1 Introduction

Over the past decades, there has been a significant increase in the incidence of skin cancers across the world potentially due to our social and lifestyle changes, as well as the depletion of the ozone layer [1]. Skin cancer, which is mostly caused by abnormal growth of skin cells, can be considered as one of the most prevalent cancer types. One in six Americans suffer from skin cancer at some point in their lives. This type of cancer accounts for one-third of all cancers in the United States. Near 75 percent of all skin cancer-related deaths are due to malignant melanoma. The most common type of skin cancer is non-melanoma with lower mortality rate. Despite the aforementioned increase in incidence of skin cancer, its diagnose is still challenging even for dermatologists [2,3,4]. Consequently, there has been a surge of interest in incorporation of computer-aided methodologies to assist the medical experts with their decision makings [5]. Recently, machine learning (ML), in particular, deep learning (DL) solutions, have achieved promising results in various application domains, encouraging their utilization for cancer screening/diagnosis [6]. Capitalizing on the fact that early diagnosis of skin cancer is of significant importance to improve life expectancy of patients, researchers strive to develop advanced DL models in this domain.

Generally speaking, DL methods have become widely popular for medical image processing and analysis tasks such as segmentation of cancerous legions. For instance, convolutional neural network (CNN) has been recently used for systematic categorization of skin legion diseases [7] and its performance has even been evaluated against a group of 58 dermatologists for skin cancer classification. As pointed out in [8], in another similar attempt, promising results have been obtained by using CNN for skin cancer classification [9,10,11]. The resulting DL model can be used as a complementary medical assistant for skin cancer classification [12]. One of the milestones of applying DL in medical applications is the lack of labeled data. This stems from the fact that labeling of medical images must be done by experts, which is both costly and time consuming. Additionally, deep neural networks (DNNs), typically, have thousands of parameters, therefore, learning appropriate parameter values from scratch may be computationally demanding. An alternative approach is transfer learning, which is generally used to fine-tune a pre-trained network to a new task and achieve acceptable performance in reasonable amount of time [13]. Deep neural networks (DNNs) have shown promising results in challenging tasks; however, their reliability is fragile when it comes to unseen data, which could lead to wrong medical diagnoses and, ultimately, patient deaths. This is due to the fact that DNNs are black-box models with deterministic behavior, lacking transparency in their decision-making process and the ability to estimate how certain they are about their own predictions. Several studies have been devoted to the important topic of uncertainty quantification for deep models [14].

Generally speaking, uncertainty is of two types: epistemic and aleatoric. Epistemic uncertainty, also known as model uncertainty, is due to limited training data or model complexity, which is reducible by providing enough data. Aleatoric uncertainty, on the other hand, arises due to the inherent noise of observations, making it data-dependent and cannot be reduced by gathering more data [?].

One possible way of uncertainty quantification in DNNs is through the Bayesian approach. Instead of using deterministic values for DNN parameters, the Bayesian approach imposes a probability distribution on them, providing a natural way of capturing uncertainty [15]. However, the computational complexity of this approach is high, hindering the estimation of posterior distribution in real time [13]. Bayesian neural networks begin with an initial value on the previous model and data parameters and use them to calculate and update the posterior distribution. For networks with thousands of parameters, finding and calculating the posterior distribution is complex and computationally costly. Estimation methods such as Monte Carlo (MC) can be used to address the computational complexity of the Bayesian approach. In this method, each sample is fed M times to the network equipped with dropout layers. Random activation/deactivation of the network neurons (due to dropout layers) yields M different outputs for the given input. These M outputs can be used to estimate mean and variance, providing a measure of uncertainty. However, the drawback of MC is the lack of calibration of forecasts, leading to performance lower than ensemble methods [16].

Fig. 1
figure 1

The density plots of predictive uncertainty predictions for correctly classified and misclassified samples of the ideal classification model

Full size image

The purpose of this paper is to propose a new method for improving the quantification of uncertainty in skin cancer detection models. Ideally, the model should assign high uncertainty to predictions that are not certain. By sorting the model’s outcomes according to their predictive entropy, we can estimate two distributions that correspond to correct and incorrect predictions, as shown in Fig. 1.

