SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\phi (\hat {\nu} _ {f} (\mathbf {x}, \cdot)) = \mathcal {N} \left(\mathbf {A B} (\mathbf {x}, [ d ]) ^ {\top} \tilde {m} (\mathbf {X}), \mathbf {A B} (\mathbf {x}, [ d ]) ^ {\top} \tilde {\mathbf {K}} _ {\mathbf {X X}} \mathbf {B} (\mathbf {x}, [ d ]) \mathbf {A} ^ {\top}\right) \tag {8}
$$
where  $\mathbf{B}(\mathbf{x},[d]) = [\mathbf{b}(\mathbf{x},[d]_1),\dots ,\mathbf{b}(\mathbf{x},[d]_{2^d})]^{\top}$
The complete algorithm, along with a discussion of computational techniques deployed to reduce computation time, such as vectorisation across  $\mathbf{X}$  using tensor operations and incorporation of the sparse GP formulation to speed up computations of quantities from Propositions 8 and 9, is provided in the appendix. It is worth noting that subsampling coalitions to reduce computational cost while estimating Shapley values is also a standard approach in SHAP algorithm implementations [2]. Empirically, this procedure has been shown to give an unbiased estimate of the true DSVs, with the variance decreasing at a rate of  $\mathcal{O}(\frac{1}{\ell})$ , where  $\ell$  is the number of coalition samples [22]. However, this approach incurs an additional source of uncertainty, due to estimation, and we propose to incorporate that as well into our framework by utilising the BayesSHAP approach proposed by Slack et al. [20].
Formulating BayesGP-SHAP. To capture the estimation uncertainty, Slack et al. [20] proposed BayesSHAP and reformulated the WLS procedure as a hierarchical Bayesian weighted least square and studied the corresponding posterior distribution over the deterministic Shapley values. We provide the hierarchical data generation process below, with an abuse of notations, for a deterministic  $f$ , we have
$$
\bar {\nu} \mid z, \bar {\phi}, \epsilon , f, \mathbf {x} \sim \bar {\phi} ^ {\top} z + \epsilon \quad \epsilon \mid z \sim \mathcal {N} (0, \sigma^ {2} w (z) ^ {- 1}) \tag {9}
$$
$$
\bar {\phi} \mid \sigma^ {2} \sim \mathcal {N} (0, \sigma^ {2} I) \quad \sigma^ {2} \sim \operatorname {I n v} - \chi^ {2} \left(\ell_ {0}, \sigma_ {0} ^ {2}\right) \tag {10}
$$
where  $w$  and  $z$  are the weight function and binary vector respectively introduced in Theorem 7, and  $\ell_0$ ,  $\sigma_0^2$  are two hyperparameters, typically set to small values to keep priors uninformative. Slack et al. [20] showed that the posterior distribution on  $\sigma^2$  and  $\bar{\phi}$  follows a scaled  $\mathrm{Inv - }\chi^2$  and normal respectively, due to their corresponding conjugacies with the likelihood.
