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Direct Uncertainty Prediction for Medical Second Opinions
Maithra Raghu, Katy Blumer, Rory Sayres, Ziad Obermeyer, Robert Kleinberg, Sendhil Mullainathan, Jon Kleinberg
TL;DR
Disagreements among medical experts make it important to identify patient cases that may benefit from a second opinion. The paper trains models to predict expert disagreement directly from patient features and compares this with classifier-based uncertainty estimation. DUP performs better than UVC in theory and across the reported medical-imaging evaluations.
Problem
Persistent disagreement among human experts, including doctors diagnosing the same patient, motivates predicting which cases would benefit from a medical second opinion.
Method
Direct Uncertainty Prediction learns an uncertainty score directly from raw patient features, unlike Uncertainty via Classification, which predicts a label distribution before postprocessing it.
Results
DUP performs better than UVC theoretically and in medical-imaging evaluations, including identifying cases associated with large doctor disagreements and outperforming UVC models on all reported adjudicated-test-set evaluations.
Takeaways & Limitations
Machine learning models can identify patient cases likely to produce high expert disagreement and potentially needing a second doctor’s opinion.
Takeaways & Limitations
The model predicts disagreement from x_i alone, whereas doctors also use medical history, patient characteristics, and other information.
Abstract
from arXiv · showhide
The issue of disagreements amongst human experts is a ubiquitous one in both machine learning and medicine. In medicine, this often corresponds to doctor disagreements on a patient diagnosis. In this work, we show that machine learning models can be trained to give uncertainty scores to data instances that might result in high expert disagreements. In particular, they can identify patient cases that would benefit most from a medical second opinion. Our central methodological finding is that Direct Uncertainty Prediction (DUP), training a model to predict an uncertainty score directly from the raw patient features, works better than Uncertainty Via Classification, the two-step process of training a classifier and postprocessing the output distribution to give an uncertainty score. We show this both with a theoretical result, and on extensive evaluations on a large scale medical imaging application.
1. Introduction
Human experts can disagree substantially on the same patient case, motivating models that predict which instances warrant medical second opinions. The paper compares direct uncertainty prediction with a classifier-based alternative and reports theoretical and empirical advantages for DUP.
- Motivation: 21% of cases in one 280-patient cohort had significant diagnostic disagreements, and this group was the most costly to treat.
- Motivation: Doctor disagreements persist because cases can contain features that generate expert uncertainty, not merely random labeling noise.
- Methods: UVC first predicts an empirical grade distribution from each instance and then postprocesses that output to obtain an uncertainty score.
- Methods: DUP instead learns a function mapping raw instance features directly to a scalar uncertainty score.
- Theory and evidence: Under a natural data model, DUP provides an unbiased estimate of true uncertainty while UVC has a bias term, and experiments support this comparison.
- Medical second opinion: DUP models identify patient cases likely to produce large disagreements among doctors in a large-scale medical imaging task.This directly targets cases that may benefit from a medical second opinion.
2. Direct Uncertainty Prediction
Direct Uncertainty Prediction (DUP) maps patient features directly to expert-disagreement scores, while Uncertainty via Classification (UVC) computes uncertainty after classification. Theory and experiments show DUP is more effective, including on synthetic mixtures and image-blurring benchmarks.
- 2. Direct Uncertainty Prediction: DUP directly predicts uncertainty from patient instances, whereas UVC first estimates a label distribution and then applies an uncertainty function.The target is a scalar score representing disagreement among doctors or noisy labels.
- 2. Direct Uncertainty Prediction: DUP gives an unbiased estimate of the target uncertainty, while UVC has a bias term for concave scoring functions such as disagreement and variance.The proof uses the tower rule for unbiasedness and Jensen’s inequality for the UVC bias.
- 2.1. Toy Example on Mixture of Gaussians: DUP performs much better than UVC at identifying high-disagreement datapoints across the evaluated Gaussian-mixture settings.Table 1 reports percentage AUC over three repeats for mixtures with different numbers of dimensions and Gaussians.
- 2.1. Toy Example on Mixture of Gaussians: UVC can output constant uncertainty despite varying true uncertainty when the observed feature mapping hides information, whereas DUP can recover the varying scores in the toy Gaussian example.With x = |o|, symmetry makes UVC’s expected posterior [0.5, 0.5] for every observed x.
