While likelihood is attractive in theory, its estimates by deep generative models (DGMs) are often broken in practice, and perform poorly for out of distribution (OOD) Detection. Various recent works started to consider alternative scores and achieved better performances. However, such recipes do not come with provable guarantees, nor is it clear that their choices extract sufficient information. We attempt to change this by conducting a case study on variational autoencoders (VAEs). First, we introduce the likelihood path (LPath) principle, generalizing the likelihood principle. This narrows the search for informative summary statistics down to the minimal sufficient statistics of VAEs' conditional likelihoods. Second, introducing new theoretic tools such as nearly essential support, essential distance and co-Lipschitzness, we obtain non-asymptotic provable OOD detection guarantees for certain distillation of the minimal sufficient statistics. The corresponding LPath algorithm demonstrates SOTA performances, even using simple and small VAEs with poor likelihood estimates. To our best knowledge, this is the first provable unsupervised OOD method that delivers excellent empirical results, better than any other VAEs based techniques. We use the same model as \cite{xiao2020likelihood}, open sourced from: https://github.com/XavierXiao/Likelihood-Regret
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