The theory of full conformal prediction uses deterministic non-conformity measure, but modern usage of full conformal prediction often relies on machine learning training, making stochasticity inevitable. A simple sufficient condition of almost sure permutation invariance of the non-conformity measure can be too restrictive, so many have suggested the relaxation to permutation in distribution as a condition for full conformal prediction validity. We, however, show that this commonly known condition is actually insufficient. We then provide a correct sufficient condition: Conditional Independence & Permutation Invariance in Distribution, which encompasses several stochastic settings that may be used in machine learning.

A.1 Proof of Proposition 4.1 Proposition 4.1 For any bounded measurable function τ(t): [0, T ] R, the following Reverse SDEs [ (1 + τ Eq. (20) is a reverse-time SDE running[ from T to 0, thus (there)are two additional minus ] signs in Eq. (21) before term A.2 Two Reparameterizations and Exact Solution under Exponential Integrator In this subsection, we will show the exact solution of SDE in both data prediction reparameterization and noise prediction reparameterization. The noise term in data prediction has smaller variance than noise prediction ones, implying the necessity of adopting data prediction reparameterization for the SDE sampler. The computation of variance uses the Itô Isometry, which is a crucial fact of Itô integral. Similar with Proposition 4.2, Eq. (37) can be solved analytically, which is shown in the following propositon: Following the derivation in Proposition 4.2, the mean of the Itô integral term is: [ A.2.4 Comparison between Data and Noise Reparameterizations In Table 1 we perform an ablation study on data and noise reparameterizations, the experiment results show that under the same magnitude of stochasticity, the proposed SA-Solver in data reparameterization has a better convergence which leads to better FID results under the same NFEs. In this subsection, we provide a theoretical view of this phenomenon.

Diffusion Probabilistic Models (DPMs) have achieved considerable success in generation tasks. As sampling from DPMs is equivalent to solving diffusion SDE or ODE which is time-consuming, numerous fast sampling methods built upon improved differential equation solvers are proposed.

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