Bayesian-Driven Multi-Kernel SVM with Statistical Feature Fusion for Skin Lesion Diagnosis

2.1 Dataset description and sampling rationale

The HAM10000 dataset serves as the foundation for experimental validation [3]. The experimental dataset comprises 500 dermoscopic images systematically sampled from the HAM10000 repository, representing seven distinct diagnostic classifications: melanocytic nevi (nv), benign keratosis-like lesions (bkl), melanoma (mel), basal cell carcinoma (bcc), actinic keratoses (akiec), vascular lesions (vasc), and dermatofibroma (df). The observed class distribution exhibits a characteristic clinical imbalance, with nv constituting 66.4% (332 instances) while df represents merely 1.0% (5 instances). For computational efficiency and balanced evaluation, we extracted a stratified sample of 500 images, maintaining the original class distribution [2, 3].

(a) Data preprocessing and feature engineering pipeline

(b) Bayesian-Enhanced Multi-Kernel Support Vector Machine (BEMK-SVM) based classification, uncertainty estimation, and evaluation framework

Figure 1. Overall system architecture

Note: Skin Lesion Analysis System Workflow

The selection of 500 samples follows established methodological principles in algorithmic development research. This sample size was strategically chosen based on several considerations. First, statistical power analysis indicates that 500 samples provide adequate power (> 0.80) for detecting medium effect sizes (Cohen's d = 0.5) in multi-class classification contexts with α = 0.05 [7]. Second, this subset enables rigorous computational experimentation with multiple kernel combinations and hyperparameter configurations while maintaining reproducibility standards. Third, the stratified sampling procedure preserves the original class distribution proportions from the complete HAM10000 dataset, ensuring representativeness of the clinical population characteristics. Fourth, comparable sample sizes have been employed in seminal methodological studies establishing foundational machine learning techniques for medical imaging applications [8, 9]. We acknowledge that this moderate sample size represents a controlled experimental setting; subsequent validation on the complete dataset and external cohorts constitutes an essential future research direction discussed in the Conclusion section.

Figure 1 illustrates the complete system architecture, displaying the complete data flow from preprocessing through classification.

Comprehensive feature engineering was involved in data preprocessing to extract clinically relevant information [36]. Seventeen handled features were expanded from eight primary features through advanced engineering techniques.

2.2 Feature engineering pipeline

• Categorical encoding: Label encoding with normalization was used to encode diagnosis type, localization, and sex variables.
• Feature summary statistics: Cross-feature interactions and polynomial feature generation.

The final feature set is included: age, sex_encoded, localization_encoded, dx_type_encoded, age_group_encoded, malignancy_risk, age_sex_interaction, and age_zscore.

2.2.1 Bayesian-Enhanced Multi-Kernel Support Vector Machine

The proposed BEMK-SVM framework extends traditional MKL by incorporating Bayesian inference for adaptive kernel weighting [14, 15]. Given a set of M base kernels {K₁, K₂, ..., K_M}, the combined kernel is defined as a convex combination with learned weights. Unlike conventional MKL approaches that optimize weights through constrained optimization, our framework employs Bayesian posterior estimation to learn a probability distribution over kernel weights [10, 11].

The Bayesian formulation provides two key advantages: first, it naturally quantifies uncertainty in kernel weight estimates, which propagates to prediction uncertainty; second, it enables adaptive kernel combination that responds to local data characteristics [23, 24]. The posterior distribution over kernel weights is computed using variational inference, which provides a tractable approximation while maintaining computational efficiency suitable for clinical applications [8, 26].

The BEMK-SVM architecture addresses fundamental limitations of single-kernel classification through a principled Bayesian framework for heterogeneous kernel integration. The theoretical foundation rests on the observation that different kernel functions capture complementary data characteristics: radial basis functions model local similarities, polynomial kernels encode global polynomial relationships, and sigmoid kernels approximate neural network decision boundaries. Rather than employing uniform or heuristically determined kernel weights, BEMK-SVM derives optimal combination coefficients from the posterior probability distribution over kernel hypotheses given observed training data. This Bayesian treatment naturally quantifies epistemic uncertainty arising from model selection ambiguity, providing calibrated confidence estimates alongside point predictions.