In our previous work [16], we addressed the drawbacks of the MCD method, including overconfidence and being non-calibrated, by proposing an uncertainty-aware loss function to optimize the model parameters. Although the use of this loss function leads to uncertainty quantification with acceptable accuracy, optimizing the model’s hyperparameters still requires manual intervention. Suboptimal choices of hyperparameter values can impede further improvements in MCD.

Inspired by the fact that MCD is very sensitive to the choice of hyperparameters and the issues discussed above, the main contribution of this paper can be listed as:

  • In addition to optimizing uncertainty accuracy (UA) and expected calibration error (ECE), our previously proposed uncertainty-aware loss function is exploited for automatic hyperparameter (e.g. dropout probability) tuning based on Bayesian optimization.

  • For the first time, uncertainty-aware diagnosis of skin cancer is investigated. Given that field of medical diagnosis is safety critical, robust uncertainty quantification is vital to successful deployment of DNNs in this field.

  • While Ensemble methods outperform MCD in terms of UA and ECE, they are computationally expensive. Achieving better uncertainty quantification but keeping computation complexity low is highly desirable. To this end, MCD is reinforced with better UA and ECE estimation as well as automatic hyperparameter tuning while computational complexity is much lower than Ensemble methods.

Through comprehensive experiments, we quantitatively evaluate our method against the existing ones such as the MCD and the Ensemble Bayesian Networks using cross-validation. We also perform a qualitative evaluation on the distributions of correctly and incorrectly categorized predictions, due to the fact that the ideal model has to separate the two distributions (their means should be as far as possible) and reduce their overlap. As a result, our model is able to detect wrong predictions which can be sent to a medical expert for further inspection.

The rest of the article is organized as follows: the background of our work is introduced in Sect. 2. The proposed method for better uncertainty quantification is presented in Sect. 3. The simulation and experimental results of our proposed method are given in Sect. 4. Finally, the paper is closed with conclusion in Sect. 5.

2 Background

In uncertainty-aware classification, two types of accuracy can be considered which are related to model prediction and model uncertainty quantification [?]. The motivation for uncertainty quantification is the fact that deterministic models are bound to predict the class for given input even if samples similar to it have not been seen during training. Under such circumstances, performing a prediction may lead to erroneous results. On the other hand, uncertainty-aware models are able to provide a measure of how certain they are about their own prediction. This way, the user will know when it is safe to trust the model’s prediction. However, solely providing an estimation of model uncertainty will not be of much use. It is vital to compute the accuracy of the model’s uncertainty estimation. A model with high uncertainty estimation accuracy and a high classification accuracy is the ideal case. Such a model is expected to assign a low uncertainty to its correct predictions and high uncertainty to its incorrect ones [17]. In the remainder of this section, the required background for uncertainty quantification is reviewed.

2.1 Predictive uncertainty quantification

For a given test sample, the predicted output of a model is either correct or incorrect. Using multiple forward passes of MCD, the predictive mean and standard deviation for the test sample can be computed. If the predictive standard deviation is less than a certain threshold, the model prediction is considered to be certain. For standard deviation higher than the threshold, the model is assumed to be uncertain about its own prediction. Considering all possible combinations of {correct, incorrect} and {certain, uncertain} yields the confusion matrix shown in Fig. 2. In this matrix, TC represents the cases in which the model has made a correct prediction that it is certain about. TU stands for the cases that the model has made a wrong prediction and the model is highly uncertain about its prediction correctness. FU is related to the cases that the model is uncertain about its own correct predictions. Finally, FC represents the worst error type which occurs when the model makes a wrong prediction with high confidence. Similar to the idea of the confusion matrix, we can propose four metrics for evaluating different Bayesian networks:

  • Uncertainty sensitivity (\(U_{{{\text{sen}}}}\)) which denotes the number of incorrect and uncertain estimates divided by the number of incorrect predictions:

    $$U_{{{\text{sen}}}} = \frac{{{\text{TU}}}}{{{\text{TU}} + {\text{FC}}}}.$$
    (1)

  • Uncertainty Specificity (\(U_{{{\text{spe}}}}\)) which is defined as the number of correct and certain predictions divided by the number of correct predictions:

    $$Uspe = \frac{{{\text{TC}}}}{{{\text{TC + FU}}}}.{\text{ }}$$
    (2)

