Deep Probability Estimation

1 INTRODUCTION . We consider the problem of building models that answer questions such as : Will it rain ? Will a patient survive ? Will a car collide with another vehicle ? Due to the inherently-uncertain nature of these real-world phenomena , this requires performing probability estimation , i.e . estimating the probability of each possible outcome of the phenomenon of interest . Models for probability prediction must be trained on observed outcomes ( e.g . whether it rained , a patient died , or a collision occurred ) , because the ground-truth probabilities are unknown . The problem is therefore analogous to binary classification , with the important difference that the objective is to estimate probabilities rather than predicting specific outcomes . In probability estimation , two identical inputs ( e.g . histopathology images from cancer patients ) can potentially result in two different outcomes ( death vs. survival ) . In contrast , in classification the class label is usually completely determined by the data ( a picture either shows a cat or it does not ) . The goal of this work is to investigate probability estimation from high-dimensional data using deep neural networks . Deep networks trained for classification often generate probabilities , which quantify the uncertainty of the estimate ( i.e . how likely the network is to classify correctly ) . This quantification has been observed to be inaccurate , and several methods have been developed to improve it ( Platt , 1999 ; Guo et al. , 2017 ; Szegedy et al. , 2016 ; Zhang et al. , 2020 ; Thulasidasan et al. , 2020 ; Mukhoti et al. , 2020 ; Thagaard et al. , 2020 ) , including Bayesian neural networks ( Gal & Ghahramani , 2016 ; Wang et al. , 2016 ; Shekhovtsov & Flach , 2019 ; Postels et al. , 2019 ) . However , these works restrict their attention almost exclusively to classification in datasets ( e.g . CIFAR10/100 Krizhevsky ( 2009 ) , or ImageNet ( Deng et al. , 2009 ) ) where the label is not uncertain , and therefore the uncertainty is completely tied to the model : it quantifies the confidence of the model in its own prediction , not the probability of an event of interest . Probability estimation from high-dimensional data is crucial in medical prognostics ( Wulczyn et al. , 2020 ) , weather prediction ( Agrawal et al. , 2019 ) , and autonomous driving ( Kim et al. , 2019 ) . In order to advance deep-learning methodology for probability estimation it is crucial to build appropriate benchmark datasets . Here we build a synthetic dataset and gather three real-world datasets , which we use to systematically evaluate existing methodology . In addition , we propose a novel approach , which outperforms current state-of-the-art methods . Our contributions are the following : • We introduce a new synthetic dataset for probability estimation where a population of people may have a certain disease connected to age . The task is to predict the probability that they contract the disease from a picture of their face . The data are generated based on the UTKFaces dataset ( Zhang et al. , 2017a ) , which contains age information . The dataset contains multiple versions of the synthetic labels , which are generated according to different distributions designed to mimic realworld probability-prediction datasets . The dataset serves two objectives . First , it allows us to evaluate existing methodology . Second , it enables us to evaluate different metrics in a controlled scenario where we have access to ground-truth probabilities . • We have used publicly available data to build probability-estimation benchmark datasets for three real-world applications : ( 1 ) precipitation forecasting from radar images , ( 2 ) prediction of cancerpatient survival from histopathology images , and ( 3 ) prediction of vehicle collisions from dashcam videos . We use these datasets to systematically evaluate existing approaches , which have been previously tested mainly on classification datasets . • We propose Calibrated Probability Estimation ( CaPE ) , a novel technique which modifies the training process so that output probabilities are consistent with empirical probabilities computed from the data . CaPE outperforms existing approaches on most metrics on synthetic and real-world data . 2 PROBLEM FORMULATION : PROBABILITY ESTIMATION . The goal of probability estimation is to evaluate the likelihood of a certain event of interest , based on observed data . The available training data consists of n examples xi , 1 ≤ i ≤ n , each associated with a corresponding outcome yi . In our applications of interest , the input data are high dimensional : each xi corresponds to an image or a video . The corresponding label yi is either 0 or 1 depending on whether or not the event in question occurred . For example , in the cancer-survival application xi is a histopathology image of a patient , and yi equals 1 if the patient survived for 5 years after xi was collected . The data have inherent uncertainty : yi , the patient ’ s survival , does not depend deterministically on the histopathology image ( due e.g . to comorbidities and other health factors ) . Instead , we assume that yi equals 1 with a certain probability pi associated with xi , , as illustrated in Figure 1 , because the input data provides key information about the patient ’ s survival chances . At inference , a probability-estimation model aims to generate an estimate p̂ of the underlying probability p , associated with a new input data point x ( e.g . the probability of survival for over 5 years for new patients based on their histopathology data ) . To summarize , this is not a classification problem , because the labels are not completely predictable . Instead , the goal is to predict the probability of the outcome , which is critical in choosing a course of treatment for the patient . 