The limitation of single-kernel SVMs is addressed by the BEMK-SVM algorithm through the grouping of multiple kernel functions via Bayesian weighting. The mathematical formulation is:

1) Multi-kernel combination

The Standard SVM optimization problem for a single kernel

$\underset{w,b,~\xi }{\mathop{\min }}\,(\frac{1}{2}{{\left| \left| w \right| \right|}^{2}}+C\underset{i}{\mathop \sum }\,\xi )~subject~to~{{y}_{i}}\left( w.\phi \left( {{x}_{i}} \right)+b \right)\ge 1-~\xi ,~~\xi \ge 0$                      (1)

${{K}_{combined}}\left( {{x}_{i}},{{x}_{j}} \right)=\underset{k=1}{\overset{K}{\mathop \sum }}\,{{w}_{k}}{{K}_{k}}\left( {{x}_{i}},{{x}_{j}} \right)$                          (2)

where, K represents the number of kernels, w_k are Bayesian-derived weights, and K_k(x_i, x_j) are individual kernel functions.

2) Individual kernel functions

Radial Basis Function (RBF) kernel:

${{K}_{RBF}}\left( {{x}_{i}},{{x}_{j}} \right)=\text{exp}(-\gamma {{\left| \left| {{x}_{i}},{{x}_{j}} \right| \right|}^{2}}$                  (3)

Polynomial kernel:

${{K}_{poly}}\left( {{x}_{i}},{{x}_{j}} \right)={{\left( \gamma \left\langle x_i, x_j\right\rangle +r \right)}^{d}}$                    (4)

Sigmoid kernel:

${{K}_{sigmoid}}\left( {{x}_{i}},{{x}_{j}} \right)=tanh{{\left( \gamma \left\langle x_i, x_j\right\rangle+r \right)}^{{}}}$                    (5)

3) Bayesian weight calculation

Posterior probability:

$P\left(M_{\mathrm{k}} \mid D\right)=\frac{P\left(D \mid M_{\mathrm{k}}\right) P\left(M_{\mathrm{k}}\right)}{\sum_{\mathrm{j}=1} 1^{\mathrm{K}} P\left(D \mid M_{\mathrm{j}}\right) P\left(M_{\mathrm{j}}\right)}$                    (6)

Likelihood estimation:

$P\left( D\text{ }\!\!|\!\!\text{ }{{M}_{k}} \right)=\underset{i=1}{\overset{n}{\mathop \prod }}\,P\left( {{y}_{i|}}{{x}_{i}},{{M}_{k}} \right)$                              (7)

Model evidence:

$logP(D|{{M}_{k}}=~\underset{i=1}{\overset{n}{\mathop \sum }}\,log\left( {{y}_{i}}\text{ }\!\!|\!\!\text{ }{{x}_{i}},{{M}_{k}} \right)-\lambda {{\left| \left| {{\theta }_{k}} \right| \right|}^{2}}$

where, λ is the regularization parameter and θₖ are model parameters.

The Bayesian weights are computed using cross-validation performance and model uncertainty:

${{w}_{k}}=P\left( {{M}_{k}}\text{ }\!\!|\!\!\text{ }D \right)\propto \text{P}\left( \text{D }\!\!|\!\!\text{ }{{M}_{k}} \right)\text{*P}\left( {{M}_{k}} \right)$

where, P(M_k|D) is the posterior probability of kernel k given data D, P(D|M_k) is the likelihood, and P(M_k) is the prior.

Algorithm implementation:

• Train multiple SVM models with different kernels (RBF, polynomial, sigmoid)
• Evaluate each model using 5-fold cross-validation
• Calculate Bayesian weights based on performance metrics and uncertainty
• Combine kernel outputs using weighted voting
• Apply uncertainty quantification for prediction confidence

The BEMK-SVM architecture (Figure 2) combines multiple kernels using Bayesian weights calculated according to Eq. (2).

Figure 2. Bayesian-Enhanced Multi-Kernel Support Vector Machine (BEMK-SVM) architecture

Table 1. Bayesian kernel weights and performance

Note: Weights normalized to sum = 1.0. Higher uncertainty (σ) indicates lower reliability.

4.1 Performance analysis and mechanistic interpretation

The observed performance improvement of BEMK-SVM (91.33%) over conventional single-kernel SVM (88.00%) warrants a mechanistic explanation beyond superficial accuracy comparison. The 3.33 percentage point accuracy gain represents a 27.75% reduction in error rate, indicating meaningful practical improvement. Several factors contribute to this enhancement. First, the multi-kernel architecture enables simultaneous exploitation of complementary feature representations. The RBF kernel effectively captures localized similarities between lesion presentations, while polynomial kernels model global interactions among clinical attributes. This heterogeneous representation proves particularly valuable for the seven-class discrimination task, where lesion subtypes exhibit varying degrees of visual and clinical overlap.

Second, the Bayesian weight optimization procedure automatically identifies the most informative kernel configurations for the specific dataset characteristics encountered. Analysis of the learned kernel weights reveals that the RBF kernel received the highest posterior probability (0.47), followed by polynomial (0.31) and sigmoid (0.22) kernels. This distribution suggests that local similarity patterns dominate the classification signal for dermatological features, while polynomial and sigmoid components contribute complementary discriminative information. Importantly, these weights were learned from data rather than manually specified, eliminating a significant hyperparameter selection burden present in conventional multi-kernel approaches.

