# Combining Human Predictions with Model Probabilities via Confusion Matrices and Calibration

arXiv: https://arxiv.org/abs/2109.14591 · HTML: https://mlsanity.com/paper/2109.14591/

## Key facts

- Cited by 40 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 69 references that are in the map.
- First posted on arXiv 29 Sep 2021.

## Builds on (most cited first, 25 of 25)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
- [Densely Connected Convolutional Networks](https://mlsanity.com/paper/1608.06993/) (2016)
- [On Calibration of Modern Neural Networks](https://mlsanity.com/paper/1706.04599/) (2017)
- [Going Deeper with Convolutions](https://mlsanity.com/paper/1409.4842/) (2014)
- [Identity Mappings in Deep Residual Networks](https://mlsanity.com/paper/1603.05027/) (2016)
- [Pyro: Deep Universal Probabilistic Programming](https://mlsanity.com/paper/1810.09538/) (2018)
- [Beyond temperature scaling: Obtaining well-calibrated multiclass probabilities with Dirichlet calibration](https://mlsanity.com/paper/1910.12656/) (2019)
- [Verified Uncertainty Calibration](https://mlsanity.com/paper/1909.10155/) (2019)
- [Consistent Estimators for Learning to Defer to an Expert](https://mlsanity.com/paper/2006.01862/) (2020)
- [Human uncertainty makes classification more robust](https://mlsanity.com/paper/1908.07086/) (2019)
- [Predict Responsibly: Improving Fairness and Accuracy by Learning to Defer](https://mlsanity.com/paper/1711.06664/) (2017)
- [Mix-n-Match: Ensemble and Compositional Methods for Uncertainty Calibration in Deep Learning](https://mlsanity.com/paper/2003.07329/) (2020)
- [Learning to Complement Humans](https://mlsanity.com/paper/2005.00582/) (2020)
- [The Algorithmic Automation Problem: Prediction, Triage, and Human Effort](https://mlsanity.com/paper/1903.12220/) (2019)
- [Beyond accuracy: quantifying trial-by-trial behaviour of CNNs and humans by measuring error consistency](https://mlsanity.com/paper/2006.16736/) (2020)
- [Differentiable Learning Under Triage](https://mlsanity.com/paper/2103.08902/) (2021)
- [Conditional independence testing based on a nearest-neighbor estimator of conditional mutual information](https://mlsanity.com/paper/1709.01447/) (2017)
- [Human-Centered Artificial Intelligence and Machine Learning](https://mlsanity.com/paper/1901.11184/) (2019)
- [CCMI : Classifier based Conditional Mutual Information Estimation](https://mlsanity.com/paper/1906.01824/) (2019)
- [Multi-Class Uncertainty Calibration via Mutual Information Maximization-based Binning](https://mlsanity.com/paper/2006.13092/) (2020)
- [Totally Looks Like - How Humans Compare, Compared to Machines](https://mlsanity.com/paper/1803.01485/) (2018)
- [Testing Conditional Independence on Discrete Data using Stochastic Complexity](https://mlsanity.com/paper/1903.04829/) (2019)
- [Human-AI Symbiosis: A Survey of Current Approaches](https://mlsanity.com/paper/2103.09990/) (2021)

