The Doob Decomposition Theorem states that any submartingale can be uniquely decomposed into a martingale and a predictable increasing process. In LLM decoding, it provides a principled method to measure a reasoning path's 'predictable advantage' for step valuation, moving beyond ad-hoc heuristics.
The Doob Decomposition Theorem is a mathematical tool that breaks down a certain type of random process into two parts: one that's unpredictable (like a fair coin flip) and one that's predictable (like a steady trend). In advanced AI, it helps models like Martingale Foresight Sampling make better decisions by precisely measuring the 'predictable advantage' of different reasoning paths.
Was this definition helpful?