@@ -217,7 +217,7 @@ To assess counterfactuals with respect to Definition~\ref{def:faithful}, we need
where $\mathbf{r}_j \sim\mathcal{N}(\mathbf{0},\mathbf{I})$ is the stochastic term and the step-size $\epsilon_j$ is typically polynomially decayed~\citep{welling2011bayesian}. The term $\mathcal{E}_{\theta}(\mathbf{x}_j|\mathbf{y}^+)$ denotes the model energy conditioned on the target class label $\mathbf{y}^+$ which we specify as the negative logit corresponding to $\mathbf{y}^{+}$. To allow for faster sampling, we follow the common practice of choosing the step-size $\epsilon_j$ and the standard deviation of $\mathbf{r}_j$ separately. While $\mathbf{x}_J$ is only guaranteed to distribute as $p_{\theta}(\mathbf{x}|\mathbf{y}^{+})$ if $\epsilon\rightarrow0$ and $J \rightarrow\infty$, the bias introduced for a small finite $\epsilon$ is negligible in practice \citep{murphy2023probabilistic}.
Generating multiple samples using SGLD thus yields an empirical distribution $\widehat{\mathbf{X}}_{\theta,\mathbf{y}^+}$ that approximates what the model has learned about the input data. While in the context of EBM, this is usually done during training, we propose to repurpose this approach during inference in order to evaluate the faithfulness of model explanations. The appendix provides additional implementation details for any tasks related to energy-based modelling\footnote{The supplementary appendix can be found here: https://github.com/pat-alt/ECCCo.jl.}.
Generating multiple samples using SGLD thus yields an empirical distribution $\widehat{\mathbf{X}}_{\theta,\mathbf{y}^+}$ that approximates what the model has learned about the input data. While in the context of EBM, this is usually done during training, we propose to repurpose this approach during inference in order to evaluate the faithfulness of model explanations. The appendix provides additional implementation details for any tasks related to energy-based modelling\footnote{The supplementary appendix can be found here: https://arxiv.org/abs/2312.10648.}.
\subsection{Quantifying the Model's Predictive Uncertainty}