Proposition 10 (BayesSHAP [20]). Given the data generation above, the posterior distribution on  $\bar{\phi}$  and  $\sigma^2$  follows:
$$
\bar {\phi} \mid \sigma^ {2}, \mathbf {Z} _ {\ell}, f, \mathbf {x}, \mathbf {D} \sim \mathcal {N} \left(\mathbf {A} _ {\ell} \bar {\mathbf {v}} _ {\mathbf {x}}, \left(\mathbf {Z} _ {\ell} ^ {\top} \mathbf {W} _ {\ell} \mathbf {Z} _ {\ell}\right) ^ {- 1} \sigma^ {2}\right) \tag {11}
$$
$$
\sigma^ {2} \mid \mathbf {Z} _ {\ell}, f, \mathbf {x}, \mathbf {D} \sim \text {S c a l e d - I n v -} \chi^ {2} \left(\ell_ {0} + \ell , \frac {\ell_ {0} \sigma_ {0} ^ {2} + \ell s ^ {2} (\bar {\mathbf {v}} _ {\mathbf {x}})}{\ell_ {0} + \ell}\right) \tag {12}
$$
where  $\ell$  is the number of coalitions  $\mathcal{S} = \{S_j\}_{j=1}^{\ell}$  we sample uniformly from  $2^{[d]}$ ,  $\mathbf{Z}_{\ell}$  is the binary matrix representing  $\mathcal{S}$ , and  $\mathbf{W}_{\ell}$  is the corresponding weight matrix, and  $\mathbf{A}_{\ell} = (\mathbf{Z}_{\ell}^{\top} \mathbf{W}_{\ell} \mathbf{Z}_{\ell})^{-1} \mathbf{Z}_{\ell}^{\top} \mathbf{W}_{\ell}$  is the WLS matrix,  $\bar{\mathbf{v}}_{\mathbf{x}} = [\bar{\nu}_f(\mathbf{x}, S_1), \dots, \bar{\nu}_f(\mathbf{x}, S_\ell)]^\top$  is the vector of deterministic payoffs, and
$$
s ^ {2} \left(\bar {\mathbf {v}} _ {\mathbf {x}}\right) = \frac {1}{\ell} \left[ \left(\bar {\mathbf {v}} _ {\mathbf {x}} - \mathbf {Z} _ {\ell} \mathbf {A} _ {\ell} \bar {\mathbf {v}} _ {\mathbf {x}}\right) ^ {\top} W _ {\ell} \left(\bar {\mathbf {v}} _ {\mathbf {x}} - \mathbf {Z} _ {\ell} \mathbf {A} _ {\ell} \bar {\mathbf {v}} _ {\mathbf {x}}\right) + \left(\mathbf {A} _ {\ell} \bar {\mathbf {v}} _ {\mathbf {x}}\right) ^ {\top} \left(\mathbf {A} _ {\ell} \bar {\mathbf {v}} _ {\mathbf {x}}\right) \right] \tag {13}
$$
measures the average weighted error in the regression and the norm of the mean explanations.
It is important to highlight that while both GP-SHAP and BayesSHAP have a notion of the "posterior of Shapley values", the two sources of uncertainty which these approaches capture are very different: GP-SHAP corresponds to the predictive uncertainty induced by the GP posterior  $p(f \mid \mathbf{D})$ , whereas BayesSHAP is the uncertainty due to having to estimate Shapley values when their exact computation is infeasible due to the exponential number of coalitions – and is in Slack et al. [20] proposed for a deterministic  $f$ . Nonetheless, by integrating the BayesSHAP posterior of Shapley values through the posterior GP  $p(f \mid \mathbf{D})$ , we can in fact incorporate both sources of uncertainty, leading to our second algorithm, BayesGP-SHAP.
Proposition 11 (BayesGP-SHAP). Continuing from Propositions 9 and 10, the posterior distribution of the stochastic Shapley values can be estimated using the Bayesian WLS approach as,
$$
\phi \mid \sigma^ {2}, \mathbf {Z} _ {\ell}, \mathbf {x}, \mathbf {D} \sim \mathcal {N} \left(\mathbf {A} _ {\ell} \mathbf {B} (\mathbf {x}, \boldsymbol {\mathcal {S}})\right) ^ {\top} \tilde {m} (\mathbf {X}), \mathbf {A} _ {\ell} \mathbf {B} (\mathbf {x}, \boldsymbol {\mathcal {S}}) ^ {\top} \tilde {\mathbf {K}} _ {\mathbf {X X}} \mathbf {B} (\mathbf {x}, \boldsymbol {\mathcal {S}}) \mathbf {A} _ {\ell} ^ {\top} + (\mathbf {Z} _ {\ell} ^ {\top} \mathbf {W} _ {\ell} \mathbf {Z} _ {\ell}) ^ {- 1} \sigma^ {2}\left. \right)
$$
where  $\sigma^2$  is sampled from  $\sigma^2\mid \mathbf{Z}_\ell \sim \mathrm{Scaled - Inv - }\chi^2\left(\ell_0 + \ell ,\frac{\ell_0\sigma_0^2 + \ell s^2(\mathbb{E}[\mathbf{v}_{\mathbf{x}}])}{\ell_0 + \ell}\right)$ .
