We study the problem of mapping parameter-space variance of a 3D Morphable Model (3DMM) to per-vertex variance. Traditionally, per-vertex variance has been used to visually interpret the learnt parameter uncertainty. However, mapping parameter-space variance to per-vertex using sampling or error propagation is computationally expensive. We address this limitation with Closed-form Local Uncertainty Estimation (CLUE), which transforms the parameter-space distribution of 3DMMs to per-vertex through a closed-form approximation. CLUE emerges from applying the law of total variance to the 3DMM formulation and is modular, computationally inexpensive, and accurate. Subsequently, we study integrating CLUE for in-the-wild human mesh recovery training and inference. We note that CLUE uses the geometry of the mesh as an inductive bias when predicting the variance, leading to improved accuracy. Through extensive qualitative and quantitative experiments, we show that when used in training, CLUE (1) outpeforms probabilistic and deterministic baselines on metrics evaluating joints, mesh and uncertainty, and (2) has the highest uncertainty-to-error spearman correlation, with applications including data mining and interpretable uncertainty. Finally, we share SAM 3D Body (CLUE) which augments the regressor with high resolution uncertainty maps. We make our code available at https://github.com/vita-epfl/CLUE

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