- 2.2. Example on SVHN and CIFAR-10: DUP also outperforms UVC on label-disagreement prediction for blurred images from SVHN and CIFAR-10, even without the theorem’s obscuring-function setting.The image-blurring task uses noise distributions that vary with blur severity, and the models learn different features.
3. Related Work
Prior work addresses expert disagreement through noisy-label modeling, generative approaches, Bayesian uncertainty, and crowdsourcing methods, while this paper compares DUP and UVC in disagreement prediction.
- Prior approaches model noisy labels using conditional-independence assumptions, domain-specific generative models, Bayesian techniques, or crowdsourcing methods.
- The paper evaluates UVC and DUP models on variance and disagreement prediction tasks, with performance reported as AUC.
- DUP models outperform baselines on both prediction settings, including cross-task variants that use a different uncertainty target from the evaluation task.
4. Doctor Disagreements in DR
The medical application uses retinal images and noisy doctor grades to predict cases with high disagreement, comparing direct uncertainty prediction against classifier-based uncertainty. DUP models outperform UVC counterparts across the reported tasks, and the performance gap is robust to training-set size and appears early in training.
- Doctor Disagreements in DR: Diabetic retinopathy is graded on five classes, with grades 3–5 representing referable disease requiring immediate specialist attention.The most costly clinical error is failing to refer a patient with referable disease.
- Doctor Disagreements in DR: The main dataset contains retinal images with typically one to three doctor grades, and more than 20% of images show large referable/non-referable disagreement.
- Models and First Experimental Results: UVC first predicts an empirical grade histogram and postprocesses it, whereas DUP directly predicts a binarized uncertainty target from patient images.
- First Experimental Results: DUP models outperform UVC counterparts on both variance prediction and disagreement prediction tasks in the diabetic-retinopathy application.The tasks identify cases with high disagreement among doctors; performance is measured by AUC.
- Models and First Experimental Results: The gold-standard adjudicated dataset differs from the main dataset by providing many specialist grades and one consensus grade per image.Three doctors grade each image independently before discussing the diagnosis and producing the adjudicated grade.
- First Experimental Results: The performance gap between DUP and UVC is robust to training-data size and appears early during training, with similar findings when entropy is used as the uncertainty function.
5. Predicting Disagreement with Consensus: Adjudicated Evaluation
The adjudicated evaluation tests whether model uncertainty identifies disagreement between individual doctor grades and a consensus diagnosis. Across agreement, ranking, and distance-based evaluations, DUP models outperform classifier-based baselines.
- Adjudicated Evaluation: The adjudicated dataset provides many individual specialist grades and a consensus grade, enabling evaluation against a more thorough diagnosis.
- Adjudicated Evaluation: All DUP models outperform all classifier-based baselines across evaluations predicting disagreement with the adjudicated diagnosis.The evaluations include multiple aggregations of individual grades and special referable/non-referable cases.
- Ranking Evaluation: DUP rankings outperform baselines under all reported distance metrics, while their average ranking corresponds to five doctor grades versus four for UVC.
- Ranking Evaluation: The continuous disagreement target is the Wasserstein distance between the empirical grade histogram and a point mass at the adjudicated grade.The score can use absolute grade difference, 2-Wasserstein distance, or a binary agreement/disagreement metric.
- Ranking Evaluation: Spearman rank correlation compares model-induced rankings with rankings based on Wasserstein distance to the adjudicated grade.
6. Discussion
The paper argues that machine learning can identify cases likely to produce high expert disagreement, supporting medical second opinions. It finds that Direct Uncertainty Prediction outperforms Uncertainty via Classification theoretically and empirically.
- Direct Uncertainty Prediction learns uncertainty directly from raw patient features and performs significantly better than Uncertainty via Classification.The comparison is supported by both a formal result and extensive experiments.
- Machine learning models can predict which data instances are likely to give rise to high expert disagreement.
- The approach targets patient cases that may benefit from a medical second opinion.
A. Proofs of Direct Uncertainty Prediction Results
The proof compares uncertainty estimated through classification with uncertainty predicted directly. Using concavity and Jensen’s inequality, it shows that the classification-based estimate overestimates true uncertainty under non-degenerate posterior variation.
- The proof establishes Direct Uncertainty Prediction’s unbiasedness by showing E[hdup] = E[U(Y)].This step uses the tower law, or law of total expectation.