  • Uncertainty precision (\(U_{pre}\)) which is related to the number of incorrect and uncertain predictions divided by the number of uncertain predictions:

    $$\begin{aligned} Upre = \frac{TU}{TU + FU}. \end{aligned}$$
    (3)

  • Uncertainty accuracy (UA) which is defined as the sum of diagonal predictions (TU and TC) divided by the total number of predictions:

    $$\ UA = \frac{{{\text{TU + TC}}}}{{{\text{TU + TC + FU + FC}}}}.$$
    (4)
Fig. 2
figure 2

The UCM and its components

Full size image

A trustworthy model has a high UA. The best value of UA is 1 and the worst value is 0. A network with UA metric near one is able to estimate its uncertainty level more accurately and identify the samples about which it is uncertain.

2.2 Uncertainty quantification techniques

The two uncertainty prediction techniques used in Bayesian neural networks are briefly described in the following subsections.

figure a

2.2.1 Monte Carlo dropout (MCD)

MCD is a Gaussian Process (GP) approximation that consists of retaining rate, model precision, and length scale parameters. According to [18], the posterior distribution of a Bayesian setting can be estimated by turning on the dropout at test time and feeding each sample to the network M times which yields M predictions. Due to the random nature of the dropout layers, the M predictions are not exactly the same and can be used to compute the mean and the standard deviation of the posterior distribution. The standard deviation is considered a measure of uncertainty. Higher standard deviation means higher uncertainty. More details on MCD method is already available in the literature [19]. The predictive mean (\(\mu _{pred}\)) for a test input x is estimated as:

$$\begin{aligned} \mu _{pred}(x, c) \approx \frac{1}{M} \sum _{m=1}^{M} p(y = c \mid x, \hat{\omega }_m), \end{aligned}$$
(5)

where \(p(y = c \mid x, \hat{\omega }_m)\) is the softmax probability of sample x belonging to class c, \(\hat{\omega }_m\) is the set of network parameters in \(m^{th}\) forward pass, and M is the number of forwarding passes of MCD. For uncertainty evaluation, the predictive entropy (PE) is calculated as follows [20, 21]:

$$\begin{aligned} PE(x) = - \sum _{c=1}^{C} \mu _{pred}(x, c) \log \left[ \mu _{pred} (x, c) \right] \end{aligned}$$
(6)

where C is the number of classes. Pseudo code of generating MCD is revealed in Algorithm 1.

figure b

2.2.2 Ensemble Bayesian networks

Probabilistic Bayesian methods are accepted as one of the best ways to quantify uncertainty. However, ensemble methods are computationally expensive leading to proposal of many approximate solutions. As an example, ensemble of neural networks is exploited to quantify epistemic uncertainties. Ensemble of networks is useful provided that the networks parameters are different enough from each other [22].

There are two types of methods for generating ensemble networks. In the first approach, for each sample multiple neural networks provide their predictions for which mean and standard deviation are computed. The mean is the final prediction and the standard deviation represents the ensemble uncertainty about the computed mean. This method is called ensembling of the models. In the second approach, a network is trained on different subsets of data, which is known as ensembling the data. In this paper, the first approach is used. Using an ensemble of N networks, the probability that input sample x belongs to class c can be estimated as follows [23]:

$$\begin{aligned} \hat{p} (y=c \mid x) = \frac{1}{N} \sum \limits _{i=1}^N p_{\theta _{i}} (y=c \mid x) \end{aligned}$$
(7)

where c is the class index and \(\theta _i\) denotes the set of parameters for \(i^{th}\) network of the ensemble. Moreover, the PE metric is defined as:

$$\begin{aligned} PE(x) = - \sum \limits _{c=1}^C \hat{p}(y=c \mid x)\ log \left[ \hat{p}(y=c\mid x) \right] \end{aligned}$$
(8)

Algorithm 2 shows the pseudo code of the ensemble setting.

