cs.LGOct 21, 2026

Conditional Flow Models Convert Epistemic Uncertainty into Apparent Aleatoric Variance

Authors: Navodita Sharma, Abhishek Shetty, Max Simchowitz, Pulkit Agrawal

Organizations: Massachusetts Institute of Technology · Georgia Institute of Technology · Carnegie Mellon University

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Abstract

Conditional Flow Models (CFMs) are increasingly being used in place of deterministic predictors across machine learning, including for tasks that were classically formulated as regression. Given their success in such tasks, we ask the question: what useful information do the conditional distributions produced by CFMs carry beyond what a point predictor would provide? We identify a systematic phenomenon: in many natural regimes, CFM samples routinely have larger variance than the true conditional distribution, even in nearly deterministic regression problems. This effect is most pronounced where the model has not resolved the dependence on the conditioning variable, and thus the additional variance can be viewed as converting epistemic uncertainty in fitting the ground truth conditional distribution into apparent aleatoric variance in the generated samples. Through controlled synthetic and real-data experiments, and comparisons with explicit smoothing estimators, we show that the induced variance tracks model underfitting of complex or hard-to-fit targets, and decreases with additional training or data. We provide a theoretical account of the phenomenon in a simplified setting and demonstrate a consequent practical use-case of conditional flow-matching: designing high-coverage predictive sets that cover the ground-truth regression target.

Figures & tables

Figure 1: Flow models vs Regression during underfitting.
Figure 2: Any smoothing bias in vector-field learning induces variance inflation. In all three cases, the average learned conditional variance (reported in each panel title) decreases monotonically as smoothing decreases.
Figure 3: Policy Evaluation: learned (purple) and true (green) conditional distributions against the first two principal components of the state-action conditioning
Figure 4: Underfitting of the velocity field leads to local mixing of conditional distributions, not just smoothing of the mean.
Figure 5: Excess variance tracks squared bias. Colors identify the mean functions f⋆f^{\star} of the synthetic target distributions X1∣C=c∼N(f⋆(c),σ2)X_{1}\mid C=c\sim\mathcal{N}(f^{\star}(c),\sigma^{2}) , and marker shapes identify vector-field estimators. For neural networks, we plot results at different training checkpoints. For roughness-penalized neural networks and kernel ridge regression, we vary the regularization strength. Lines connect results for the same target–estimator pair. Squared bias and excess conditional variance are both averaged over CC . The corresponding conditional distributions are shown in Fig. 2 (blue) and Figs. 16 – 18 (green, red, and purple, respectively).
Figure 6: Policy Evaluation: Antmaze-Large
Figure 7: Cell-pertubation dataset: Norman
Figure 8: Behavioral Cloning
Appendix figures & tables11 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 9: Kernel approximation error and velocity field visualization
Figure 10: Visualization of density-weighted smoothing in the learned velocity field and the kernel approximation
Figure 11: Effect of kernel approximation on the endpoint distribution
DatasetnMCn_{\mathrm{MC}}HHγ\gammadim⁡s\dim sdim⁡a\dim adim⁡c\dim c∣D∣|\mathcal{D}|
Cube-single202000.9928533200,000
AntMaze-large2010000.99829837200,000
Humanoid-Maze-medium2020000.999692190200,000
Appendix
Table 1: nMCn_{\mathrm{MC}} is the number of independent on-policy rollouts averaged into Q^(s,a)\hat{Q}(s,a) , HH is the fixed rollout horizon, and γ\gamma is the discount factor.
Figure 12: Coverage on Cube-Single . Layout matches Figure 6 .
Figure 13: Coverage on Humanoidmaze-medium . Layout matches Figure 6 .
Figure 14: Coverage on PBMC. Layout matches Figure 7 .
Figure 15: Coverage on SciPlex3 (drug-dose perturbations). Layout matches Figure 7 .
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