Perturbative adaptive importance sampling for Bayesian LOO cross-validation
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arXiv
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| Format: | Preprint |
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2024
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| author | Chang, Joshua C Li, Xiangting Su, Tianyi Xu, Shixin Yao, Hao-Ren Porcino, Julia Chow, Carson |
| author_facet | Chang, Joshua C Li, Xiangting Su, Tianyi Xu, Shixin Yao, Hao-Ren Porcino, Julia Chow, Carson |
| contents | 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 $T(θ)=θ+ h Q(θ)$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_08151 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Perturbative adaptive importance sampling for Bayesian LOO cross-validation Chang, Joshua C Li, Xiangting Su, Tianyi Xu, Shixin Yao, Hao-Ren Porcino, Julia Chow, Carson Methodology Artificial Intelligence Machine Learning Spectral Theory Statistics Theory 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 $T(θ)=θ+ h Q(θ)$ 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. |
| title | Perturbative adaptive importance sampling for Bayesian LOO cross-validation |
| topic | Methodology Artificial Intelligence Machine Learning Spectral Theory Statistics Theory |
| url | https://arxiv.org/abs/2402.08151 |