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 paper "Combining Human Predictions with Model Probabilities via Confusion Matrices and Calibration" proposes a method to obtain a well-performing ensemble model of both an ML classifier (which outputs a probability for every class) and a human observer (who outputs a categorical decision). This is a setting that traditional ensemble methods do not account for; e.g. majority vote would not be possible here since there are only two decision makers in total (1 model, 1 human). The problem may not be common in standard ML but since ML applications are increasingly being used in real-world settings, there are areas where this approach can be applied (such as high-stakes medical decisions). Combining model and human predictions is shown to lead to lower error rates than either of the two alone.