Deep neural networks (DNNs) have proven their effectiveness on numerous tasks. However, research into the reliability of DNNs falls behind their successful applications and remains to be further investigated. In addition to prediction, it is also important to evaluate how confident a DNN is about its predictions, especially when those predictions are being used within medical applications. In this paper, we quantify the uncertainty of DNNs for the task of Chest X-Ray (CXR) image classification. We investigate uncertainties of several commonly used DNN architectures including ResNet, ResNeXt, DenseNet and SENet. We then propose an uncertaintybased evaluation strategy that retains subsets of held-out test data ordered via uncertainty quantification. We analyze the impact of this strategy on the classifier performance. In addition, we also examine the impact of setting uncertainty thresholds on the performance. Results show that utilizing uncertainty information may improve DNN performance for some metrics and observations.
Neural networks have been very successful in many fields such as
natural language processing [
Neural network-based deep learning algorithms are also getting
popular for medical X-ray image processing [
In recent years Bayesian learning and estimation of prediction
uncertainty have gained more and more attention in neural networks
context due to the wide application of deep neural networks in many
areas [
The authors in [
[
Despite the above works in Bayesian deep neural network learning and uncertainty quantification, there are few works on evaluating the effects of uncertainty-based evaluation strategies for medical image classification. To the best of our knowledge, we are the first to apply uncertainty quantification strategies for chest X-ray image classification using deep neural networks and evaluate their impacts on performances. The main contributions of this paper are: We apply uncertainty quantification to five deep neural network models for chest X-ray image classification and analyze their performances.
We investigate the impact that uncertainty information has on classification task performance by evaluating subsets of held-out test data ordered via uncertainty quantification. 3
In this section, we will introduce the basic ideas of Bayesian Neural
Networks and one of its approximations – SWAG [
In the ordinary deterministic neural networks, we get point estimation of the network weights w which are regarded as fixed values and will not be changed after training. During inference, for each input xi we get one deterministic prediction p(yijxi) = p(yijxi; w) without getting the uncertainty information.
In the Bayesian neural network settings, in addition to the target prediction, we also want to get the uncertainty for the prediction. To do so we regard the neural network weights as random variables that subject to some form of distribution and try to estimate the posterior distribution of the network weights given the training data during training. We then integrated out the weights and get the distribution over the prediction during inference. From the prediction distribution we can further calculate the prediction output and corresponding uncertainty. More specifically, let D = f(X; Y )g and w be the training data and weights of a neural network, respectively. The ordinary deterministic neural network methods try to get a point estimate of w by either maximum likelihood estimator (MLE) w = arg maxw p(Djw) or maximum a posterior (MAP): w = arg maxw p(wjD) where p(wjD) = p(wp)p(D(D) jw) / p(w)p(Djw). The w are fixed after training and used for inference for the new data. In Bayesian learning, we estimate the posterior distribution p(wjD) during training and marginalize out w during the inference to get a probability distribution of the prediction.
p(yjx; D) = Ew p(wjD)[p(yjx; w)] = R p(yjx; w)p(wjD)dw (1) After getting the p(yjx), we can calculate the statistical moments of the predicted variable and regard the first and second moment (i.e., mean and variance) as the prediction and uncertainty, respectively.
However, in practice there are two major difficulties. The first one is that p(D) = R p(w)p(Djw)dw is usually intractable and thus we cannot get exact p(wjD). The second lies in that Eq. (1) is also usually intractable for neural networks. One common approach to deal with the first difficulty is to use a simpler form of distribution q(wj ) with hyperparameters to approximate p(wjD) by minimizing the Kullback-Leibler (KL) divergence between q(wj ) and p(wjD). This turns the problem into an easier optimization problem: = arg min KL[q(wj )jjp(wjD)] = arg min
Z q(wj )log q(wj ) dw p(wjD) For the second difficulty, the usual approach is to use sampling to estimate Eq. (1), and it becomes p(yjx)
Ew q(wj )[p(yjx; w)] 1 PT T i=1 p(yjx; w(i)) (3) where w(i) q(wj ).