3 EVALUATION METRICS . Probability estimation shares similar target labels and network outputs with binary classification . However , classification accuracy is not an appropriate metric for evaluating probability-estimation models due to the inherent uncertainty of the outcomes . This is illustrated by the example in Figure 2a where a perfect probability estimate would result in a classification accuracy of just 75 % .1 Metrics when ground-truth probabilities are available . For synthetic datasets , we have access to the ground truth probability labels and can use them to evaluate performance . Two reasonable metrics are the mean squared error or ` 2 distance MSEp , and the Kullback–Leibler divergence KLp between the estimated and ground-truth probabilities : MSEp = 1 N N∑ i=1 ( p̂i − pi ) 2 , and KLp = 1 N N∑ i=1 ( p̂i log ( p̂i pi ) + ( 1− p̂i ) log ( 1− p̂i 1− pi ) ) . ( 1 ) N is the number of data , and pi , p̂i are the ground-truth and predicted probabilities respectively . Calibration metrics . In real-world data , ground-truth probabilities are not available . In order to evaluate the probabilities estimated by a model , we need to compare them to the observed probabilities . To this end , we aggregate the examples for which the model output equals a certain value ( e.g . 0.5 ) , and verify what fraction of them have outcomes equal to 1 . If the fraction is close to the model output , then the model is said to be well calibrated . Definition 3.1 . A model f is well calibrated if P ( y = 1 | f ( x ) ∈ I ( q ) ) = q , ∀ 0 ≤ q ≤ 1 , ( 2 ) where y is the observed outcome , f ( x ) is the probability predicted by model f for input x , and I ( q ) is a small interval around q . Model calibration can be evaluated using the expected calibration error ( ECE ) ( Guo et al. , 2017 ) ( note however that the definition Guo et al . ( 2017 ) is specific to classification ) . Given a probabilityestimation model f and a dataset of input data xi and associated outcomes yi , 1 ≤ i ≤ N , we partition the examples into B bins , I1I2 , · · · , IB , according to the probabilities assigned to the examples by the model . Let Q1 , . . . , QB−1 the B-quantiles of the set { f ( x1 ) , . . . , f ( xN ) } , we have Ib : = [ Qb−1 , Qb ] ∩ { f ( xi ) } Ni=1 ( setting Q0 = 0 ) . For each bin , we compute the mean predicted and empirical probabilities , p ( b ) emp = E ( y | f ( x ) ∈ Ib ) = 1 |Ib| ∑ i∈Index ( Ib ) yi , ( 3 ) q ( b ) = 1 |Ib| ∑ i∈Index ( Ib ) f ( xi ) , ( 4 ) 1A perfect model ( in terms of probability estimation ) , assigns 0.25 to the blue class and 0.75 to the red class . To maximize classification accuracy , we predict when the model outputs 0.75 ( red examples ) and 0 when it outputs 0.25 ( blue examples ) . However , 25 % of red examples have an outcome of 0 , and 25 % of blue examples have an outcome of 1 . As a result , the model would only have 75 % accuracy . where Index ( Ib ) = { i | f ( xi ) ∈ Ib } . The pairs ( q ( b ) , p ( b ) emp ) can be plotted as a reliability diagram , shown in the second row of Figure 4 and in Figure 6 . ECE is then defined as ECE = 1 B B∑ b=1 ∣∣∣p ( b ) emp − q ( b ) ∣∣∣ . ( 5 ) Other metrics for calibration include the maximum calibration error ( MCE ) defined as MCE = max b=1 , ... , B ∣∣∣p ( b ) emp − q ( b ) ∣∣∣ , and the Kolmogorov-Smirnov error ( KS-error ) ( Gupta et al. , 2021 ) , a metric based on the cumulative distribution function , which is described in more detail in Appendix B. Brier score . Crucially , a model without any discriminative power can be perfectly calibrated ( see Figure 2b ) . The Brier score is a metric designed to evaluate both calibration and discriminative ability . It is the mean squared error between the predicted probability and the observed outcomes : Brier = 1 N N∑ i=1 ( p̂i − yi ) 2 . ( 6 ) This score can be decomposed into two terms associated to calibration and discrimination ability , as shown in Appendix C. Using the synthetic data in Section 6.1 , where the ground-truth probabilities are known , we show that Brier score is indeed a reliable proxy for gold-standard MSE metric based on ground-truth probabilities MSEp , in contrast to calibration metrics such as ECE , MCE or KSerror , and to classification metrics such as AUC ( see Figure 3 and Appendix D ) . 