  In-context learning (ICL) has emerged as a particularly remarkable
characteristic of Large Language Models (LLM): given a pretrained LLM and an
observed dataset, LLMs can make predictions for new data points from the same
distribution without fine-tuning. Numerous works have postulated ICL as
approximately Bayesian inference, rendering this a natural hypothesis. In this
work, we analyse this hypothesis from a new angle through the martingale
property, a fundamental requirement of a Bayesian learning system for
exchangeable data. We show that the martingale property is a necessary
condition for unambiguous predictions in such scenarios, and enables a
principled, decomposed notion of uncertainty vital in trustworthy,
safety-critical systems. We derive actionable checks with corresponding theory
and test statistics which must hold if the martingale property is satisfied. We
also examine if uncertainty in LLMs decreases as expected in Bayesian learning
when more data is observed. In three experiments, we provide evidence for
violations of the martingale property, and deviations from a Bayesian scaling
behaviour of uncertainty, falsifying the hypothesis that ICL is Bayesian.