- Uncertainty via Classification overestimates true uncertainty whenever posteriors conditioned on g(O) are not concentrated at a single point.The strict inequality follows from the concavity of U(·) and Jensen’s inequality.
- For specific uncertainty functions, the bias is computed from the expected difference huvc − hdup.The derivation takes expectations over values of g(O).
B. Mixture of Gaussians Setting
The mixture-of-Gaussians experiment trains DUP and UVC on synthetic data with uncertainty defined from multiple posterior labels. Fully connected networks are used for the comparison.
- DUP and UVC are trained on a synthetic mixture-of-Gaussians task with five labels sampled per instance from the posterior over Gaussian centers.The Gaussian means are drawn from N(0, 1/d), the variance is the identity, and x = g(o) = |o|.
- The experiment uses uniform Gaussian mixtures whose mean vectors have expected norm 1.The mean-vector distribution is N(0, 1/d), with d denoting the dimension.
- Both models use fully connected networks with two hidden layers of width 300, optimized with SGD with momentum.The initial learning rate is 0.01.
C. SVHN and CIFAR-10 Setting
The SVHN and CIFAR-10 experiments vary image blur to control label noise and compare DUP with UVC. DUP outperforms UVC on both datasets, while saliency maps examine differences in learned features.
- Each image receives a Gaussian filter variance of 0, 1, 2, or 3, and three noisy labels are sampled according to that variance.The filter variance controls the label-noise distribution.
- DUP outperforms UVC on both SVHN and CIFAR-10.The UVC model predicts from the empirical label histogram, whereas DUP uses a binary agree/disagree target.
- Learned Features: SmoothGrad and Integrated Gradients are used to compare the input features emphasized by DUP and UVC.
D. Details of DUP in the Medical Domain
In the medical domain, DUP models are trained from noisy doctor labels to predict disagreement-related uncertainty, using thresholding, pretrained representations, calibration, and regularization. Additional experiments compare model variants and visual explanations, with entropy-targeted models also favoring DUP.
- Prelogits, Calibration and Regularization: Medical DUP models use pretrained Inception-v3 prelogit embeddings because the multi-label training data is too small for effective end-to-end training.Smaller models are trained on embeddings for both DUP and the UVC baseline.
- Additional Results: Entropy-targeted models again show DUP outperforming UVC.The result is reported in Table 8.
- Prelogits, Calibration and Regularization: Prelogit embeddings and calibration produce the strongest baseline and DUP variants, while replacing prelogits with logits fails to match performance.The comparison indicates that information is lost between the prelogit and logit layers.
- Additional Model: A two-head Variance-E2E variant shows a small performance drop when its variance head is evaluated, despite enabling use of all data.The variant adds a classification head alongside the variance-prediction head.
E. Additional Results: Entropy, Finite Sample Behavior and Convergence Analysis
Additional experiments test DUP against UVC with entropy targets, varying training-set size, and monitoring convergence. Across these analyses, DUP retains its advantage, including with finite data and early in training.
- Entropy: DUP outperforms UVC when entropy is used as the target uncertainty function.The comparison is reported in Table 8 for Disagree-P and Histogram-PC.
- Finite Sample Behavior: The performance gap between DUP and UVC is robust to training-set size, with both methods relatively constant when at least 30% of training data is used.The analysis averages results over five subsampling and optimization repeats.
- Convergence Analysis: The DUP–UVC performance gap appears very early in training and gradually widens as training continues.This convergence behavior is shown in the right plot of Figure 5.
F. Background on the Wasserstein Distance
The appendix introduces Wasserstein distance as a minimum-cost way to transform one probability distribution into another. It then specifies distance choices and derives the transport plan when one distribution is a point mass.
- Wasserstein Distance: Wasserstein distance measures the minimum cost of moving mass from distribution f to distribution g under a metric d.The transport amount from r to t is represented by π(r, t).
- Wasserstein Distance: When g is a point mass at t0, the transport plan π is uniquely determined.The result follows because all mass in g must be assigned to t0.
- Distance Choices: The paper considers absolute-value and squared distance functions, with squared distance imposing a higher penalty on larger grade differences.Absolute value treats successive grade differences equally, while squared distance is adjusted to form a metric.
- Binary Disagreement: For binary disagreement, the metric assigns zero distance to matching grades and one to mismatches.This defines disagreement as a binary distance.