\section{Related Works}
\label{section: Limitation-existing-work}
Several works have studied different approaches to mitigate the inconsistency of LIME. ALIME \citep{shankaranarayana2019alime} incorporates an autoencoder as a weighting mechanism to assess the proximity of sampled points to the instance being explained (IE), thereby improving coefficient stability. DLIME \citep{zafar2019dlime} applies hierarchical clustering to segment the dataset into multiple clusters and selects representative points from the cluster nearest to the IE, ensuring alignment with LIME’s locality principle. S-LIME \citep{zhou2021s} enhances the consistency of LIME explanations by introducing a hypothesis testing framework, which utilizes the Central Limit Theorem (CLT) to determine the required sample size for stable explanations. BayLIME \citep{zhao2021baylime} adopts a Bayesian approach for local surrogate modeling, where explanations are generated by combining prior knowledge with estimates from newly sampled data through a Bayesian-weighted summation. SLICE \citep{Bora_2024_CVPR} uses a two stage strategy of using adaptive Gaussian Blur (Adaptive-blur) followed by feature selection algorithm to remove inconsistent superpixels from explanations. The feature selection algorithm uses bootstrapping to estimate the probability of sign flips of coefficients (i.e., sign entropy) and eliminate superpixels with a high likelihood of sign flips. A surrogate model is built using Ridge Regression, similar to the original LIME \citep{ribeiro2016should}. SLICE \citep{Bora_2024_CVPR} achieved high consistency and fidelity as compared to LIME \citep{ribeiro2016should} and BayLIME \citep{zhao2021baylime} at the cost of a considerable increase in execution time due to the use of bootstrapping for feature selection. 
 
 \section{Our Contributions}
 \label{sec:contribution}
 We propose a new method, BELIEF, to achieve consistency of explanations similar to SLICE \citep{Bora_2024_CVPR}, using a Bayesian Regularization approach, eliminating the need for external feature selection and offering faster computation time. Specifically, we propose a novel Sign Entropy regularization, modeling it in Bayesian paradigm instead of the bootstrapping-based frequentist approach. Thus, our approach achieves consistency and fidelity similar to SLICE but with much faster computing time ($\approx10\times$) by eliminating bootstrapping based feature selection. 
 
 %This enables us to use the distribution of coefficients instead of the point estimates used in the frequentist approach for feature selection. Thus, unlike $SLICE$ to integrate the feature selection of $SLICE$ into the Bayesian Regression.
 
We first demonstrate our proposed method for reducing the uncertainty associated with the sign of the coefficients in linear models, thereby enhancing consistency on tabular datasets. We then show the broader use of our proposed method in stabilizing LIME for consistent explanations for DL-based image classification applications. With extensive experiments on multiple tabular and image datasets, we evaluate and compare our approach with State-Of-The-Art (SOTA) counterparts for consistent explanation. Additionally, we perform statistical validation using multiple tests to provide empirical evidence supporting our claims of consistent explanation, high fidelity, and lower execution time.