People had proposed different methods to approximate the
posterior p(wj ) or to get the samples of w [
The basic idea of SWAG [
During inference, for each input (image) xi, sample the weights from the Gaussian ws N (wSW A; SW A) then update the batch norm statistics by performing one epoch of forward pass, and then the sample prediction is given by p(y^isjxi) = p(yijxi; ws). Repeat the precedure for S times and we get S predictions y^i1, y^i2, , y^is, , y^iS for the same input xi. By using these S predictions we can get the final prediction and uncertainty. For regression problem, the final prediction will be y^i = S1 PS
s=1 y^is. 3.3
Some methods had been proposed to quantify the uncertainty in
classification [
For a classification problem, suppose there are C classes, denote ps , [ps1; ps2; ; psc] = p(yjx; s), s 2 f1; 2; ; Sg as the softmax (or sigmoid in binary case if C = 2) output of the neural network for a same repeated input x for S times, then the predicted “probability” is the average of those S sample outputs p = 1 PS S s=1 ps The predicted class label index is y^ = arg maxc p. The aleatoric uncertainty Ua and the epistemic uncertainty Ue are Ua = S1 PsS=1[diag(ps) pspsT ], Ue = S1 PsS=1(ps p)(ps p)T The total uncertainty is Utotal = Ua + Ue. For binary classification, the sigmoid output is a scalar and the uncertainty equations are reduced to
Ua = Ue =
S 1 X ps(1 S s=1
S 1 X(ps S s=1 ps) p)2 y^ = (1 p
0:5 0 p < 0:5 where p = S1 PS
s=1 ps and ps = p(y = 1jx; s) = 1 0jx; s). The predicted label is: In this way, we can get uncertainties for all the instances. 3.4
Transfer learning is a widely used technique to help improve performance for deep neural networks in image classification. Here we can also benefit from transfer learning by loading pre-trained neural network models trained by ImageNet (http://image-net.org) dataset. The SWAG method has one advantageous characteristic that it does not require to modify any architecture of the original neural networks and therefore we can fully utilize pre-trained models trained by ImageNet dataset to speed up training process and get better predictions. In the initialization stage, we download the pretrained model parameters and use them to initialize our models to be trained. 3.5
Basically we follow the method in [
Algorithm 1 Uncertainty Quantification
D = f(X; Y )g / Xi: training / evaluating chest X-ray images and corresponding observation labels
load pre-trained neural network (NN) models by ImageNet
Fine-tune NN models using cheXpert dataset
i) train NN models using SGD for some epochs with D ii) save statistics of the weights for those epochs iii) calculate wSW A and SW A using Eq. 4 and 5 vi) fit a Gaussian using wSW A as mean and SW A as covariance Prediction for s from 1 to S draw weights ws N (wSW Aj SW A) update batch norm statistics using D p(yisjXi) = p(yisjXi; ws) end for
p(yijXi) = S1 PS
s=1 p(yisjXi) Calculate y^i, Ua and Ue using Eq. (8), (6) and (7).
Utotal = Ua + Ue 6: Return:
y^i, Ua, Ue, Utotal
from which we can get wSW A and SW A using Eq. 4 and 5. Then
we fit a multivariate Gaussian using wSW A as mean and SW A as
covariance and get an approximated distribution for the neural
network weights. When doing a prediction, an input chest X-ray image
is repeatedly fed into the network for S times, each time with a new
set of weights sampled from the Gaussian distribution. The S
output probabilities are used to calculate the final predicted label y^i and
uncertainty Utotal = Ua + Ue. It is worthwhile to note that, after
drawing sample weights the network batch norm statistics need to
be updated for the models that use batch normalization. It can be
achieved by running one epoch with partial or full training set D.
