Indeterminate Probability Theory

Tao Yang, Chuang Liu, Xiaofeng Ma, Weijia Lu, Ning Wu, Bingyang Li, ZHIFEI YANG, Peng Liu, Lin Sun, xiaodong Zhang, Can Zhang

Primary Area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)

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Keywords: Indeterminate Probability Theory, Analytical Solution, General Posterior, MTS

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TL;DR: We propose a new probability theory which is the analytical solution for any general complex posterior.

Abstract:

Currently, there is no mathematical analytical form for a general posterior. We have discovered a new theory to address this issue, which is called Indeterminate Probability Theory. This is a big discovery in the field of probability, and it is an extension of classical probability theory, and makes classical probability theory a special case to our theory. In this paper, we propose a new perspective for understanding probability theory by introducing Observers and treating the outcome of each random experiment as an indeterminate probability distribution, which leads to probability calculations being a combination of ground truth and observation errors. We then discover three conditional mutual independent assumptions as Candidate Axioms and divide the probability process into two phase: observation phase and inference phase. In the observation phase, a general equation for any complex posterior is derived. In the inference phase, the inference probability equation with the posterior is derived. Base on this theory, we propose a new general model called IPNN - Indeterminate Probability Neural Network to validate our theory. Furthermore, in one of our another papers, this new theory is successfully applied to the task of multivariate time series (MTS) forecasting without relying on any neural models, and it outperforms LSTM models as well as some transformer-based models. In addition, further applications of this new theory are also discussed in this paper. Validations of this theory are reflected in experimental results.

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Submission Number: 4295