We note that in the above proposition, instead of integrating  $p(\sigma^2 \mid \mathbf{Z}_\ell, f, \mathbf{x}, \mathbf{D})$  with respect to the posterior GP, which leads to a complex scaled mixture of Gaussians, we simplify the expression and construct a scaled inverse chi-square distribution with  $s^2(\mathbb{E}[\mathbf{v}_x])$  instead, which represents the error of the weighted regression with respect to the mean payoffs  $\mathbb{E}[\mathbf{v}_x]$ .
Conditionally on  $\sigma^2$ , the posterior variance in BayesGP-SHAP is therefore the sum of the variance from GP-SHAP and BayesSHAP due to Gaussian conjugacies.
4 Predictive explanations using the Shapley prior
In this section, we move beyond standard GP explanations by formulating explanation functions of a broader class of models as Gaussian processes. We introduce a Shapley prior over the space of vector-valued explanation functions  $\phi : \mathcal{X} \to \mathbb{R}^d$ , which correspond to Shapley values for arbitrary functions  $f$ . By treating previously obtained Shapley values as regression labels, we can predict explanations for new data without relying on the standard procedure to compute Shapley values. Our framework is not limited to explanations generated from GP-SHAP, but can also be applied to other explanation methods such as TreeSHAP, DeepSHAP, or model-agnostic KernelSHAP. The proposed approach learns the direct mapping from  $\mathcal{X}$  to the space of Shapley values, without the need to access the underlying model  $f$  during training, which is different from previous predictive approaches such as FastSHAP [39]. In contrast to prior work, Hill et al. [40] also considered fitting a GP regression model directly to explanation functions, but their focus is on incorporating the corresponding predictive uncertainty to explanations from black-box classifiers rather than predicting Shapley values for unseen data.
To achieve this, we leverage the key ideas behind the GP-SHAP algorithm, which uses the facts that GPs remain tractable under conditional expectations (to build the s-game) and linear combinations (to compute the stochastic Shapley values). By applying these properties to a GP prior  $\mathcal{GP}(0,k)$  instead of a posterior, we have an induced prior over the space of Shapley functions  $\phi$ .
Proposition 12 (The Shapley prior over  $\phi$ ). The prior  $f \sim \mathcal{GP}(0, k)$  and the corresponding stochastic game  $\nu_{f}(\mathbf{x}, S) = \mathbb{E}[f(X) \mid X_{S} = \mathbf{x}_{S}]$  induce a vector-valued GP prior over the explanation functions  $\phi \sim \mathcal{GP}(0, \kappa)$  where  $\kappa: \mathcal{X} \times \mathcal{X} \to \mathbb{R}^{d \times d}$  is the matrix-valued covariance kernel
$$
\kappa (\mathbf {x}, \mathbf {x} ^ {\prime}) = \mathcal {A} (\mathbf {x}) ^ {\top} \mathcal {A} (\mathbf {x} ^ {\prime}), \quad \mathcal {A} (\mathbf {x}) = \Psi (\mathbf {x}) \mathbf {A} ^ {\top} \tag {14}
$$
where  $\Psi (\mathbf{x}) = \left[\mathbb{E}[k(\cdot ,X)\mid X_{S_1} = x_{S_1}],\ldots ,\mathbb{E}[k(\cdot ,X)\mid X_{S_{2^d}} = x_{S_{2^d}}]\right]$ , and the  $\top$  sign refers to taking inner products in the RKHS of  $k$ .
By utilizing the Shapley function prior, we can apply our approach to predict explanations for a wide range of models, including trees, deep neural networks, or RKHS functions, by treating their explanations as noisy samples from this prior and using them as regression labels. Our approach is based on the perspective that there exists a true data generating mechanism  $f_{*}: \mathcal{X} \to \mathcal{Y}$  and that any model  $f$  we use is an approximation of  $f_{*}$ . Therefore, any explanations derived from  $f$  can be seen as an attempt to reveal the true explanations under  $f_{*}$ . This perspective has also been adopted in the work of Chen et al. [36] and Marx et al. [14]. We present the predictive posterior below.
