Goal-directed optimisation requires information about the world

A research post on LessWrong extends a classic information-theory result to goal-directed optimization. The author proves that any agent that reduces the KL divergence between its environment and a goal distribution must have mutual information with the world's initial state. The theorem parallels the Touchette-Lloyd theorem for entropy reduction, and the post provides an explicit algorithm for computing the bound. The work, part of a research fellowship, used AI language models to assist with proofs, though the writing is entirely human.

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