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A New Class of Explanations for Classifiers with Non-binary Features

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Logics in Artificial Intelligence (JELIA 2023)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 14281))

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Abstract

Two types of explanations have been receiving increased attention in the literature when analyzing the decisions made by classifiers. The first type explains why a decision was made and is known as a sufficient reason for the decision, also an abductive explanation or a PI-explanation. The second type explains why some other decision was not made and is known as a necessary reason for the decision, also a contrastive or counterfactual explanation. These explanations were defined for classifiers with binary, discrete and, in some cases, continuous features. We show that these explanations can be significantly improved in the presence of non-binary features, leading to a new class of explanations that relay more information about decisions and the underlying classifiers. Necessary and sufficient reasons were also shown to be the prime implicates and implicants of the complete reason for a decision, which can be obtained using a quantification operator. We show that our improved notions of necessary and sufficient reasons are also prime implicates and implicants but for an improved notion of complete reason obtained by a new quantification operator that we also define and study.

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Notes

  1. 1.

    We will use sufficient reasons and PI/abductive explanations interchangeably.

  2. 2.

    See, e.g., [13, 44, 49] for some approaches that can be viewed as approximating sufficient reasons and [26] for a study of the quality of some of these approximations.

  3. 3.

    We will use necessary reasons and contrastive explanations interchangeably in this paper. Counterfactual explanations are related but have alternate definitions in the literature. For example, as defined in [5], they correspond to length-minimal necessary reasons; see [18]. But according to some other definitions, they include contrastive explanations (necessary reasons) as a special case; see Sect. 5.2 in [34]. See also [1] for counterfactual explanations that are directed towards Bayesian network classifiers and [2] for a relevant recent study and survey.

  4. 4.

    Interestingly, the axiomatic study of explanations in [3] allows non-binary features, yet Axiom 4 (feasibility) implies that explanations must be simple.

  5. 5.

    For example, we can use it to provide forgetting semantics for the dual operator \(\overline{\exists }\,x_i \cdot \varDelta = \overline{\overline{\forall }\,x_i \cdot \overline{\varDelta }}\). Using Definition 2, we get \(\overline{\exists }\,x_i \cdot \varDelta = \varDelta +\varDelta | x_i\). Using Proposition 4, we get \(\overline{\exists }\,x_i \cdot \varDelta = \varDelta | x_i +\sum _{j \not = i} (x_j \cdot \varDelta | x_j)\). We can now easily show that (1) \(\varDelta \models \overline{\exists }\,x_i \cdot \varDelta \) and (2) \(\overline{\exists }\,x_i \cdot \varDelta \) is equivalent to an NNF whose X-literals do not mention state \(x_i\). That is, \(\overline{\exists }\,x_i\) can be understood as forgetting the information about state \(x_i\) from \(\varDelta \). This is similar to the dual operator \(\exists x_i \cdot \varDelta = \overline{\forall x_i \cdot \overline{\varDelta }}\) studied in [19, 32] except that \(\overline{\exists }\,x_i\) erases less information from \(\varDelta \) since one can show that \(\varDelta \models \overline{\exists }\,x_i \cdot \varDelta \models \exists x_i \cdot \varDelta \).

  6. 6.

    Unlike SRs, two GSRs may mention the same set of variables. Consider the class formula \(\varDelta = (x_1 \cdot y_{12}) +(x_{12} \cdot y_1)\) and instance \(\mathcal{I}= x_1 \cdot y_1\). There are two GSRs for the decision on \(\mathcal{I}\), \(x_1 \cdot y_{12}\) and \(x_{12} \cdot y_1\), and both mention the same variables X, Y.

  7. 7.

    A dual notion, contrastive path explanation (CPXp), was also proposed in [27].

  8. 8.

    An NNF circuit is a DAG whose leaves are labeled with \(\bot , \top \), or literals; and whose internal nodes are labelled with \(\cdot \) or \(+\).

  9. 9.

    The condition \(V \cap (vars(\tau ) \setminus vars(\tau ')) \ne \emptyset \) is trivially satisfied when \(\varDelta \) is the root of the NNF circuit since V will include all circuit variables in this case.

  10. 10.

    The number of clauses in this CNF will be no more than the number of NNF nodes if the NNF is the general reason of a decision tree (i.e., the NNF has a tree structure).

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Acknowledgments

This work has been partially supported by NSF grant ISS-1910317.

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Authors and Affiliations

  1. University of California, Los Angeles, Los Angeles, CA, 90095, USA

    Chunxi Ji & Adnan Darwiche

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  1. Chunxi Ji
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Correspondence to Chunxi Ji .

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  1. TU Dresden, Dresden, Germany

    Sarah Gaggl

  2. Artificial Intelligence Research Institute - IIIA CSIC, Barcelona, Spain

    Maria Vanina Martinez

  3. Umeå University, Umeå, Sweden

    Magdalena Ortiz

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Ji, C., Darwiche, A. (2023). A New Class of Explanations for Classifiers with Non-binary Features. In: Gaggl, S., Martinez, M.V., Ortiz, M. (eds) Logics in Artificial Intelligence. JELIA 2023. Lecture Notes in Computer Science(), vol 14281. Springer, Cham. https://doi.org/10.1007/978-3-031-43619-2_8

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