More detailed justification for the necessity was given in the original
paper [
We perform experiments using the CheXpert data set [
The original dataset consists of training set and validation set and
we do not have access to test set. The labels for the training set were
generated by automated rule-based labeler which extract
information from radiology reports. This was done by the Stanford research
group who released the dataset. There are three possible values for
the label of an instance for a given observation, i.e., 1, 0 and 1. 1
means the observation is positive (or exists), 0 means negative (or
not exists), and 1 means not certain about whether the observation
exists. The labels for the validation set were determined by the
majority vote from three board-certified radiologists and only contains
positive (1) or negative (0) values. The original paper [
Specifically, the uncertain labels of cardiomegaly, consolidation and pleural effusion are replaced with 0, while edema and atelectasis with 1. Therefore the problem becomes a multi-label binary image classification problem. The predicted result is a five dimensional vector with element value being 1 or 0, where 1 means that the network predicts existence for the corresponding observation while 0 means the network predicts not existence of the corresponding observation. We follow the official training set / validation set split given by the data set provider. After removing invalid instances, we get a total number of 223,414 instances for training and 234 instances for validation. We first initialize the neural network’s parameters with corresponding downloaded pre-trained model parameters, and then train the neural network using the training set and test their performance on the validation set. We will use the original training set as the training set and original validation set as the evaluation set in our experiments.
In Figure 2 we show the patient statistics of the 5 observations after replacing the uncertain labels in the training set. The prevalence is the ratio of the number of positive instances over the total number of instances. From the figure we can see that all five observations are imbalance as the prevalence being under 50%. Besides, there is a gap in the prevalence for the training and evaluation sets in all observations, which will probably affect the performance of the neural network models. 5
In this section, we perform experiments and present the
investigation results of uncertainty quantification and strategy on five
different neural network models using PyTorch implementation. These
neural networks are DenseNet [
(d) Validation set age histogram
skip connections to mitigrate the gradient vanishment problem and
was the winner of ILSVRC 2015 [
All networks are trained as binary classifiers for multi-label classification instead of training separate models for each class.
The pipeline of the experiment is shown in Figure 4. We use PyTorch implementation. The neural network models and pretrained parameters are from torchvision (except SENet154 which is from pretrainedmodels, https://github.com/Cadene/ pretrained-models.pytorch).
In our experiment we set the number of sample weights T = 5, the number of columns of the deviation matrix K = 10 and the number of repeated prediction samples S = 10. During training, we use Adam optimizer with weight decay regularizer and ReduceLROnPlateau learning rate scheduler. The the initial learning rate is 1 10 5 and weight decay coefficient is 0:005. The maximum number of fine-tuning epoch is 50 epochs. The original chest X-ray images are resized and randomly cropped to 256 256 (except for SENet154 which has a fixed input size 224 224). We stop finetuning the model when the AUC (explained below) does not increase for consecutive 10 epochs and save the model with the best AUC as the optimal trained model.
We use four metrics to evaluate the network classification performance: Area under curve (AUC), Sensitivity, Specifity and Precision. Those metrics are widely used for machine learning and medicine community. The AUC is often used to measure the quality of a classifier and is defined as the area under the Receiver Operating Characteristic (ROC) curve which plots the sensitivity against the false positive rate. The sensitivity (or true positive rate or recall) is defined as the ratio of the number of correctly predicted positive instances over the number of total positive instances. The specificity is defined as the ratio of the number of correctly predicted negative instances over the total number of negative instances. And the precision is defined as the ratio of the number of correctly predicted positive instances over the number of instances that are predicted as positive. 5.1
First we compare the AUC of the original ordinary deterministic neural networks with the AUC corresponding neural networks after performing SWAG but before applying any uncertainty strategies. The results are shown in Table 1. The “Average” column is the average over all 5 observations. The bold font indicates better performance. For edema and pleural effusion, the original neural network performs better than SWAG for most of the networks. For cardimegaly, consolidation and atelectasis, the performances are mixed. This maybe because edema and pleural effusion are harder to detect and more sensitive to network weights perturbation. On the whole the SWAG algorithm does not outperform the original neural network. These might be accountable because SWAG uses a Gaussian to approximate the distribution over the optimal weights and then draws sample weights from the approximated Gaussian distribution, and may deviate from the optimal weights if the approximation is inaccurate. Therefore we need to adopt some strategy to prevent the performance from deterioration. The benefit lies in that we can get the uncertainty estimation for each prediction while keeping similar or even better prediction results. Next we utilize the uncertainty quantification information to determine if the performances can be improved. One strategy is to sort instances according to uncertainty in an ascending order, and then take those instances with less uncertainty into consideration and discard the rest. In clinical practice, the discarded instances could be flagged for further evaluation by a physician.
