# Indeterminate Probability Theory

arXiv: https://arxiv.org/abs/2303.11536 · HTML: https://mlsanity.com/paper/2303.11536/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 48 references that are in the map.
- First posted on arXiv 21 Mar 2023.

## Builds on (most cited first, 16 of 16)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Generating Diverse High-Fidelity Images with VQ-VAE-2](https://mlsanity.com/paper/1906.00446/) (2019)
- [An Introduction to Variational Autoencoders](https://mlsanity.com/paper/1906.02691/) (2019)
- [Unsupervised Anomaly Detection via Variational Auto-Encoder for Seasonal KPIs in Web Applications](https://mlsanity.com/paper/1802.03903/) (2018)
- [Convergence Analysis of Two-layer Neural Networks with ReLU Activation](https://mlsanity.com/paper/1705.09886/) (2017)
- [Deep AutoRegressive Networks](https://mlsanity.com/paper/1310.8499/) (2013)
- [Denoising Criterion for Variational Auto-Encoding Framework](https://mlsanity.com/paper/1511.06406/) (2015)
- [Recent Advances in Zero-shot Recognition](https://mlsanity.com/paper/1710.04837/) (2017)
- [Sparse Probabilistic Circuits via Pruning and Growing](https://mlsanity.com/paper/2211.12551/) (2022)
- [A Tutorial on Deep Latent Variable Models of Natural Language](https://mlsanity.com/paper/1812.06834/) (2018)
- [Effectiveness of Hierarchical Softmax in Large Scale Classification Tasks](https://mlsanity.com/paper/1812.05737/) (2018)

## Abstract (arXiv)

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.