As a favorable tool for explainable artificial intelligence (XAI), Shapley value has been widely used to interpret deep learning based predictive models. However, accurate and efficient estimation of Shapley value is difficult since the computation load grows exponentially with the increase of input features. Most existing accelerated estimation methods have to compromise on estimation accuracy with efficiency. In this article, we present EmSHAP(Energy-based model for Shapley value estimation) to estimate the expectation of Shapley contribution function under arbitrary subset of features given the rest. The energy-based model estimates the conditional density in the Shapley contribution function, which involves an energy network for approximating the unnormalized conditional density and a GRU (Gated Recurrent Unit) network for approximating the partition function. The GRU network maps the input features onto a hidden space to eliminate the impact of input orderings. In order to theoretically evaluate the performance of different Shapley value estimation methods, Theorems 1, 2 and 3 analyzed the error bounds of EmSHAP as well as two state-of-the-art methods, namely KernelSHAP and VAEAC. It is proved that EmSHAP has tighter error bound than KernelSHAP and VAEAC. Finally, case studies on two application examples show the enhanced estimation accuracy of EmSHAP.

Explainable artificial intelligence (XAI) and interpretable machine learning (IML) have become active research fields in recent years (Adadi and Berrada 2018; Molnar 2019). This is a natural consequence as complex machine learning (ML) models are now applied to solve supervised learning problems in many high-risk areas: cancer prognosis (Kourou et al. 2015), credit scoring (Kvamme et al. 2018), and money laundering detection (Jullum, Løland, et al. 2020). The high prediction accuracy of complex ML models often comes at the expense of model interpretability. As the goal of science is to gain knowledge from the collected data, the use of black-box models hinders the understanding of the underlying relationship between the features and the response, and thereby curtail scientific discovery. Model explanation frameworks from the XAI field extract the hidden knowledge about the underlying data structure captured by a black-box model, and thereby make the model's decision-making process transparent. This is crucial for, e.g., medical researchers that apply an ML model to obtain well-performing predictions, but who simultaneously also strive to discover important risk factors. Another driving factor is the Right to Explanation legislation in EU's General Data Protection Regulation (GDPR) (European Commission 2016).

Undirected graphical models are compact representations of joint probability distributions over random variables. To solve inference tasks of interest, graphical models of arbitrary topology can be trained using empirical risk minimization. However, to solve inference tasks that were not seen during training, these models (EGMs) often need to be re-trained. Instead, we propose an inference-agnostic adversarial training framework which produces an infinitely-large ensemble of graphical models (AGMs). The ensemble is optimized to generate data within the GAN framework, and inference is performed using a finite subset of these models. AGMs perform comparably with EGMs on inference tasks that the latter were specifically optimized for. Most importantly, AGMs show significantly better generalization to unseen inference tasks compared to EGMs, as well as deep neural architectures like GibbsNet and VAEAC which allow arbitrary conditioning. Finally, AGMs allow fast data sampling, competitive with Gibbs sampling from EGMs.