Ideally we would expect a decreasing trend for the metrics when data coverage increase as shown in Figure 6. The horizontal axis “Data coverage” is the percentage of instances being considered. For example, a data coverage of 20% means that only the top twenty percent of the least uncertain (or the most confident) instances are taken into consideration and the rest are discarded.
Figure 3 shows the comparison of performances with regard to the foure metrics (AUC, sensitivity, specificity and precision) between the original deterministic networks and Bayesian neural networks with uncertainty strategy. The solid lines are the Bayesian neural network with uncertainty strategy, while the dashed lines are the original ordinary deterministic networks without any uncertainty strategy. Different colors represent different observations.
From Figure 3 we can see that for edema and pleural effusion, the AUC decreases as the coverage increases, and are above the corresponding original AUC until around 45% and 90% coverage, respectively. This means that applying the uncertainty strategy can improve AUC for these two observations. The highest AUC gain can be 8% and 6% for edema and pleural effusion, respectively. We also observe similar trend in sensitivity, specificity and precision for both edema and pleural effusion. Three observations (cardiomegaly, atelectasis and consolidation) have low sensitivity as most of the predictions are negative. On the contrary the specificity is high.
The highest gains for applying the uncertainty strategy are shown in the Table 2. The effect of the uncertainty strategy over the five observations with the model DenseNet201 can be summarized as in the Table 3. The symbols p, , and represents helpful, not helpful, mixed behavior and missing value, respectively. For edema and pleural effusion, applying uncertainty strategy is beneficial for improving all four metrics. However, for other observations, it does not show benefits or only limited benefits for some metrics. The reason why it show varied behavior may be interesting and needs further investigation. Similarly, we summarize the effect of applying uncertainty strategy for different neural network architectures and the results are shown in Table 4 to Table 7. From the tables we can see that applying
Despite that for some observations (e.g., pleural effusion), several metrics performance benefit a lot from applying the uncertainty strategy, we should also notice that the strategy does not help to improve performance for some other observations with regard to these metrics, and in some cases even degrade the performance. The reasons behind might be varied and needs more investigation. For example, this may be that the neural network weight distribution approximated by the SWAG algorithm does not capture the true distribution, or even the uncertainty quantification formulas are inappropriate. 5.3
We also plot the total uncertainty distribution for each observation, as shown in Figure 5. From the figure we can see that for cardiomegaly, the estimated uncertainty tends to be smaller, while for edema, atelectasis and plueral effusion, the proportion of larger estimated uncertainty is higher. Consolidation has a relatively even distribution for estimated uncertainty. This suggest that edema, atelectasis and pleural effusion are more prone to be affected by setting an uncertainty threshold. Combining this finding with the results in Table 2, we set thresholds for both edema and pleural effusion to check the influence on metric performance. We only consider the instances whose estimated uncertainty is smaller than the threshold to compute the performance metrics. We vary the threshold from 0:2 to 0:24 by a step of 0:01 and the results are shown in Figure 7. The black dashed line is the average metric values of the original deterministic neural network, while the solid color thin lines are metric values for each observation, and the thick brown line is the average metric values of all five observation after applying threshold only to edema and pleural effusion. Comparing the thick brown line with the dash black line, we can see that the average specificity and precision have been improved while the average AUC and sensitivity roughly keep the same. This means that applying uncertainty threshold to edema and pleural effusion is beneficial. 6
In this paper we investigate uncertainty quantification in medical image classification using Bayesian deep neural networks. We train five different deep neural network models on the CheXpert X-ray image data for five clinical observations and quantify the model uncertainty. Then we analyze the performance of the network for situations with and without applying uncertainty strategy. The results show that the uncertainty quantification and strategy improve several performance metrics for some observations. This suggests that uncertainty quantification is helpful in medical image classification using neural networks. However, the results also show that in some cases the strategy is not helpful, or can even deteriorate the performance. Further analysis may be needed to examine this phenomenon.