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While likelihood is attractive in theory, its estimates by deep generativemodels (DGMs) are often broken in practice, and perform poorly for out ofdistribution (OOD) Detection. Various recent works started to consideralternative scores and achieved better performances. However, such recipes donot come with provable guarantees, nor is it clear that their choices extractsufficient information.  We attempt to change this by conducting a case study on variationalautoencoders (VAEs). First, we introduce the likelihood path (LPath) principle,generalizing the likelihood principle. This narrows the search for informativesummary statistics down to the minimal sufficient statistics of VAEs'conditional likelihoods. Second, introducing new theoretic tools such as nearlyessential support, essential distance and co-Lipschitzness, we obtainnon-asymptotic provable OOD detection guarantees for certain distillation ofthe minimal sufficient statistics. The corresponding LPath algorithmdemonstrates SOTA performances, even using simple and small VAEs with poorlikelihood estimates. To our best knowledge, this is the first provableunsupervised OOD method that delivers excellent empirical results, better thanany other VAEs based techniques. We use the same model as\cite{xiao2020likelihood}, open sourced from:https://github.com/XavierXiao/Likelihood-Regret

Rethinking Test-time Likelihood: The Likelihood Path Principle and Its Application to OOD Detection