## 3 Methodology  
### Problem Formulation  
Let \(\mathcal{S}=\{(\mathbf{x}_{i},\mathbf{y}_{i})\}_{i=1}^{\mathcal{N}}\subset \mathcal{Z}\) be a sample from a distribution \(\mathcal{D}\) in a domain \(\mathcal{Z}=\mathcal{X}\times\mathcal{Y}\), where \(\mathcal{X}\) is the instance space and \(\mathcal{Y}\) is the label space. We learn a differentiable predictive function \(f\in\mathcal{F}:\mathcal{X}\rightarrow\mathcal{Y}\) together with a transparent explainer function \(g\in\mathcal{G}:\mathcal{X}\rightarrow\mathcal{Y}\) defined over a functional class \(\mathcal{G}\). We refer to functions \(f\) and \(g\) as the predictor and the explainer, respectively, throughout the paper. \(\mathcal{G}\) is strictly constrained to be an inherently explainable functional set, such as a set of linear functions or decision trees. We assume that we have a distance function \(d:\mathcal{X}\times\mathcal{Y}\rightarrow\mathbb{R}_{\geq 0}\) such that \(d(y,\hat{y})=0\longleftrightarrow y=\hat{y}\), which measures the point-wise similarity between two probability distributions in \(\mathcal{Y}\) and can be used to optimize \(f\) and \(g\). Instead of learning a post hoc white-box model, our idea is to learn a white-box model that is explainable from the start and then let this explainer model guide the learning of the black-box predictor model. To accomplish this goal, we design two different frameworks: (1) a Pre Hoc Explainable Predictive Framework, where the white-box model regularizes the black-box model for optimized fidelity, and (2) a Co Hoc Explainable Predictive Framework, where the white-box and black-box models are optimized simultaneously with a shared loss function that enforces fidelity. Enforcing Fidelity  
Given an inherently interpretable white-box model \(g\) with parameters \(\phi\), let its predictions result in a probability distribution \(p_{\phi}\). Given the black-box model \(f\) with parameters \(\theta\), let its predictions result in probability distribution \(p_{\theta}\) over \(K\) classes \(y\in\mathcal{Y}=\{1,2,\ldots,K\}\). We propose a fidelity objective function, which measures the point-wise probability distance between \(p_{\phi}\) and \(p_{\theta}\), which are, respectively, the outputs of \(g\) and \(f\) for all given input data \(\mathcal{X}\). The optimization problem is formulated as follows:  
\[\min_{f\in\mathcal{F}}\frac{1}{N}\sum_{i=1}^{N}D(f(x_{i}),g(x_{i})), \tag{7}\]  
where function \(D\) is a divergence distance measurement, such as the Jensen-Shannon divergence [57]. We use \(D_{JS}\), Jensen-Shannon divergence, to measure the point-wise deviation of the predictive distributions \(f_{\theta}\) and \(g_{\phi}\). Denote by \(\mathcal{P}\) the set of probability distributions. The Kullback-Leibler divergence (KL). \(\text{KL}:\mathcal{P}\times\mathcal{P}\rightarrow[0,\infty]\) is a fundamental distance between probability distributions in \(\mathcal{D}\)[58], defined by  
\[D_{\text{KL}}(p||q):=\int p\log\frac{p}{q}\;\mathrm{d}\mu, \tag{8}\]  
where \(p\) and \(q\) denote probability measures \(P\) and \(Q\), respectively, with respect to measure \(\mu\). In our context, \(P\) represents the probability distribution from the white-box explainer model \(g_{\phi}\), and \(Q\) represents the distribution from the black-box predictor model \(f_{\theta}\). Let \(p,q\in\Delta^{K-1}\) have the corresponding weights \(\mathbf{\pi}=[\pi_{1},\pi_{2}]^{T}\in\Delta\). Then, the Jensen-Shannon divergence between \(p\) and \(q\) is given by  
\[ \begin{split} D_{\text{JS}}(p,q)&:=H(m)-\pi_{1}H(p )-\pi_{2}H(q)\\ &=\pi_{1}D_{\text{KL}}(p\|m)+\pi_{2}D_{\text{KL}}(q\|m),\end{split}  \tag{9}\]  
with \(H\) the Shannon entropy, and \(m=\pi_{1}p+\pi_{2}q\). Unlike the Kullback-Leibler divergence \((D_{\text{KL}}(p\|q))\), JS is symmetric, bounded, and does not require absolute continuity.