### What Is XAI? In 2016, DARPA initiated the explainable artificial intelligence (XAI) challenge. The goal is to build suites of machine learning algorithms that are interpretable without sacrificing prediction performance (Figure 1). The trade-off between learning performance and the effectiveness of explanation is illustrated in Figure 1. The work in this paper delivers a novel interaction-based methodology that produces the power of interpretability and explainability while maintaining state-of-the-art learning performance. A popular description of interpretability states the essential element for XAI is the ability to explain or to present in understandable terms to a human [16]. Another popular version defines interpretability as the degree to which a human can understand the cause of a decision [17]. Though intuitive, these definitions lack mathematical formality and rigorousness as certain perspectives [18]. In this paper, we regard the core issue of an XAI problem to be how features or variables are used to produce the prediction performance. In other words, the effectiveness of explanations in the DARPA document (Figure 1) is innately a variable set assessment and selection problem. This means the explainability and interpretability of a machine learning algorithm directly refer to what measures statisticians use to assess how the features or variables affect the final prediction results. In order to establish accountability, responsibility, and transparency in an AI system, one must first address an explainable and interpretable measure to assess feature importance. We define the following three dimensions, \(\mathcal{D}1\), \(\mathcal{D}2\), and \(\mathcal{D}3\), for a measure to be interpretable and explainable. In other words, these three dimensions serve as the key premises of the definition of an explainable and interpretable measure. This means that a measure that assesses the importance of a variable set needs to satisfy all three dimensions in order to be considered interpretable and explainable. We regard AI systems that explain features using an explainable measure such as explainable AI frameworks. This can be regarded as a heuristic framework for the existing concepts of interpretability and explainability. \(\mathcal{D}1\): A quantifiable measure for interpretability and explainability does not need to rely on the knowledge of correct specification of the underlying model. In other words, any interpretable explanation of features used in making predictions should be directly associated with the response variable. If a model is used in explaining the feature importance and mistakes are carried over from model fitting to providing explanations, it is exceptionally challenging to differentiate the mistakes occurring in feature selection from mistakes created in the fitted model. Suppose a data set has input variables \(X\) and response variable \(Y\), let us assume we use a fitted model \(f(\cdot)\), an ultra-deep neural network, to compute estimated label \(\hat{Y}=f(X)\). Then, the conventional approach is to compute the loss between \(\hat{Y}\) and ground truth \(Y\), i.e., \(\mathcal{L}(\hat{Y},Y)\). In this case, the explanations of how input variables \(X\) affect the response variable \(Y\) have contributions from both the input variables \(X\) and the fitted model \(f(\cdot)\). In the cases of using ultra-deep neural networks as the fitted model, there are hundreds of millions of parameters, which makes expressing \(f(\cdot)\) exceedingly challenging or an almost impossible task. This implies that the explanation is clouded by this fitted model and we cannot explicitly establish how the explanatory variables \(X\) affect the outcome variable (or response variable) \(Y\). The requirement for this dimension \(\mathcal{D}1\) is to directly create an understanding between \(X\) and outcome variable \(Y\) without searching for the fitted model \(f(\cdot)\).