4.2 Comparison with prior art

Contextualizing the present results within existing literature requires careful consideration of methodological differences. Esteva et al. [7] achieved dermatologist-level performance using deep convolutional networks trained on substantially larger datasets (> 100,000 images), demonstrating the potential of end-to-end learning approaches. However, their methodology lacks explicit uncertainty quantification and requires computational resources unsuitable for resource-constrained clinical settings. Gal et al. [8] reported comparable accuracy on similar lesion classification tasks using transfer learning, though their approach similarly provides only point predictions without confidence calibration.

The present BEMK-SVM framework addresses a complementary niche: providing interpretable, uncertainty-aware predictions using classical machine learning foundations suitable for clinical deployment where computational constraints exist. The 91.33% accuracy achieved on our controlled 500-sample dataset demonstrates methodological validity, while acknowledging that direct comparison with deep learning approaches trained on orders-of-magnitude larger datasets would be methodologically inappropriate. The key contribution lies not in absolute performance maximization but in the principled integration of uncertainty estimation with multi-kernel classification, enabling confidence-aware decision support in clinical workflows.

A distinguishing feature of the BEMK-SVM framework is its ability to provide calibrated uncertainty estimates [23, 24]. Analysis of prediction confidence reveals a strong correlation between model confidence and classification accuracy. High-confidence predictions (probability > 0.9) achieved 96.2% accuracy, while low-confidence predictions (< 0.5) showed significantly reduced accuracy, demonstrating effective uncertainty calibration [25, 26]. This capability enables clinical workflows where uncertain cases can be flagged for additional expert review [28].

The classification performance of all evaluated models on the HAM10000 test set is summarized in Table 1.

4.2.1 Classification performance

The proposed BEMK-SVM framework was assigned a classification accuracy of 91.33% on the HAM10000 test set, with significant performance being achieved over baseline methods, including standard single-kernel SVM (88.00%) and Random Forest (88.67%). These results are aligned with recent findings in deep learning-based skin lesion classification, in which accuracies are reported to range from 85% to 95% depending on dataset characteristics and model architecture [31, 33]. The improvement over conventional approaches is demonstrated by the effectiveness of Bayesian kernel combination and information-theoretic feature weighting.

Competitive performance is revealed by comparative analysis with existing literature. Dermatologist-level accuracy was achieved by Esteva et al. [7] using deep convolutional networks trained on substantially larger datasets. While impressive results are achieved by their methodology, explicit uncertainty quantification is lacking, and computational resources unsuitable for resource-constrained clinical settings are required. Recent ensemble methods have also been discussed.

Table 4 shows that BEMK-SVM achieves the highest performance across all evaluation metrics, including accuracy, precision, recall, F1-score, and AUC, demonstrating its superior classification capability. The baseline models exhibit comparatively lower performance, with the ensemble meta-learner showing the weakest results.

Table 4. Model performance comparison

Note: BEMK-SVM = Bayesian-Enhanced Multi-Kernel Support Vector Machine; SVM = Support Vector Machine; AUC = Area Under the Curve.

Table 5. Confusion matrix analysis (BEMK-SVM)

Note: BEMK-SVM = Bayesian-Enhanced Multi-Kernel Support Vector Machine.

Table 6. Statistical significance tests

Note: BEMK-SVM = Bayesian-Enhanced Multi-Kernel Support Vector Machine; SVM = Support Vector Machine; CI = Confidence Interval.

The classification accuracy was 91.33%, with 137 out of 150 predictions being correct.

The confusion matrix of the proposed BEMK-SVM model is illustrated in Table 5, showing the relationship between actual and predicted class labels. Accurate classification performance is indicated by high values along the diagonal, particularly for the nv and akiec classes. Limited misclassifications are observed among visually similar lesion categories such as bkl, mel, and bcc. Overall, the effectiveness and robustness of the BEMK-SVM classifier are demonstrated by the results.

The results of statistical significance tests comparing the proposed BEMK-SVM with baseline classifiers are reported in Table 6. Statistically significant performance improvements over Standard SVM, Random Forest, and Ensemble Meta models are indicated by the mean difference, confidence intervals, and t-statistics related to BEMK-SVM. The robustness of these improvements is confirmed by very low p-values. In contrast, no significant difference is observed between Standard SVM and Random Forest.

The highest accuracy of 91.33% was attained by the BEMK-SVM, representing a 3.33% upgrading over Standard SVM and a 2.66% upgrading over Random Forest. The effectiveness of Bayesian kernel weighting and uncertainty quantification is established by the superior performance.

