Combining Human Predictions with Model Probabilities via Confusion Matrices and Calibration

1 Introduction . One of the main goals of machine learning is to develop algorithms that can operate robustly in an autonomous fashion without human supervision . However , there are many applications where hybrid human-machine approaches are likely to be a preferred mode of operation for a variety of different reasons , such as improving trust between humans and machines , and allowing for a human or a model to take over in situations where the one or the other lacks expertise [ Kamar , 2016 , Vaughan , 2018 , Kleinberg et al. , 2017 , Riedl , 2019 , Trouille et al. , 2019 , Johnson and Vera , 2019 , Zahedi and Kambhampati , 2021 ] . The performance benefits of combining multiple predictors , rather than relying on a single predictor , have been clearly demonstrated in past work in several fields . For example , in machine learning there is a rich vein of research over the past few decades on combining models using a variety of different estimation and algorithmic approaches [ Kittler et al. , 1998 , Dietterich , 2000 , Kuncheva , 2014 , Sagi and Rokach , 2018 ] . This existing line of work emphasizes that combinations of models that have diversity in how they make predictions can systematically outperform a single model . In parallel , in the behavioral science literature , there has been extensive prior work studying combinations of human opinions where , again , diverse combinations tend to outperform any single individual [ Hong and Page , 2004 , Lamberson and Page , 2012 ] . This naturally leads to questions about hybrid combinations of human and machine predictions , rather than just combining one type or the other . For example , one motivation for hybrid combinations is empirical evidence that human and machine classifiers do not make the same types of errors for problems such as image classification [ Geirhos et al. , 2020 , Rosenfeld et al. , 2018 , Serre , 2019 ] , i.e. , they are diverse in their predictions . These ideas have begun to have impact in real-world applications , where hybrid human-machine teams have been found to be effective in areas such as crowdsourcing [ Kamar et al. , 2012 ] , citizen science [ Beck et al. , 2018 ] , speech transcription [ Gaur et al. , 2015 ] , 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . face identification [ Phillips et al. , 2018 ] , and clinical radiology [ Bien et al. , 2018 , Patel et al. , 2019 , Rajpurkar et al. , 2020 ] . In this paper , we focus on a simple yet realistic instantiation of the general problem of hybrid combinations of human and machine predictions . In particular , we consider a K-way classification task such as image classification , with a single human making categorical classification decisions ( i.e . predicting only a label ) and a single classification model predicting a distribution over labels . While humans can provide confidence estimates with their label predictions , calibrated self-assessment of confidence can be difficult [ Keren , 1991 , Kahneman and Tversky , 1996 , Klayman et al. , 1999 ] as well as time-consuming . A key question in this context is whether the non-probabilistic information from a human predictor can be effectively combined with the probabilistic information from a machine learning model . We answer this question in the affirmative and show both theoretically and empirically how relatively simple probabilistic combination techniques can robustly outperform each of a human and machine on their own . In particular we show that a human can augment their predictions with those of a classification model , improving classification accuracy and producing calibrated predictions , even when the model is less accurate than the human . Similarly , from the model ’ s perspective , accuracy can often be significantly improved by augmenting it ’ s class probabilities with a human ’ s labels , while improving calibration performance , even when the human is less accurate than the model . Figure 1 shows an example using our proposed methodology for an image from the ImageNet-16H dataset [ Steyvers et al. , 2022 ] . The human incorrectly predicts the label dog . The classification model ( a deep network ) predicts the correct label bear with a probability of 0.86 ( uncalibrated ) and 0.78 ( calibrated ) . The combined prediction is for bear with a confidence of 0.74 , with dog having a higher confidence given the human ’ s prediction . The histograms on the right show the overall average reduction in error rate on test images : even though the classifiers are on average less accurate ( 6.10 % , 11.1 % error ) than the human ( 4.62 % , 9.99 % error ) , the resulting combinations have lower error rates than either ( 2.85 % , 6.62 % error ) . The primary contributions of our work are as follows : • We propose and investigate a general framework for combining predictions using instance-level confidence from a model and class-level information from a human . The methods we propose are straightforward to implement in practice and label-efficient.1 • We empirically validate our approach on the CIFAR-10H [ Peterson et al. , 2019 ] and ImageNet-16H [ Steyvers et al. , 2022 ] image classification datasets and show that human-machine combinations in this context are systematically more accurate and better calibrated than either alone . • We develop a theoretical understanding , for this framework , of the key tradeoffs related to calibration and accuracy for both the individual human and model , introducing the notion of model and human confidence ratios . We illustrate how these factors affect the combination , showing for 1Code for our estimation methods and experiments is available at : https : //github.com/GavinKerrigan/conf_matrix_and_calibration . example how two models with the same accuracy but with different calibration properties can have different performance when combined with human predictions . The paper begins in Section 2 by introducing notation and deriving our combination method . Section 3 discusses related work and in Section 4 we propose a number of different estimation methods . Section 5 describes our experimental results with two image classification datasets , with individual human labelers , individual models , and combinations . Complementing the experimental results in Section 6 we develop theoretical results that characterize combination performance . Section 7 discusses the potential societal impact and limitations of the work , and Section 8 concludes the paper . 