A Probabilistic Circuit-Induced Pseudo-Metric for Out-of-Distribution Detection

Traditional out-of-distribution (OOD) detection often relies on scalar likelihoods computed at the root of a generative model. This approach frequently discards the rich, hierarchical probabilistic information encoded within the model's internal structure. The authors investigate whether this internal hierarchy can be leveraged to create a more robust, statistically calibrated OOD detector that does not require held-out in-distribution data for calibration.

To address this, the authors introduce the Hierarchical Likelihood Vector (HLV), which represents an observation as a vector of likelihoods evaluated at selected nodes within a Probabilistic Circuit (PC). By comparing the expectations of these vectors between the learned distribution and a test batch, they define the Hierarchical Likelihood Distance (HLD). Because the PC is a tractable generative model, the authors demonstrate that the population mean and covariance of these HLVs can be computed exactly and analytically. This allows for the construction of a goodness-of-fit hypothesis test with a closed-form decision threshold, bypassing the need for reference samples at deployment.

The study shows that HLD functions as an integral probability metric over a function class naturally induced by the PC's structure. Experiments on tabular and MNIST datasets indicate that this method consistently outperforms traditional root-likelihood, uncertainty-based, and kernel-based baselines. Furthermore, the HLV representation provides a natural mechanism for localizing distribution shifts, as specific PC nodes can be identified as being responsible for the discrepancy between the ID and OOD distributions.

This work provides a principled, computationally efficient framework for OOD detection that fully exploits the architectural advantages of Probabilistic Circuits. By removing the requirement for held-out ID data, the method is particularly well-suited for real-world deployment scenarios where access to clean, representative in-distribution data is limited or impossible to obtain after the model is trained.