A Human-guided AI approach for Symbolic Cumulant Calculation

Cumulants play an important role in probability and statistics, but explicit formula for high-order cumulants are hard to derive, especially in multi-variate case. To tackle the problem, we develop a human-guided AI (hugAI) approach. First, we outline a clear strategy by developing a graph-based theoretical framework. Second, we divide the task into many small steps, and for each step, we communicate with AI (sometimes for several rounds) by carefully written prompts. Finally, the AI execute the formula with a python code. This way, our hugAI approach is able to execute formulas for cumulants of multivariate random variables at any given order. We have carefully validated the formulas with several approaches to make sure they are correct. As applications, we apply our results to Edgeworth expansion for multivariate random variables. Our study primarily uses GPT but we also investigate a handful other LLMs.