Proposition 13 (Predictive explanations as multi-output GPs). Given  $\mathbf{D}_{\phi} = \{(\mathbf{x}_i,\phi_i)\}_{i = 1}^n = (\mathbf{X},\Phi_{\mathbf{X}})$  where  $\phi_i\in \mathbb{R}^d$  are the Shapley values computed under predictive model  $f$  and  $\Phi_{\mathbf{X}} = [\phi_1,\dots,\phi_n]^\top$ , the predictive explanations for new data  $\mathbf{x}'$  is distributed as,
$$
\phi \left(\mathbf {x} ^ {\prime}\right) \mid \mathbf {D} _ {\phi} \sim \mathcal {N} \left(\tilde {m} _ {\phi} \left(\mathbf {x} ^ {\prime}\right), \quad \kappa \left(\mathbf {x} ^ {\prime}, \mathbf {x} ^ {\prime}\right) - \kappa \left(\mathbf {x} ^ {\prime}, \mathbf {X}\right) b _ {\kappa} \left(\mathbf {x} ^ {\prime}, \mathbf {X}\right)\right) \tag {15}
$$
where  $\tilde{m}_{\phi}(\mathbf{x}^{\prime}) = b_{\kappa}(\mathbf{x}^{\prime},\mathbf{X})^{\top}\operatorname {vec}(\Phi_{\mathbf{X}}),b_{\kappa}(\mathbf{x}^{\prime},\mathbf{X}):= (\mathcal{X}_{\mathbf{XX}} + \sigma_{\phi}^{2}I)^{-1}\kappa (\mathbf{X},\mathbf{x}^{\prime}),\mathcal{X}_{\mathbf{XX}}$  is the gram matrix for kernel  $\kappa$  of size  $nd\times nd,\kappa (\mathbf{x}^{\prime},\mathbf{X}) = [\kappa (\mathbf{x}^{\prime},\mathbf{x}_{1}),\dots ,\kappa (\mathbf{x}^{\prime},\mathbf{x}_{n})]$  is of size  $d\times nd$  and  $\sigma_{\phi}^{2}$  is the noise parameter for regression.
In fact, we see that the posterior mean  $\tilde{m}_{\phi}(\mathbf{x}^{\prime})$  from (15) are Shapley values of the following payoff vector  $\tilde{\mathbf{v}}_{\mathbf{x}^{\prime}}$ , computed based on the observed explanations  $\Phi_{\mathbf{X}}$ ,
Proposition 14 (Posterior mean as Shapley values for payoff vector  $\tilde{\mathbf{v}}_{\mathbf{x}'}$ ). The posterior mean  $\tilde{m}_{\phi}(\mathbf{x}')$  corresponds to Shapley values for the payoff vector  $\tilde{\mathbf{v}}_{\mathbf{x}'}$ , i.e.,  $\tilde{m}_{\phi}(\mathbf{x}') = \mathbf{A}\tilde{\mathbf{v}}_{\mathbf{x}'}$ , where  $\tilde{\mathbf{v}}_{\mathbf{x}'} = \sum_{i=1}^{n} \Psi(\mathbf{x}')^\top \Psi(\mathbf{x}_i) \mathbf{A}^\top \alpha_i$  and  $\alpha_i \in \mathbb{R}^d$  is the  $[i, \dots, i + (d-1)]$  subvector of  $(\mathcal{X}_{\mathbf{XX}} + \sigma_{\phi}^2 I)^{-1} \operatorname{vec}(\Phi_{\mathbf{X}})$ .
5 Illustrations
We demonstrate the proposed approaches through three sets of illustrations. We first conduct an ablation study to examine how predictive and estimation uncertainty captured by our explanation algorithms vary under different configurations. We then move on to discuss various exploratory tools that can assist practitioners in utilizing the stochastic explanations. Finally, we demonstrate the
Figure 1: Ablation study on different uncertainties captured by GP-SHAP, BayesSHAP, and BayesGP-SHAP when computing local explanations (SSVs) using the California housing dataset [41].  $95\%$  credible intervals around explanations are shown.
effectiveness of the Shapley prior for the predictive explanations problem. Our code is included in the supplementary material, and we provide implementation details in the appendix. For all experiments, we pick the radial basis function (RBF) as our covariance kernel  $k$  for the GPs.
