Navigating Explanatory Multiverse Through Counterfactual Path Geometry

TL;DR

The paper introduces explanatory multiverse, a framework that models the full geometry of counterfactual explanations as multiple interconnected journeys rather than isolated points. It formalises two complementary implementations: a vector-space version for theoretical grounding and a graph-based FACElift for scalable retrieval, introducing the opportunity potential metric to quantify how much following one path helps reach alternatives. Across tabular and image datasets, FACElift demonstrates higher opportunity potential at a controlled distance cost, supporting greater explainee agency and interactive decision-making. The work advances human-centered explainability by incorporating path geometry, branching dynamics, and monotonicity constraints, with open-source code to enable reproducibility and further exploration.

Abstract

Counterfactual explanations are the de facto standard when tasked with interpreting decisions of (opaque) predictive models. Their generation is often subject to technical and domain-specific constraints that aim to maximise their real-life utility. In addition to considering desiderata pertaining to the counterfactual instance itself, guaranteeing existence of a viable path connecting it with the factual data point has recently gained relevance. While current explainability approaches ensure that the steps of such a journey as well as its destination adhere to selected constraints, they neglect the multiplicity of these counterfactual paths. To address this shortcoming we introduce the novel concept of explanatory multiverse that encompasses all the possible counterfactual journeys. We define it using vector spaces, showing how to navigate, reason about and compare the geometry of counterfactual trajectories found within it. To this end, we overview their spatial properties -- such as affinity, branching, divergence and possible future convergence -- and propose an all-in-one metric, called opportunity potential, to quantify them. Notably, the explanatory process offered by our method grants explainees more agency by allowing them to select counterfactuals not only based on their absolute differences but also according to the properties of their connecting paths. To demonstrate real-life flexibility, benefit and efficacy of explanatory multiverse we propose its graph-based implementation, which we use for qualitative and quantitative evaluation on six tabular and image data sets.

Figures (8)

• Figure 1: Example of explanatory multiverse constructed for tabular data with two continuous (numerical) features. It demonstrates various types of counterfactual path geometry such as the affinity, branching, divergence and convergence of these explanations. Each journey terminates in a (possibly the same or similar) counterfactual explanation but the characteristics of the steps leading there make some explanations more attractive targets, e.g., by giving the explainee more agency through multiple actionable choices towards the end of the path.
• Figure 2: Demonstration of how the number of vectors $o$ into which a path is split affects counterfactual trajectories. This parameter must be carefully selected, which may require domain knowledge and familiarity with the underlying data.
• Figure 3: Demonstration of how the branching (i.e., divergence) threshold $\epsilon$ affects counterfactual trajectories and branching points. Here, the paths are split into $o=10$ vectors; $\epsilon$ must be carefully selected, which may require domain knowledge and familiarity with the underlying data. In this example, if $\epsilon=0.1$, the yellow path diverges from the red path after its third step since their distance $d$ exceeds $\epsilon$. When $\epsilon=0.25$, the divergence occurs after the yellow path's fifth step. If $\epsilon \geq 0.5$, the two paths do not diverge given that $\epsilon$ is greater than the distance separating them at every point along their way
• Figure 4: Visual depiction of calculating opportunity potential (left) and its example values (right). Each element $l_{a,b}$ of the metric table conveys how far along the reference counterfactual path from $\mathring{x}$ to $\check{x}_a$ we can travel while still getting closer -- albeit in a possibly sub-optimal way -- to another path between $\mathring{x}$ and $\check{x}_b$; the numbers at the bottom of the table (best in bold) capture the overall opportunity potential of a path in relation to all the other paths under consideration. In this case the paths are assumed to be standalone direct vectors between the factual data point and counterfactual instances. For example, if travelling along $z_1$ from $\mathring{x}$ to $\check{x}_1$, we can move toward the target ($\check{x}_1$) and at the same time get closer to $\check{x}_2$; on the other hand, travelling the same path allows us to only contribute to a small fraction of the $\check{x}_3$ path, thus $l_{1,2} > l_{1,3}$.
• Figure 5: Example counterfactual journeys identified in the MNIST data set of handwritten digits. Paths 1 (blue) and 2 (green) are counterfactual explanations for an instance $\mathring{x}$ classified as digit $1$ ($\mathring{y} = 1$) with the desired counterfactual class being digit $9$ ($\check{y} = 9$). Paths leading to alternative counterfactual instances of classes other than $9$, i.e., $y \in \mathcal{Y} \setminus (1, 9)$, are also possible (shown in grey). Path 1 is shorter than Path 2 at the expense of explainees' agency, which is reflected by the branching factor score of 0.38 versus 0.41; therefore, switching to alternative paths that lead to different classes while travelling along Path 1 is more difficult, i.e., more costly in terms of distance.