Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Scaling the Explanation of Multi-Class Bayesian Network Classifiers

Yaofang Zhang, Adnan Darwiche

Abstract

We propose a new algorithm for compiling Bayesian network classifier (BNC) into class formulas. Class formulas are logical formulas that represent a classifier's input-output behavior, and are crucial in the recent line of work that uses logical reasoning to explain the decisions made by classifiers. Compared to prior work on compiling class formulas of BNCs, our proposed algorithm is not restricted to binary classifiers, shows significant improvement in compilation time, and outputs class formulas as negation normal form (NNF) circuits that are OR-decomposable, which is an important property when computing explanations of classifiers.

Scaling the Explanation of Multi-Class Bayesian Network Classifiers

Abstract

We propose a new algorithm for compiling Bayesian network classifier (BNC) into class formulas. Class formulas are logical formulas that represent a classifier's input-output behavior, and are crucial in the recent line of work that uses logical reasoning to explain the decisions made by classifiers. Compared to prior work on compiling class formulas of BNCs, our proposed algorithm is not restricted to binary classifiers, shows significant improvement in compilation time, and outputs class formulas as negation normal form (NNF) circuits that are OR-decomposable, which is an important property when computing explanations of classifiers.
Paper Structure (18 sections, 3 theorems, 7 equations, 8 figures, 3 tables, 2 algorithms)

This paper contains 18 sections, 3 theorems, 7 equations, 8 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Bayesian Network Classifier
  4. Explaining Decisions using Class Formulas
  5. Prior Work: Block-Order Based Compilation
  6. Feature Tree
  7. Splitting Features
  8. Classifying Partial Feature Instantiation
  9. Compiling Class Formulas using f-Trees
  10. Complexity Analysis
  11. Experimental Results
  12. Explanations in a Multi-Class BNC
  13. Complete Reasons
  14. General Reasons
  15. Contrastive Explanations
  16. ...and 3 more sections

Key Result

theorem 1

Let $\mathcal{T}$ be an f-tree of BNC $(\mathcal{N}, \mathbf{X}, Y)$, $e$ be an edge in $\mathcal{T}$ that splits features $\mathbf{X}$ into partition $(\mathbf{U},\mathbf{V})$. For a partial feature instantiation $\mathbf{u}$, if $\Delta_\mathbf{S}^{ij}(\mathbf{u})$ is non-positive for some $j \neq

Figures (8)

  • Figure 1: An example Bayesian network. The root variable A is in red, and the leaf variables D, E, F are in green. For simplicity, all variables in this network are binary.
  • Figure 2: (a) An AND-decomposable NNF for the negative instances of $y_1$, and (b) its negation, which is an OR-decomposable NNF of the class formula for $y_1$.
  • Figure 3: (a) An OBDD compiled by shih_compiling_2019. Solid (or dashed) arrows denote feature set to true (or false). (b) An equivalent OR-decomposable NNF.
  • Figure 4: A jointree and an f-tree (the sub-tree in the box) for the network in \ref{['fig:examplebn']}. The nodes are labeled by their clusters and the edges are labeled by their separators. The nodes that include the target variable A are in red, and the nodes that include the features are in green.
  • Figure 5: Cactus plot of runtime.
  • ...and 3 more figures

Theorems & Definitions (6)

  • definition 1: Feature tree (f-tree)
  • definition 2: Feature-Splitting Edge
  • theorem 1
  • theorem 2
  • theorem 3
  • proof