5.1 Ablation study on different notions of uncertainties captured
We conducted a comparison between GP-SHAP, BayesSHAP, and BayesGP-SHAP to demonstrate the differences between model predictive uncertainty and estimation uncertainty captured in the stochastic Shapley values. BayesSHAP is applied to the predictive posterior mean of GP to obtain its explanation. For this purpose, we used the California housing dataset [41] from the StatLib repository, which includes 20640 instances and 8 numerical features, with the goal of predicting the median house value for California districts, expressed in hundreds of thousands of dollars. We trained our GP model using  $25\%$  and  $100\%$  of the data and calculated local stochastic explanations from GP-SHAP, BayesSHAP, and BayesGP-SHAP using  $50\%$  and  $100\%$  of coalitions. The results, shown in Figure 1, demonstrate that the magnitude of BayesSHAP uncertainties (green bars) are uniform across features as it is designed to capture the overall estimation performance and not feature-specific uncertainty. In contrast, GP-SHAP (red bars) exhibits varying uncertainties across features due to the propagation of variation from the posterior  $\mathrm{GP}f$  to the attributions. This allows practitioners to make more granular statements about the uncertainty around specific feature explanations. We also observe that increasing the number of coalitions and training data generally leads to a decrease in both estimation and predictive uncertainties, but the estimation uncertainty drops more significantly. BayesGP-SHAP (yellow bars) consistently provides a more conservative uncertainty estimation by considering both predictive and estimation uncertainties.
5.2 Exploratory analysis of the stochastic explanations
We propose several exploratory analysis methods to aid practitioners in comprehending the stochastic explanations generated for their downstream tasks. To this end, we use BayesGP-SHAP to explain a Gaussian process model trained on the breast cancer dataset [42]. The dataset consists of 569 patients, 30 numeric features, and the objective is to predict whether a tumor is malignant or benign.
Local explanation. To visualize local explanations, we can plot the mean and standard deviations of stochastic Shapley values in a bar plot. This approach allows us to not only understand the degree of contribution a feature has to the prediction but also the corresponding credible interval as a measure of explanation uncertainty. Figure 2a displays that both "worst fractal dimension" and "worst perimeter" features have a similar contribution in terms of their absolute mean stochastic Shapley values. However, the model is much more uncertain about the former, allowing the user to calibrate their trust in model explanations.
(a) Local explanation
(b) Global explanation
(c) Local explanation correlation
Graphical model for top 10 explanations for patient 1
(d) Local explanation graphical model
Figure 2: Illustrations of possible exploratory analyses utilising the BayesGP-SHAP covariance structure, applied on the breast cancer dataset.
Heuristic global explanation as means of absolute stochastic Shapley values. It is common to compute the mean absolute DSVs as a proxy to global feature contributions [43] based on computed local explanations. Although this heuristic does not result in any explanations that are themselves DSVs of any game at the global level, they provide a quick summary to practitioners about the overall contributions. We can similarly compute the average of the absolute stochastic Shapley values, which are now distributed according to a folded multivariate Gaussian distribution [44] with a tractable covariance structure. However, it is important to highlight a key distinction from the previous approach: when computing the mean of absolute SSVs, we consider the uncertainty in the local stochastic explanations, while directly calculating the absolute values of mean SSVs (equivalent to computing absolute DSVs) disregards this uncertainty. We believe the former approach is more appropriate than the latter as practitioners utilising Gaussian processes would want their global explanations to take the predictive uncertainty into account. The blue and yellow bars in Figure 2b depict these two approaches, respectively. Notably, we observe that the blue bars suggest that the "worst fractal dimension" feature is considered more influential than "worst concave points" because it accounts for the higher variability in the former feature's explanation. In contrast, the yellow bar overlooks this variability, leading to a different conclusion.
Correlations and dependencies across explanations. As we have access to the explanation covariance, we can explore their correlation and visualise them as in Figure 2c. For instance, the stochastic Shapley values of "concavity error" are less correlated with other features. Moreover, as our explanations are multivariate Gaussians, we can build an undirected Gaussian graphical model [45] using the precision matrix (inverse of the covariance matrix) to study the independence across the local stochastic explanations. We visualise the corresponding graphical model for explanations from patient 1 in Figure 2d by setting a sparsity threshold of  $90\%$ , following the approach in [46].