3 Proposed method

Despite being powerful learners, DNNs performance is directly affected by the right choice of hyperparameter values. Moreover, DNNs are deterministic in nature and incapable of quantifying uncertainty about their own predictions. As mentioned before, methods like MCD can be used to make DNNs uncertainty-aware. However, MCD is known to provide miscalibrated uncertainty estimates [23] which we have tackled by proposing an uncertainty-aware multi-objective loss function [16]:

$$\begin{aligned} \hat{\mu }^{pred}_{b,c}&= \frac{1}{M} \sum _{m=1}^{M} \hat{y}^{(m)}_{b,c}, b=1:B, c=1:C, \end{aligned}$$
(9)
$$\begin{aligned} \hat{\mu }^{pred}_{b}&= \underset{c}{\textrm{argmax}}\,\, \hat{\mu }^{pred}_{b,c}, \end{aligned}$$
(10)
$$\begin{aligned} Loss&= \frac{1}{B} \sum _{b=1}^{B}\nonumber \\&\quad \left\{ \underbrace{-\left[ y_b \log \left( \hat{\mu }^{pred}_{b}\right) + (1 - y_b) \log \left( 1- hat{\mu }^{pred}_{b} \right) \right] }_{Binary \, cross \, entropy}\right. \nonumber \\&\qquad \quad \left. + \sum _{m=1}^{M} PE(x_{b}^{(m)})\right\} , \end{aligned}$$
(11)

where M is the number of forward passes of MCD, B is the batch size, C is the number of classes, \(y_b\) is the desired label for input sample \(x_b\), \(\hat{y}^{(m)}_{b,c}\) is a \([B \times C]\) matrix such that its \(b^{th}\) row is the network prediction (softmax output) corresponding to \(x_b\). The left-hand side of equation 9 is a \([B \times C]\) matrix and the left-hand side of equation 10 is a \([B \times 1]\) vector due to applying argmax operator. Finally, \(PE(x_b^{(m)})\) is the predictive entropy for \(x_b\) in the \(m^{th}\) forward pass of MCD. The loss function in equation 11 is the sum of traditional cross entropy and PEs aiming to minimize the overlapped region depicted in Fig. 1. According to the definition of the PE uncertainty metric, lower/higher PE means that the network has higher/lower confidence in its prediction. Using the loss function in equation 11 leads to the uncertainty aware training approach called MCD+entropy which improves MCD in terms of providing uncertainty estimates with better calibration.

The pseudo-code of MCD+Entropy method is available in Algorithm 3. In line 2, the T training epochs are started. In each epoch, the training set is traversed batch by batch. For each batch M, forward passes are computed (line 5) which are used to compute the PEs in line 7. After computing the loss value (line 8), the network parameters are updated using backpropagation at line 9. At the end of each epoch, the evaluation of the model is done on the test set (line 11).

figure c

To further improve the performance of the method in algorithm 3, it is necessary to find appropriate values for dropout probabilities which can be done using Bayesian Optimization (BO). In general, hyper-parameter tuning can be done by either random search, grid search or BO. The grid search is a brute force method which is very time consuming but achieves good results. As the name implies, random search explores the space of hyper-parameters completely randomly, which may come up with reasonable values but not necessarily optimal ones. Contrary to random search and grid search, the BO takes a wiser approach by controlling the search direction based on obtained observations. In the BO approach, an initial probability distribution is assumed in the space of hyper-parameter values. A sample is drawn from the aforementioned distribution and used as hyper-parameter values during the training. The BO distribution is then updated based on observed training loss. The update is done in an attempt to reduce the loss during the next training process. By repeating this process multiple times, the distribution in the hyper-parameter space keeps getting better and better.

Table 1 The resulted optimized parameters for each dataset by MCD+Entropy BO algorithm
Full size table

4 Simulation and results

In this section, the proposed method is compared with three Bayesian architectures in terms of epistemic uncertainty quantification. To this end, Moons and Blobs synthetic datasets from Scikit Learn library and MNIST HAM-10000 (real) dataset related to skin cancer have been used in the experiments. HAM10000 (Human Against Machine with 10000 training images) dataset is available at Kaggle [24] and includes 10015 dermoscopy images of cancer and non-cancerous cases in different age ranges.

For training and evaluation of the four Bayesian approaches, six-fold cross validation has been used. Given that deep models are data hungry and medical data are usually limited, using pre-trained models are advantageous. In our experiments, a DenseNet 121 pre-trained on ImageNet dataset was used for feature extraction. As preprocessing step, the images were resized to \(224 \times 224\) and standardized (normalized). Relying on DenseNet121, 50176 convolutional features were extracted for each image which were fed to two dense layers. The activation functions were set to Relu. Optimization was done using Adam [25] algorithm. The experiments were run on Google Colab with its default settings (GPU: Tesla K80, 12GB GDDR5 VRAM).

