# Enhancing Neural Network Interpretability Through Deep Prior-Guided Expected Gradients
## 4 Experiments
### Results
#### 4.3.2 Improvement of Model Robustness to Noise via Attribution Priors  
In traditional deep learning training, the optimization objective is to minimize the loss function \(L(\theta;X,y)\), where \(\theta\) represents the model parameters, \(X\) are the input data, and \(y\) is the labels. Regularization terms are often added to prevent overfitting, and the optimization problem can be expressed by Equation (24).  
\[\theta=\underset{\theta}{\text{argmin}}L(\theta;X,y)+\lambda^{\prime}\Omega^{ \prime}(\theta) \tag{24}\]  
where \(\Omega^{\prime}(\theta)\) is the regularization term for the model parameters, and \(\lambda^{\prime}\) controls the regularization strength.  
When incorporating attribution priors, the model must minimize both the loss function and constraints on feature attributions. Let \(\Phi(\theta,X)\) represent the feature attribution matrix, where each element \(\phi_{li}\) denotes the importance of feature \(i\) in sample \(l\). The prior attribution is introduced as a penalty function \(\Omega(\Phi(\theta,X))\), with \(\lambda\) controlling the regularization strength. The optimization objective is modified according to the mathematical expression Equation (25).  
\[\theta=\underset{\theta}{\text{argmin}}L(\theta;X,y)+\lambda\Omega(\Phi( \theta,X)) \tag{25}\]  
This formulation ensures that the model not only minimizes prediction error but also adheres to attribution constraints, enhancing robustness and interpretability.  
For image tasks, we adopt the pixel-wise smoothness prior (\(\Omega_{\text{pixel}}\)) proposed by [PERSON] et al. [22], which enforces local consistency in feature attributions by penalizing differences between adjacent pixels, shown in Equation (26).  
\[\Omega_{\text{pixel}}(\Phi(\theta,X))=\sum_{\ell}\sum_{i,j}|\phi_{i+1,j}^{ \ell}-\phi_{i,j}^{\ell}|+|\phi_{i,j+1}^{\ell}-\phi_{i,j}^{\ell}| \tag{26}\]  
The strength of the prior \(\lambda\) is selected via grid search over a logarithmic scale (\(10^{-20}\) to \(10^{-10}\)), where we choose the optimal \(\lambda\) that simultaneously minimizes \(\Omega_{\text{pixel}}\) on the training set while maintaining test accuracy within 1% of the baseline model to avoid significant performance degradation. For instance, in the MNIST experiments, the optimal \(\lambda\) was empirically set to 0.01 for both DeepBGMM-EG and DeepMGM-EG, effectively balancing attribution smoothness and model accuracy. This approach ensures that the prior sufficiently enforces the desired spatial consistency in pixel attributions without compromising predictive performance.  
In this experiment, we assessed the noise robustness of different models by progressively introducing Gaussian noise into the test data of the MNIST handwritten digit dataset. Four models were evaluated: DeepBGMM-EG (BGMM prior-based Expected Gradients), DeepMGM-EG (MGM prior-based Expected Gradients), EG (standard Expected Gradients), and a baseline model with no attribution priors. The objective was to observe how the classification accuracy of each model changed as the noise level increased.  
As shown in Figure 6, the test precision of the four models was plotted against different noise levels, ranging from 0% to 100%. For noise levels below 0.3, all models exhibit similar performance, maintaining a test accuracy close to 0.9, with DeepBGMM-EG and DeepMGM-EG performing slightly better. However, as the noise percentage increased, both DeepBGMM-EG and DeepMGM-EG significantly outperformed the base and EG models, particularly at higher noise levels.  
The base and EG models showed a steep decline in accuracy when the noise level exceeded 0.5, indicating poor performance in high-noise environments. In contrast, models trained with prior attribution--especially those using DeepBGMM-EG and DeepMGM-EG--showed notable improvements in noise robustness. These models effectively maintained higher accuracy rates as noise increased from 0.3 to 1.0, highlighting the impact of incorporating prior attribution on improving the model's resilience to noisy data.