Anomaly detection goal is to detect unexpected behaviours in the data. Because anomaly detection is usually an unsupervised task, traditional anomaly detectors learn a decision boundary by employing heuristics based on intuitions, which are hard to verify in practice. This introduces some uncertainty, especially close to the decision boundary, which may reduce the user trust in the detector's predictions. A way to combat this is by allowing the detector to reject examples with high uncertainty (Learning to Reject). This requires employing a confidence metric that captures the distance to the decision boundary and setting a rejection threshold to reject low-confidence predictions. However, selecting a proper metric and setting the rejection threshold without labels are challenging tasks.

Providing a very low critical probability pc means that certification occurs when the simulation ends after a large number of iterations m. We introduce `c the threshold associated to pc s.t. Table 5 shows that this approximation is excellent even for large pc. This shows that mis a little larger than mc = log(pc)/log(1 1/N). This section assumes that X = xo + σ X with X N(0n; In) and that h(x) = x>g τ with g Rn and kgk= 1 (w.l.o.g.).

Diffusion models [1, 2, 3] have emerged as a powerful class of generative models, achieving state-of-the-art performance across a wide range of applications, including imaging [2] and scientific-data synthesis [4]. From a statistical perspective, they can be viewed as flexible nonparametric estimators of a (conditional) distribution via score estimation and reverse-time stochastic differential equations (SDEs) [5, 6]. Despite this expressive power, standard diffusion models are typically causality-agnostic: they learn a joint law without encoding the directional asymmetries required for causal interpretation. As a consequence, they do not, on their own, provide principled answers to interventional queries or support broader causal analyses, which are central to structural causal models (SCMs) [7]. When a causal ordering (or a directed acyclic graph) is available, it is natural to construct generative procedures that sample variables sequentially according to the causal factorisation. Such iterative, ordering-respecting approaches have been proposed using a variety of generative models, including generative adversarial networks [8], variational autoencoders [9], normalising flows [10], and diffusion-based constructions such as DDIM [11]. However, a rigorous statistical understandingof the advantages of exploitingsuch causalstructureand the inferential use of the resulting generator remain less developed.

Unsafe operations described in natural text lead to a lower rejection rate than thoseincodeformat.

Providing guarantees on the decision-making processes of autonomous systems, often based on complex black-box machine learning models, is paramount for their safe deployment. This need motivates efforts towards responsible artificial intelligence, which broadly entails questions of reliability, robustness, fairness, and interpretability.