2 Combining Human Labels and Model Probabilities . Notation . We consider a K-ary classification problem , where the goal is to predict a label y ∈ Y = { 1 , . . . , K } from features x ∈ X . The random variable ( x , y ) follows an unknown distribution with support X × Y . We assume access to an individual human labeler represented by the function h : X → Y , where h ( x ) ∈ Y is the label predicted by the human . In addition , we have access to a trained machine classifier m : X → RK , where m ( x ) ∈ RK is the normalized probability vector output by the classifier . Both the human and classifier are assumed to be noisy labelers relative to the ground truth y . The true labels could be determined , for example , by expert labelers or additional information not contained in x . Combining Predictions . Given an input x ∈ X , our goal is to predict a true label conditioned on the predictions h ( x ) and m ( x ) . The key challenge in combining human and classifier predictions is simultaneously leveraging both the class-level outputs from the human and the predictive distributions output by the classifier . Although we focus on the particular case where h is a human and m is a classifier , our setup could be applied more generally to combinations of a non-probabilistic labeler ( whose output is categorical ) and a probabilistic labeler ( whose output is a distribution over classes ) . There are a variety of functional forms that could be used to combine the predictions . We pursue a probabilistic approach , where the conditional distribution over labels that we seek can be factored via Bayes ’ rule as p ( y|h ( x ) , m ( x ) ) ∝ p ( h ( x ) |y , m ( x ) ) p ( y|m ( x ) ) . ( 1 ) In this work , we pursue a conditional independence ( CI ) approach where the human labels h ( x ) and the probabilistic predictions m ( x ) are assumed to be conditionally independent given y . Under this assumption , the quantity of interest factors as p ( y|h ( x ) , m ( x ) ) ∝ p ( h ( x ) |y ) p ( y|m ( x ) ) . ( 2 ) In the above , the term p ( h ( x ) |y ) can be interpreted as calibrated probabilities at the class level . We parameterize the term p ( h ( x ) |y ) by the confusion matrix for the labeler h , which we denote by ϕ with entries ϕij = p ( h ( x ) = i|y = j ) . The second term p ( y|m ( x ) ) can be interpreted as calibrated probabilities at the instance level . However , the probabilistic output of the classifier m ( x ) may differ from p ( y|m ( x ) ) . For example , modern neural networks tend to be overconfident in their predictions [ Guo et al. , 2017 ] . To remedy this , post-hoc calibration maps m ( x ) to well-calibrated probabilities via a learned calibration map with parameters θ . We note that there are many possible choices for such a post-hoc calibration map [ Guo et al. , 2017 , Zhang et al. , 2020 , Kull et al. , 2019 , Patel et al. , 2021 ] , but our method is agnostic to this choice . In this work , we use mθ ( x ) to denote the output of the classifier after applying such a calibration map . The second term in Equation ( 2 ) is then parameterized by the calibrated classifier probabilities mθ ( x ) . Altogether , our method expresses the predicted probability for class j , given that the human predicts class i and the model produces a class probability vector m ( x ) as : p ( y = j|h ( x ) = i , m ( x ) ) = ϕijm θ j ( x ) ∑K k=1 ϕikm θ k ( x ) . ( 3 ) The CI assumption above is common ( both implicitly and explicitly ) in prior work on combining predictions , such as additive classifier ensembles [ Kuncheva , 2014 , Sagi and Rokach , 2018 ] and ( log- ) linear opinion pools [ Genest and Zidek , 1986 , Jacobs , 1995 ] . As our primary motivation is to develop a relatively simple and robust methodology for combining human and model predictions , the additional functional or parametric assumptions ( and parameters ) required to specify a joint model for p ( h ( x ) , m ( x ) |y ) are beyond the scope of the work in this paper . In addition , although the CI assumption is unlikely to hold exactly , prior work [ Kuncheva , 2006 ] notes that a CI model can be an optimal discriminant even when the CI assumption is violated . As further motivation , for the two datasets we use in this paper , CIFAR-10H and ImageNet-16H , the conditional dependence of the predictions h ( x ) by the human and m ( x ) by the model appears to be relatively weak ( see Appendix A for details ) .

Summary: The current work focuses on the interesting prospect and problem of combining human and machine classification information per input, especially in the small-data regime, for the sake of ultimately improving accuracy. The authors take a probabilistic approach to motivating their strategy for combining human-machine information, leveraging a carefully chosen prior to enable inference of a reliable human confusion matrix when the number of annotators is low. They ultimately find that the resulting combination (evaluated in the context of popular image classification tasks) improves accuracy over both humans and models alone, and with drastically less data than previous work. They show more formally that this success is in part driven by both the accuracy of humans/models as well as the confidence values of each.