Exchangeability and randomness for infinite and finite sequences

TL;DR

This paper explores the relationship between exchangeability and randomness in sequences, highlighting their equivalence in infinite sequences but emphasizing their differences in finite sequences, which impacts statistical modeling.

Contribution

It clarifies the distinction between exchangeability and randomness for finite sequences, an area less understood compared to infinite sequences.

Findings

Exchangeability and randomness are nearly equivalent for infinite sequences.

Differences between the assumptions become significant in finite sequences.

Implications for nonparametric statistics and machine learning models.

Abstract

Randomness (in the sense of being generated in an IID fashion) and exchangeability are standard assumptions in nonparametric statistics and machine learning, and relations between them have been a popular topic of research. This short paper draws the reader's attention to the fact that, while for infinite sequences of observations the two assumptions are almost indistinguishable, the difference between them becomes very significant for finite sequences of a given length.