4 PROPOSED METHOD : CALIBRATED PROBABILITY ESTIMATION ( CAPE ) . Prediction models based on deep learning are typically trained by minimizing the cross entropy between the model output and the training labels ( Goodfellow et al. , 2016 ) . This cost function is a proper scoring rule , which means that it evaluates probability estimates in a consistent manner and is therefore guaranteed to be well calibrated in an infinite-data regime ( Buja et al. , 2005 ) , as illustrated by Figure 4 ( first column ) . Unfortunately , in practice prediction models are trained on finite data . This is crucial in the case of deep neural networks , because these models are highly overparametrized and therefore prone to overfitting ( Goodfellow et al. , 2016 ) . In classification , networks have been shown to be capable of fitting arbitrary random labels ( Zhang et al. , 2017a ) . In probability estimation , we observe that neural networks indeed eventually overfit the observed outcomes completely . Moreover , the estimated probabilities collapse to 0 or 1 ( Figure 4 , second column ) , a phenomenon that has also been reported in classification ( Mukhoti et al. , 2020 ) . However , calibration is preserved during the first stages of training ( Figure 4 , third column ) . This is reminiscent of the early-learning phenomenon observed for classification from partially corrupted labels ( Yao et al. , 2020 ; Xia et al. , 2020 ) , where neural networks learn from the correct labels before eventually overfitting the false ones ( Liu et al. , 2020 ) . Here , we propose to exploit the training dynamics of cross-entropy minimization through a method that we name Calibrated Probability Estimation ( CaPE ) . Our starting point is a model obtained via early stopping using validation data on the cross-entropy loss . CaPE is designed to further improve the discrimination ability of the model , while ensuring that it remains well calibrated . This is achieved by alternatively minimizing the following two loss functions : Discrimination loss : Cross entropy between the model output and the observed binary outcomes , LD = − N∑ i=1 [ yi log ( f ( xi ) ) + ( 1− yi ) log ( 1− f ( xi ) ) ] . Calibration loss : Cross entropy between the output probability of the model and the empirical probability of the outcomes conditioned on the model output : LC = − N∑ i=1 [ piemp log ( f ( xi ) ) + ( 1− piemp ) log ( 1− f ( xi ) ) ] , where piemp is an estimate of the conditional probability P [ y = 1|f ( x ) ∈ I ( f ( xi ) ) ] where I ( f ( xi ) ) is a small interval centered at f ( xi ) . As explained in Section 3 if f ( xi ) is close to this value , then the model is well calibrated . We consider two approaches for estimating piemp . ( 1 ) CaPE ( bin ) where we divide the training set into bins , select the bin bi containing f ( xi ) and set piemp = p ( bi ) emp in equation 3 . ( 2 ) CaPE ( kernel ) where piemp is estimated through a moving average with a kernel function ( see Appendix E for more details ) . Both methods are efficiently computed by sorting the predictions p̂i . The calibration loss requires a reasonable estimation of the empirical probabilities p ( i ) emp , which can be obtained from the model after early learning . Therefore using the calibration loss from the beginning is counterproductive , as demonstrated in Section J. Algorithm 1 Pseudocode for CaPE Require : f . early stopped model Require : m . freq . of training with LC Require : { xi , yi } Ni=1 . training set Require : K ( p , q ) : = exp [ − ( p− q ) 2 /σ2 ] . Gaussian kernel for t = 1 to num epochs do if t mod m = 0 then p̂i ← f ( xi ) , ∀i Update piemp , ∀i , with BIN or KERNEL L ← LC . compute discrimination loss else L ← LD . compute calibration loss end if f ← backprop with L . train with loss end for function BIN ( B ) . B-number of bins I1 , · · · IB ← partitions by quantile of { p̂j } Nj=1 Find b such that p̂i ∈ Ib Index ( Ib ) ← { j|p̂j ∈ Ib } . get indices in bin b piemp ← 1|Ib| ∑ i∈Index ( Ib ) yi . empirical mean of bin b end function function KERNEL ( r , K ) . r-window size ; kernel Nr ( i ) ← r-nearest neighbor of p̂i ( output probability space ) Z ← ∑ p̂j∈Nr ( i ) K ( p̂i , p̂j ) . normalization factor piemp ← ∑ p̂j∈Nr ( i ) K ( p̂i , p̂j ) yj/Z . kernel smooth end function CaPE is summarized in Algorithm 1 . Figures 4 and 5 show that incorporating the calibration-loss minimization step indeed preserves calibration as training proceeds ( this is not necessarily expected because CaPE minimizes a calibration loss on the training data ) , and prevents the model from overfitting the observed outputs . This is beneficial also for the discriminative ability of the model , because it enables it to further reduce the cross-entropy loss without overfitting , as shown in Figure 5 . The experiments with synthetic and real-world data reported in Section 6 suggest that this approach results in accurate probability estimates across a variety of realistic scenarios .

Summary: This work tackles the problem of probability estimation. Current machine learning models do not fully reflect the uncertainty of the outcome but only of the model. The authors propose a loss that enforces both calibration and discrimination. Additionally, they present a semi-synthetic dataset for further study of this problem: from an image dataset of faces with associated age, they estimate the risk of developing a given disease.