SymmPI: Predictive Inference for Data with Group Symmetries

TL;DR

SymmPI extends predictive inference beyond exchangeability by leveraging distributional invariance under general group actions $\mathcal{G}$ and distributionally equivariant transforms $V$. The method constructs prediction regions using the $1-\alpha$ quantile of the test function $\psi$ evaluated over random group actions, ensuring finite-sample coverage while preserving the symmetry. It provides rigorous coverage guarantees, including under distribution shift, and offers a non-symmetric extension to handle shifts. The framework is demonstrated across exchangeable, network, rotationally invariant, and two-layer hierarchical settings with both simulations and real data, showing improved efficiency and adaptivity to heterogeneity relative to classical conformal prediction. The work also develops practical computational strategies, graph-based implementations, and supplementary theory to support broad applicability in complex, symmetry-rich data domains.

Abstract

Quantifying the uncertainty of predictions is a core problem in modern statistics. Methods for predictive inference have been developed under a variety of assumptions, often -- for instance, in standard conformal prediction -- relying on the invariance of the distribution of the data under special groups of transformations such as permutation groups. Moreover, many existing methods for predictive inference aim to predict unobserved outcomes in sequences of feature-outcome observations. Meanwhile, there is interest in predictive inference under more general observation models (e.g., for partially observed features) and for data satisfying more general distributional symmetries (e.g., rotationally invariant or coordinate-independent observations in physics). Here we propose SymmPI, a methodology for predictive inference when data distributions have general group symmetries in arbitrary observation models. Our methods leverage the novel notion of distributional equivariant transformations, which process the data while preserving their distributional invariances. We show that SymmPI has valid coverage under distributional invariance and characterize its performance under distribution shift, recovering recent results as special cases. We apply SymmPI to predict unobserved values associated to vertices in a network, where the distribution is unchanged under relabelings that keep the network structure unchanged. In several simulations in a two-layer hierarchical model, and in an empirical data analysis example, SymmPI performs favorably compared to existing methods.

Figures (4)

• Figure 1: Left: A $1-\alpha$ prediction region extracted from the orbit of $\tilde{z}$, where $\tilde{z}=V(z)$. Right: A $95\%$-prediction region $T^{\mathrm SymmPI}(z_{1:12})$ for $z_{13}$ defined in \ref{['rotation_example']}. Here we let $z_i^\top, i\in [13]$ be i.i.d. random vectors generated from $\mathcal{N}(0,30\cdot I_2)$, where $I_2$ is the $2\times 2$ identity matrix. The blue stars represent 12 observed values, while the white region is the $95\%$-prediction region, as defined in Equation \ref{['rotation_example']}. Thus, the pink region depicts the complement of the prediction region, which is "safe", in the sense of not having blue stars with 95% probability. In contrast, the 95% prediction region constructed via conformal prediction is the whole two-dimensional space. Consequently, using conformal prediction does not yield any "safe" area; and is thus not informative here.
• Figure 4: Empirical data example: Prediction set lengths and coverage probabilities for various methods with level $\alpha=0.10$. Left: Prediction set lengths; Right: Empirical coverage probabilities.
• Figure 5: Illustration of aircraft types from the MTARSI dataset.
• Figure 6: Prediction set lengths and coverage probabilities for various methods with level $\alpha=0.20$. Left: Prediction set lengths; Right: Empirical coverage probabilities.

Theorems & Definitions (22)

• Example 2.1: Permutation action
• Definition 2.2: Distributional equivariance
• Definition 4.1: Randomized SymmPI Prediction Set
• Theorem 4.2: Coverage guarantee under distributional invariance
• Proposition 4.3: Coverage upper bound
• Theorem 4.4: Coverage guarantee under distribution shift
• Proposition 4.5
• Proposition 5.1
• Proposition 5.2
• Theorem 5.3