# The limits of distribution-free conditional predictive inference

## TL;DR

This paper explores the feasibility of distribution-free predictive inference that guarantees conditional coverage, analyzing the limitations and potential relaxations of existing marginal coverage methods like conformal prediction.

## Contribution

It investigates the theoretical boundaries of achieving conditional coverage guarantees without distributional assumptions, proposing relaxations to address practical concerns.

## Key findings

- Exact conditional coverage is impossible without assumptions
- Relaxations of conditional coverage can be achieved in a distribution-free setting
- The paper clarifies the trade-offs between marginal and conditional coverage

## Abstract

We consider the problem of distribution-free predictive inference, with the goal of producing predictive coverage guarantees that hold conditionally rather than marginally. Existing methods such as conformal prediction offer marginal coverage guarantees, where predictive coverage holds on average over all possible test points, but this is not sufficient for many practical applications where we would like to know that our predictions are valid for a given individual, not merely on average over a population. On the other hand, exact conditional inference guarantees are known to be impossible without imposing assumptions on the underlying distribution. In this work we aim to explore the space in between these two, and examine what types of relaxations of the conditional coverage property would alleviate some of the practical concerns with marginal coverage guarantees while still being possible to achieve in a distribution-free setting.