What does this equation mean?

Constitutional AI keeps RLHF’s reward-model-and-KL-penalty backbone intact and changes where the comparison labels come from. Bai and colleagues describe a two-phase method: a supervised phase in which the model critiques and revises its own responses against a written set of principles, and a reinforcement phase in which a model, rather than a human, judges which of two candidate responses better satisfies those principles — producing an AI-generated preference dataset that trains the reward model [ 6 ] . Formally, this changes only the source of the comparison label. The Bradley–Terry equation above is unchanged in form; what changes is that the probability being fitted is now $P_{\mathrm{AI}}(y_A \succ y_B \mid x, C) = \sigma\big(r_\psi(x,y_A,C) - r_\psi(x,y_B,C)\big)$. a judgment conditioned on a written constitution C rather than on an anonymous rater’s unstated standards. Lee and colleagues tested this substitution directly at scale and reported that AI-generated feedback achieved comparable performance to human feedback across summarization and dialogue tasks, and that a direct variant skipping the intermediate reward model entirely outperformed the standard two-stage form [ 7 ] .