology on classification ($n\ll p$), count regression (Poisson and zero-inflated negative binomial), and survival analysis problems, finding that no single transformation dominates but their combination nearly eliminates the need to refit.