The Geometry of Ignorance: LLMs Know When to Temper Bayesian Priors
TL;DR
The direction of ignorance is causally active: raising or lowering $\lambda$ at the final prediction state steers the prediction toward or away from the unigram prior in KL divergence, with larger models generally exhibiting lower prior reliance in the high-context limit.
Abstract
What does a language model predict when it has few clues? The answer lurks in its unembedding geometry: a single direction of the unembedding matrix encodes the unigram distribution of the training corpus, which serves as the Bayesian prior the model falls back on when uncertain. This structure --- which we term the \emph{direction of ignorance} --- appears in all four model families examined (\texttt{Llama}, \texttt{Qwen}, \texttt{Gemma}, and \texttt{Pythia}), ranging from 0.4B to 405B parameters. Projecting the final prediction state onto this direction yields a per-token \emph{prior loading factor} $\lambda$, which, empirically, declines steadily as the context becomes more informative. Formally, the same projection decomposes the prediction state into two orthogonal vectors that correspond exactly to the two factors of a tempered Bayesian update: a unigram prior raised to the exponent $\lambda$ and a context-driven likelihood. This geometric-probabilistic interpretation calibrates $\lambda$, making it meaningfully comparable across model sizes and families, with larger models generally exhibiting lower prior reliance in the high-context limit. Finally, we show that the direction of ignorance is causally active: raising or lowering $\lambda$ at the final prediction state steers the prediction toward or away from the unigram prior in KL divergence.
