Indeterminate Probability Theory
Complex continuous or mixed joint distributions (e.g., P(Y | z_1, z_2, ..., z_N)) generally lack closed-form solutions, often necessitating approximations such as MCMC. This paper proposes Indeterminate Probability Theory (IPT), which makes the following contributions: (1) An observer-centered framework in which experimental outcomes are represented as distributions combining ground truth with observation error; (2) The introduction of three independence candidate axioms that enable a two-phase probabilistic inference framework; (3) The derivation of closed-form solutions for arbitrary complex joint distributions under this framework. Both the Indeterminate Probability Neural Network (IPNN) model and the non-neural multivariate time series forecasting application demonstrate IPT's effectiveness in modeling high-dimensional distributions, with successful validation up to 1000 dimensions. Importantly, IPT is consistent with classical probability theory and subsumes the frequentist equation in the limit of vanishing observation error.
View on arXiv@article{yang2025_2303.11536,
title={ Indeterminate Probability Theory },
author={ Tao Yang and Chuang Liu and Xiaofeng Ma and Weijia Lu and Ning Wu and Bingyang Li and Zhifei Yang and Peng Liu and Lin Sun and Xiaodong Zhang and Can Zhang },
journal={arXiv preprint arXiv:2303.11536},
year={ 2025 }
}