5.3 Predictive explanations
Finally, we showcase the effectiveness of our Shapley prior by comparing the regression performances of a multi-output GP using the Shapley prior with multi-output random forest and neural networks when predicting withheld explanations generated from GP-SHAP, Tree-SHAP, and DeepSHAP, respectively. We use the diabetes dataset [47] from the UCI repository, which contains 442 patients with 10 numerical features and the goal is to predict disease progression. We first train a GP, a random forest, and a neural network, to obtain the
subsequent explanations from GP-SHAP, TreeSHAP, and DeepSHAP. Next, we feed  $70\%$  of the explanations to our predictive model as training data and the remaining  $30\%$  as test data.
We repeat this process over 10 seeds and compute the root mean squared error between the predicted explanation and the exact explanations for respective models, as shown in Figure 3. We observe that our GP-model with the Shapley prior consistently outperforms random forest and neural network model for predicting all three types of explanations. This demonstrates that the inductive bias that comes from our choice of covariance structure  $\kappa$  allows to build more accurate predictive explanation models. We also observe that the overall average prediction error for explanations generated from GP-SHAP is lower than that of TreeSHAP [11] and DeepSHAP [2], suggesting that GP-SHAP produce explanations that are easier to learn. This follows our intuition as GPs are typically smoother functions than trees and neural networks.
Figure 3: Predictive performance of using Shapley prior to predict explanations generated from different explanation algorithms on the diabetes dataset.
In this work, we presented a novel and principled approach for explaining Gaussian process models using stochastic Shapley values. The proposed algorithm GP-SHAP, and its variant BayesGP-SHAP, allow practitioners to consider both predictive and estimation uncertainties when reasoning about the explanations and calibrate their trust in the model accordingly. Furthermore, we consider the setting of predictive explanations, where we introduced a Shapley prior over explanation functions, enabling us to model and predict Shapley values for a wider range of predictive models. We demonstrated the effectiveness of our methods through a number of illustrations and discussed various exploratory tools for practitioners to analyse the stochastic explanations.
Limitations and future outlook. While the general framework of using stochastic Shapley values for stochastic explanations can be applied beyond Gaussian process models, the specific estimation algorithms presented in this paper are tailored to GPs and may not be directly applicable to other probabilistic models like Bayesian neural networks. These alternative models would require different estimation procedures as the resulting explanations would not be GPs anymore. Another promising avenue for future research involves leveraging the uncertainty obtained from predicting explanations and incorporating it into Bayesian optimization for guiding experimental design. For example, this could allow practitioners to explore data regions that produce significant Shapley values for specific features. At last, we would like to highlight that although SHAP is popular, as an explanation tool it contains several limitations, specifically Kumar et al. [48] have shown that SHAP violates several expected properties of explanations from a human-centric perspective.
The authors would like to thank Jean Francois Ton, Shahine Bouabid, Jake Fawkes, and Simon Foll for insightful discussions. The authors are also indebted to anonymous reviewers whose feedback has significantly improved the manuscript.
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SUPPLEMENTARY MATERIAL: Explaining the uncertain: Stochastic Shapley values for Gaussian process models
A The GP-SHAP algorithm and discussion on computation techniques
We present the complete algorithm for both GP-SHAP and BayesGP-SHAP in Algorithm 1.
Algorithm 1 GP-SHAP / BayesGP-SHAP
Input: Posterior mean function  $\tilde{m}$ , posterior covariance function  $\tilde{k}$ , inducing locations  $\tilde{\mathbf{X}}$ , explanation instances  $\mathbf{X}$ , number of coalition samples  $n_Z$ , hyperparameter  $\lambda, n_0, \sigma_0^2$ , base kernel  $k$ , algorithm algo,
1: Compute  $n_I =$  number of inducing location,  $n =$  number of explanation instances,  $d =$  number of features.
2: Compute Cholesky decomposition on posterior covariance  $\mathbf{L}\mathbf{L}^{\top} = \tilde{\mathbf{K}}_{\tilde{\mathbf{X}}\tilde{\mathbf{X}}}$
3: Sample coalitions  $\mathcal{S} = \{S_1,\dots,S_{n_Z}\}$  from [d], build binary matrix  $\mathbf{Z} = \{0,1\}^{n_Z\times d}$  from  $\mathcal{S}$ , and compute weights  $W = \mathrm{diag}[w_1,\dots,w_{n_Z}]$  with  $w_{i} = \frac{d - 1}{\binom{d}{|S_{i}|}|S_{i}|(d - |S_{i}|)}$ .