## Built on by (most cited first, 30 of 40)

- [Who Should Predict? Exact Algorithms For Learning to Defer to Humans](https://mlsanity.com/paper/2301.06197/) (2023)
- [Learning to Defer to Multiple Experts: Consistent Surrogate Losses, Confidence Calibration, and Conformal Ensembles](https://mlsanity.com/paper/2210.16955/) (2022)
- [Improving Expert Predictions with Conformal Prediction](https://mlsanity.com/paper/2201.12006/) (2022)
- [Sample Efficient Learning of Predictors that Complement Humans](https://mlsanity.com/paper/2207.09584/) (2022)
- [Human-Algorithm Collaboration: Achieving Complementarity and Avoiding Unfairness](https://mlsanity.com/paper/2202.08821/) (2022)
- [Human Uncertainty in Concept-Based AI Systems](https://mlsanity.com/paper/2303.12872/) (2023)
- [Principled Approaches for Learning to Defer with Multiple Experts](https://mlsanity.com/paper/2310.14774/) (2023)
- [Human-AI Collaboration: The Effect of AI Delegation on Human Task Performance and Task Satisfaction](https://mlsanity.com/paper/2303.09224/) (2023)
- [Regression with Multi-Expert Deferral](https://mlsanity.com/paper/2403.19494/) (2024)
- [In Defense of Softmax Parametrization for Calibrated and Consistent Learning to Defer](https://mlsanity.com/paper/2311.01106/) (2023)
- [Realizable H-Consistent and Bayes-Consistent Loss Functions for Learning to Defer](https://mlsanity.com/paper/2407.13732/) (2024)
- [A Two-Stage Learning-to-Defer Approach for Multi-Task Learning](https://mlsanity.com/paper/2410.15729/) (2024)
- [Mastering Multiple-Expert Routing: Realizable H-Consistency and Strong Guarantees for Learning to Defer](https://mlsanity.com/paper/2506.20650/) (2025)
- [Counterfactual Inference of Second Opinions](https://mlsanity.com/paper/2203.08653/) (2022)
- [Crowd-Calibrator: Can Annotator Disagreement Inform Calibration in Subjective Tasks?](https://mlsanity.com/paper/2408.14141/) (2024)
- [Budgeted Multiple-Expert Deferral](https://mlsanity.com/paper/2510.26706/) (2025)
- [Utilizing Human Behavior Modeling to Manipulate Explanations in AI-Assisted Decision Making: The Good, the Bad, and the Scary](https://mlsanity.com/paper/2411.10461/) (2024)
- [From Text to Trust: Empowering AI-assisted Decision Making with Adaptive LLM-powered Analysis](https://mlsanity.com/paper/2502.11919/) (2025)
- [AI, Meet Human: Learning Paradigms for Hybrid Decision Making Systems](https://mlsanity.com/paper/2402.06287/) (2024)
- [Learning to Complement and to Defer to Multiple Users](https://mlsanity.com/paper/2407.07003/) (2024)
- [Why Ask One When You Can Ask k? Learning-to-Defer to the Top-k Experts](https://mlsanity.com/paper/2504.12988/) (2025)
- [Learning to Complement with Multiple Humans](https://mlsanity.com/paper/2311.13172/) (2023)
- [Adversarial Robustness in One-Stage Learning-to-Defer](https://mlsanity.com/paper/2510.10988/) (2025)
- [Factors that influence the adoption of human-AI collaboration in clinical decision-making](https://mlsanity.com/paper/2204.09082/) (2022)
- [How to Enable Effective Cooperation Between Humans and NLP Models: A Survey of Principles, Formalizations, and Beyond](https://mlsanity.com/paper/2501.05714/) (2025)
- [Narrowing Action Choices with AI Improves Human Sequential Decisions](https://mlsanity.com/paper/2510.16097/) (2025)
- [Consistent Learning-to-Defer with Expert-Conditional Advice](https://mlsanity.com/paper/2603.14324/) (2026)
- [Bayesian Online Learning for Consensus Prediction](https://mlsanity.com/paper/2312.07679/) (2023)
- [Learning to Help in Multi-Class Settings](https://mlsanity.com/paper/2501.13810/) (2025)
- [Conformal Set-based Human-AI Complementarity with Multiple Experts](https://mlsanity.com/paper/2508.06997/) (2025)

## Abstract (arXiv)

An increasingly common use case for machine learning models is augmenting the abilities of human decision makers. For classification tasks where neither the human or model are perfectly accurate, a key step in obtaining high performance is combining their individual predictions in a manner that leverages their relative strengths. In this work, we develop a set of algorithms that combine the probabilistic output of a model with the class-level output of a human. We show theoretically that the accuracy of our combination model is driven not only by the individual human and model accuracies, but also by the model's confidence. Empirical results on image classification with CIFAR-10 and a subset of ImageNet demonstrate that such human-model combinations consistently have higher accuracies than the model or human alone, and that the parameters of the combination method can be estimated effectively with as few as ten labeled datapoints.