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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.