Supplementary Material Density-driven Regularization for Out-of-distribution Detection A.1 Proof of lemma 1

If Eq.(3) holds, then null Lemma 2. If Eq.(3) holds, then Υ For independent and identically distributed (i.i.d.) random vector Let g ([x, y,z ]) = y/x z. Proposition 1. subtracting a fixed constant from the classification logits leads to the same consistency OOD datasets to verify the effectiveness of the proposed two regularization terms. The result is the average value across all OOD datasets.ablation Fig.4 shows the distribution of log-likelihood values