Conditional Flow Models Convert Epistemic Uncertainty into Apparent Aleatoric Variance

Counterfactual generation seeks to sample outcomes under a hypothetical intervention or decision using observational data collected under the factual assignment mechanism. We develop a flow-matching approach that combines a sample-split, doubly robust training objective with a learned coupling between observed source outcomes and target outcomes drawn from a fitted conditional outcome model. To enable finite-step generation, we leverage a score-corrected stochastic sampler based on a Gaussian-smoothed interpolation. Our main theoretical contribution is a coupling-sensitive KL bound for constant-step Euler discretization: the error is controlled by moments of the source--target displacement under the chosen coupling, rather than by global uniform regularity of the velocity field, and has near-linear dependence on the ambient dimension. We also establish finite-sample non-parametric guarantees for the learned velocity and score fields when both the conditional outcome model and the source-target coupling are estimated from data. These bounds separate approximation, coupling-replacement, nuisance-estimation, generalization, and Monte Carlo errors and, combined with the sampler analysis, yield an end-to-end guarantee for counterfactual generation. Experiments on synthetic and semi-synthetic image benchmarks support the coupling-dependent theory and show that, at finite discretization budgets, the stochastic sampler can outperform the corresponding deterministic ODE sampler.

Stream-level flow matching replaces the linear interpolant of conditional flow matching (CFM) by a Gaussian-process (GP) stream connecting each source--target pair, and reports lower sample error than \icfm{} on 2-Gaussian, MNIST and CIFAR-10 benchmarks. We ask what such a stream model actually contributes. Three results answer the question. (i)Reduction. The stream-level CFM objective depends on the stream law only through the per-time joint law of $(s_t,\sdot_t)$, so the conditional paths a Gaussian stream can reach are exactly the Gaussian conditional paths CFM already parametrises; in the coordinate-wise, shared-scalar-kernel construction gpcfm actually uses, the entire design space collapses to two scalar curves $(m_t,v_t)$, and cross-time covariance affects only estimator variance. (ii)The GP is a constrained chart of that space. One kernel sets both $m_t$ and $v_t$, so the paper's own recipe for widening coverage-shrinking the SE length-scale---destroys the interpolant (the midpoint mean weight falls from $1.03$ to $0.00$). On the 2-Gaussian benchmark this makes the GP chart diverge on $15/200$ runs at high coverage against $0/200$ for a decoupled $(m_t,v_t)$ chart ($p=6.6\times10^{-5}$), and crossing the two curves shows the divergence tracks the mean, not the variance. On MNIST the same sweep does not diverge and the ordering reverses, so whether the coupling is harmful is benchmark-dependent; what holds on both is that the recipe buys nothing---no coverage level beats the paper's own, and past $\max_t\sqrt{v_t}\approx0.6$ both charts degrade. (iii)~Audit. The released code does not implement the mechanism it describes: state and velocity are drawn independently ($\mathrm{corr}=0.00\pm0.01$ against an intended $\pm0.83$--$0.99$).

Conformal prediction (CP) is a post-hoc calibration method that returns prediction sets with distribution-free coverage guarantees, and the size of the output set is often interpreted as a notion of uncertainty. We show that the difference in set size between full and split CP is directly connected to Bayesian posterior variance, which is the epistemic uncertainty of the model. Essentially, this difference in set size is an aggregation of two opposite effects: (1) better model accuracy and therefore smaller set size achieved by training over extra calibration points, and (2) the "formability" of the model at test point towards every possible label. The latter effect vanishes with stronger models or more calibration data (as epistemic uncertainty is known for that behavior). It turns out that the model's formability itself (without the quantization effect of prediction sets, and through estimating the posterior variance) is a useful tool to quantify the epistemic uncertainty. In our empirical evaluation it performs on par with the state of the art in many setups.