4.3 Feature importance analysis

The feature attribution analysis reveals clinically meaningful patterns. Malignancy risk indicators emerged as the dominant predictive features (64.21% aggregate importance), consistent with dermatological practice where risk stratification guides diagnostic pathways. Age-related factors collectively contributed 21.45% importance, reflecting the established epidemiological relationship between patient age and skin lesion presentation profiles. The relatively modest contribution of sex-encoded features (9.12%) suggests that biological sex plays a secondary role in lesion classification for this cohort, potentially indicating that sex-related variations are already captured through age-sex interaction terms.

Malignancy risk indicators were identified as the dominant predictors (64.21% relative importance) by feature importance analysis, followed by diagnosis type encoding and age-related factors [17, 19]. Established dermatological diagnostic criteria and clinical decision-making processes are aligned with these findings [28, 29]. The interpretability provided by feature importance analysis is enhancing clinical acceptance by allowing the basis for model predictions to be understood by practitioners [9].

Table 7 illustrates feature importance analysis showing that malignancy risk is the strongest predictor, followed by diagnosis type and age-related features. Demographic and anatomical factors contribute moderately, indicating both clinical and demographic relevance in the model’s decision-making.

Table 7. Feature importance weights

Table 8. Feature correlation matrix

Strong correlations (|r| > 0.8) are shown: age age_zscore (r = 1.0), dx_type_encoded malignancy_risk (r = 0.823). The optimal feature subset is provided by the top 5 features, which account for 99.3% of full performance with 37.5% fewer features.

Table 9 illustrates feature ablation results for BEMK-SVM, showing that the full feature set achieves the highest accuracy. Removing malignancy risk or limiting features to age-only subsets causes a significant performance drop, highlighting the critical role of clinical features, particularly malignancy risk, in model performance.

Table 9. Feature selection ablation study

Figure 3. Comparison of feature importance across different methods

Figure 4. Bayesian kernel weight evolution during training

The comparison of feature major rankings from dissimilar methods: PFW, Random Forest, and Mutual Information is characterized by Figure 3. The malignancy risk is validated as the most discriminative feature by the consensus ranking.

The evolution of Bayesian weights for different kernels during cross-validation is represented by Figure 4. Dominant contributors to the group have been revealed through RBF kernels with C = 1.0 and C = 10.0.

A very high significance (64.21%) was generated, showing poor specificity of the risk assessment feature, which confirms its medical utility as a primary diagnostic indicator. Significant contributions to classification performance were made by age-related features collectively.

4.4 Computational efficiency

It has been indicated by computational complexity analysis that higher training time (127.3 +/− 8.4 seconds) and memory usage (45.2 MB) are required by BEMK-SVM compared to Standard SVM (23.1 +/− 2.1 seconds, 12.8 MB) and Random Forest (18.7 +/− 1.9 seconds, 28.4 MB) [12, 13]. However, low prediction latency (2.8 +/− 0.3 ms) is maintained, making the framework suitable for clinical deployment where real-time inference is required [16]. The additional computational cost during training is justified by improved accuracy and uncertainty quantification capabilities.

4.5 Class-specific performance

Varying classification accuracy across different lesion types was revealed by a detailed analysis of per-class performance:

Melanocytic nevi (nv) had 94.2% accuracy (best performing class):

• Benign keratosis (bkl) had 89.1% accuracy
• Melanoma (mel) had 87.5% accuracy
• Basal cell carcinoma (bcc) had 85.3% accuracy
• Actinic keratoses (akiec) had 82.1% accuracy
• Vascular lesions (vasc) had 75.0% accuracy
• Dermatofibroma (df) had 60.0% accuracy (challenging due to small sample size)

Table 10 illustrates a computational efficiency comparison showing that BEMK-SVM incurs higher training time and memory usage than Standard SVM and Random Forest, but maintains fast prediction time, making it suitable for deployment scenarios where inference speed is critical.

Table 10. Computational complexity analysis

4.5.1 Impact on performance metrics

In the left subfigure of Figure 5, a significant imbalance in the class distribution of the HAM10000 dataset is shown. In the right subfigure, the correlation between sample size and classification accuracy is revealed by per-class performance metrics.

Figure 5. Class distribution impact on performance metrics

In Table 11, class-wise performance of the proposed model shows high precision, recall, and AUC for common lesion classes, with strong weighted-average performance (F1-score = 0.913). Lower scores in rare classes reflect class imbalance, while high specificity across all classes indicates reliable discrimination.

Table 11. Detailed class-wise performance metrics

Table 12. Confusion matrix analysis (BEMK-SVM)