Imaginary Probabilities: Extending Bayesian Inference to the Complex Plane for Modeling Cognitive Dissonance

Richard L. Douglas (2015) provocatively suggests incorporating imaginary numbers into Bayesian reasoning, but this remains largely speculative. Building on this, I propose developing a rigorous mathematical formulation where beliefs and posterior updates can take values in the complex plane. The real part would represent traditional probability, while the imaginary part could encode “doubt,” internal conflict, or cognitive dissonance—phenomena poorly handled by standard Bayesian updating. This could be validated against psychological experiments where humans hold contradictory beliefs (see conflict-heuristic, e.g., Pan et al. 2020; Xu et al. 2024). Such a model would challenge the orthodoxy of probability theory, potentially offering a powerful new tool for computational psychology, behavioral economics, and even artificial intelligence systems that need to reason about inconsistent information.

References:

1. Improved Fuzzy Bayesian Network-Based Risk Analysis With Interval-Valued Fuzzy Sets and D–S Evidence Theory. Yue Pan, Limao Zhang, Zhiwu Li, L. Ding (2020). IEEE transactions on fuzzy systems.
2. A New Fuzzy Bayesian Inference Approach for Risk Assessments. Jintao Xu, Yang Sui, Tao Yu, Rui Ding, Tao Dai, Mengyan Zheng (2024). Symmetry.
3. Imaginary Number Probability in Bayesian-type Inference. Richard L. Douglas (2015).