COBALT: Censored Optimization and Bayesian Active Learning Techniques

Andrea Karlova, Rishabh Kabra, Daniel Augusto de Souza, Brooks Paige
Proceedings of the 42nd Conference on Uncertainty in Artificial Intelligence, PMLR 337:2713-2743, 2026.

Abstract

We target {Bayesian} Active Learning ({AL}) and Optimization (BO) for censored data regimes. While the Tobit likelihood accurately models such clipped observations, its mixed continuous-discrete nature impedes the analytical evaluation of information-theoretic acquisition functions. To address this, we investigate Censored Optimization via {Bayesian} Active Learning Techniques ({COBALT}). We establish rigorous theoretical guarantees for this framework, proving posterior consistency and the asymptotic normality of the {MAP} estimator under greedy maximization. Central to our framework is the derivation of a closed-form entropy for the Censored Normal distribution, enabling an analytical {BALD} ({Bayesian} Active Learning by Disagreement) score compatible with any {Gaussian} posterior approximations. We further underpin this method by deriving a numerically stable Evidence Lower Bound ({ELBO}) for censored atoms, utilizing robust approximations of the log-normal cumulative density. Empirical evaluations using our open-source implementation demonstrate {COBALT}’s best accuracy–compute trade-off among censored-likelihood methods in learning GP posteriors and effectiveness on a variety of benchmarks.

Cite this Paper

BibTeX
@InProceedings{pmlr-v337-karlova26a, title = {{COBALT}: Censored Optimization and {Bayesian} Active Learning Techniques}, author = {Karlova, Andrea and Kabra, Rishabh and de Souza, Daniel Augusto and Paige, Brooks}, booktitle = {Proceedings of the 42nd Conference on Uncertainty in Artificial Intelligence}, pages = {2713--2743}, year = {2026}, editor = {Perković, Emilija and Malinsky, Daniel}, volume = {337}, series = {Proceedings of Machine Learning Research}, month = {17--21 Aug}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v337/main/assets/karlova26a/karlova26a.pdf}, url = {https://proceedings.mlr.press/v337/karlova26a.html}, abstract = {We target {Bayesian} Active Learning ({AL}) and Optimization (BO) for censored data regimes. While the Tobit likelihood accurately models such clipped observations, its mixed continuous-discrete nature impedes the analytical evaluation of information-theoretic acquisition functions. To address this, we investigate Censored Optimization via {Bayesian} Active Learning Techniques ({COBALT}). We establish rigorous theoretical guarantees for this framework, proving posterior consistency and the asymptotic normality of the {MAP} estimator under greedy maximization. Central to our framework is the derivation of a closed-form entropy for the Censored Normal distribution, enabling an analytical {BALD} ({Bayesian} Active Learning by Disagreement) score compatible with any {Gaussian} posterior approximations. We further underpin this method by deriving a numerically stable Evidence Lower Bound ({ELBO}) for censored atoms, utilizing robust approximations of the log-normal cumulative density. Empirical evaluations using our open-source implementation demonstrate {COBALT}’s best accuracy–compute trade-off among censored-likelihood methods in learning GP posteriors and effectiveness on a variety of benchmarks.} }
Endnote
%0 Conference Paper %T COBALT: Censored Optimization and Bayesian Active Learning Techniques %A Andrea Karlova %A Rishabh Kabra %A Daniel Augusto de Souza %A Brooks Paige %B Proceedings of the 42nd Conference on Uncertainty in Artificial Intelligence %C Proceedings of Machine Learning Research %D 2026 %E Emilija Perković %E Daniel Malinsky %F pmlr-v337-karlova26a %I PMLR %P 2713--2743 %U https://proceedings.mlr.press/v337/karlova26a.html %V 337 %X We target {Bayesian} Active Learning ({AL}) and Optimization (BO) for censored data regimes. While the Tobit likelihood accurately models such clipped observations, its mixed continuous-discrete nature impedes the analytical evaluation of information-theoretic acquisition functions. To address this, we investigate Censored Optimization via {Bayesian} Active Learning Techniques ({COBALT}). We establish rigorous theoretical guarantees for this framework, proving posterior consistency and the asymptotic normality of the {MAP} estimator under greedy maximization. Central to our framework is the derivation of a closed-form entropy for the Censored Normal distribution, enabling an analytical {BALD} ({Bayesian} Active Learning by Disagreement) score compatible with any {Gaussian} posterior approximations. We further underpin this method by deriving a numerically stable Evidence Lower Bound ({ELBO}) for censored atoms, utilizing robust approximations of the log-normal cumulative density. Empirical evaluations using our open-source implementation demonstrate {COBALT}’s best accuracy–compute trade-off among censored-likelihood methods in learning GP posteriors and effectiveness on a variety of benchmarks.
APA
Karlova, A., Kabra, R., de Souza, D.A. & Paige, B.. (2026). COBALT: Censored Optimization and Bayesian Active Learning Techniques. Proceedings of the 42nd Conference on Uncertainty in Artificial Intelligence, in Proceedings of Machine Learning Research 337:2713-2743 Available from https://proceedings.mlr.press/v337/karlova26a.html.

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