The Geometry of Persuasion: Quantifying Belief Change in a Latent Embedding Space

Beliefs are not isolated attitudes but elements of structured systems, where the relative position of competing ideas shapes decision and belief revision. Yet, it remains unclear how the distance between beliefs governs belief choice and change in real-world settings. Here, we represent these processes within a geometric embedding space constructed from thousands of online debates. By embedding users and opposing belief positions into a shared latent space using a fine-tuned language model, we define a user-conditioned distance margin that captures the relative proximity between competing beliefs. We find that both belief choice and belief change follow simple and highly regular patterns: the probability of selecting the farther belief decreases monotonically with this margin and is well described by a logistic function. Analysis of belief change reveals that persuasion events are rare and become increasingly unlikely as the distance margin grows. Yet, the expected magnitude of belief change peaks at intermediate margins, yielding a robust inverted-U relationship consistent with Social Judgment Theory (SJT). Across domains, the same geometric mechanism applies, but both baseline persuasion rates and sensitivity to belief distance vary substantially, leading to domain-specific optimal distances at which persuasion is most effective. Together, our findings suggest that belief change can be understood as probabilistic, distance-dependent transitions in a structured belief space.