  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.