To generate our ensemble model, number of neurons in the first and second dense layers were randomly selected from the sets \(\{256, 257,..., 512\}\) and \(\{32, 33,..., 64\}\), respectively. The parameters of the two dense layers for each network in the ensemble were initialized randomly. The dropout probability was set to 0.25 in order to prevent overfitting during training. The hyperparameters of MCD and MCD+entropy i.e. P1, P2, L1, and L2 were set to 0.25, 0.25, 128, and 64, respectively. It should be noted that P and L stand for dropout probability and number of neurons in each dense layer, respectively.

In our algorithm, the parameters of the model are optimized, similar to MCD+entropy algorithm. Additionally, the hyperparameters are optimized using BO in each training epoch. The resulting hyperparameters are reported in Table 1 for each dataset.

Fig. 3
figure 3

UA metric for four algorithms on MNIST HAM-10000 dataset. UA is calculated for different uncertainty thresholds for different folds

Full size image
Fig. 4
figure 4

UA metric for different algorithms and different thresholds for Two moons and Blobs dataset. The upper row denotes to two moons dataset and the below one is for Blobs dataset for different noise levels

Full size image
Table 2 UA and ECE of different Bayesian models averaged over six folds
Full size table
Table 3 The qualitative comparison of four Bayesian models: \(\mu _1\) and \(\mu _2\) are the estimated centers of correct and incorrect distributions and Dist is the distance between the two centers
Full size table

The estimated UA for different noise thresholds applied to the two synthetic datasets (Two Moons and Blobs) is available in Fig. 3. The top row is associated with the Two Moons dataset and the bottom row is related to the Blobs dataset with different values of standard deviation. Large standard deviation values mean that the two classes are heavily overlapped. The results for synthetic datasets reveal the superiority of our algorithm (MCD+entropy BO) over the other evaluated methods.

Since the real datasets are usually imbalanced, the performance of different architectures may vary for different subsets of data (splitting of data to test and train may affect). To tackle this issue, we use six fold cross validation to measure methods performance. The UA for all four algorithms has been depicted in Fig. 4 revealing that our algorithm outperforms its rivals by achieving better UA for different thresholds as we compare the results with others. Obviously, our algorithm (using optimized hyperparameters) improves the MCD+entropy algorithm for differentiating between correct and incorrect predictions (according to the UA’s definition).

Table 2 reports expected values of UA and ECE computed over six folds (UA near one and ECE near zero is favorable). The lowest ECE belongs to MCD+entropy BO, which shows that this model is calibrated better than the other methods (ECE shows how much a specific model could produce calibrated predictions). Also, the UA of our method is higher than other methods revealing our model’s ability to assign low PE to correct predictions and high PE to incorrect ones.

All four Bayesian methods have been compared in Table 3. Similar to Fig. 1, the expected correct and incorrect predictions (\(\mu _1\) and \(\mu _2\)) and their distance (Dist) have been computed. Large Dist values for a model show its ability to distinguish between correct and incorrect distributions. As can be seen in Table 3, our proposed method has achieved higher Dist values demonstrating its superiority over the other three methods.

5 Conclusion

In this paper, we improved the uncertainty quantification performance of MCD method by proposing a novel approach for automatic optimization of deep model hyper-parameters. The proposed approach was compared against three other Bayesian algorithms, namely, the MCD, the Ensemble Bayesian networks, and the MCD+entropy method for quantifying uncertainty associated with the skin cancer dataset. Experiments on the MNIST HAM10000 dataset showed that MCD+entropy BO method outperforms its rivals by offering reliable uncertainty without sacrificing the classification accuracy achieving 85.65±0.18 prediction accuracy, 83.03±0.25 uncertainty accuracy, and 1.93±0.3 expected calibration error outperforming vanilla MCD and MCD with a loss enhanced based on predicted entropy. Uncertainty quantification is of paramount importance for making trustworthy decisions under uncertainty in critical tasks of biomedical engineering. For the future works, we will use different heuristic optimization algorithms such as Whale and Grey Wolf to find out whether they can improve our proposed method.