Core Concepts

This page introduces the key concepts behind DFI (Disentangled Feature Importance) and FDFI (Flow-DFI), and how they relate to other interpretability methods.

What is Feature Importance?

Feature importance quantifies how much each input feature contributes to a model’s predictions. Given a model \(f(x)\) and an input \(x = (x_1, ..., x_d)\), we want to compute attributions \(\\phi = (\\phi_1, ..., \\phi_d)\) where \(\\phi_j\) represents the importance of feature \(j\).

SHAP and Shapley Values

SHAP (SHapley Additive exPlanations) computes feature importance using Shapley values from cooperative game theory. For a prediction \(f(x)\), SHAP values satisfy:

Efficiency: \(\\sum_j \\phi_j = f(x) - E[f(X)]\)
Symmetry: Features with equal contributions get equal attributions
Null: Features that don’t affect the output get zero attribution
Linearity: Attributions combine linearly for ensemble models

DFI is inspired by SHAP but uses optimal transport methods to compute feature importance.

Disentangled Feature Importance (DFI)

DFI introduces disentangled feature importance using optimal transport to create counterfactual distributions. The key insight is:

To measure the importance of feature j, compare the model output when feature j comes from the data distribution vs. when it’s replaced by an independent sample.

Mathematically, let \(Z = L^{-1}(X - \\mu)\) be the whitened (disentangled) representation where features are uncorrelated. The Unit Effect Independent Feature (UEIF) for feature \(j\) is:

\[\begin{split}\\text{UEIF}_j(x) = \\left( f(x) - E[f(\\tilde{X}^{(j)})] \\right)^2\end{split}\]

where \(\\tilde{X}^{(j)}\) has feature \(j\) replaced with an independent sample from the marginal distribution.

Gaussian vs Entropic OT

DFI provides two main approaches:

Gaussian OT (OTExplainer)

Assumes data is approximately Gaussian
Uses closed-form Gaussian optimal transport: \(Z = L^{-1}(X - \\mu)\)
Fast and stable
Best for continuous, roughly normal data

Entropic OT (EOTExplainer)

Relaxes Gaussian assumption using Sinkhorn algorithm
Adaptive regularization via median distance heuristic
Supports mixed data types (continuous + categorical)
Better for non-Gaussian or mixed-type data

Flow-Disentangled Feature Importance (Flow-DFI)

FlowExplainer uses normalizing flows to learn a flexible, data-driven transformation between the original feature space X and a disentangled latent space Z where features are approximately independent.

CPI (Conditional Permutation Importance)

Averages predictions first, then computes squared difference:

\[\phi_{Z,j}^{CPI} = (Y - \mathbb{E}_b[f(\tilde{X}_b^{(j)})])^2\]

where \(\tilde{X}_b^{(j)} = T^{-1}(\tilde{Z}_b^{(j)})\) and \(\tilde{Z}_b^{(j)}\) has the j-th component replaced with sample b.

SCPI (Sobol-CPI)

Computes squared differences first for each Monte Carlo sample, then averages:

\[\phi_{Z,j}^{SCPI} = \mathbb{E}_b[(Y - f(\tilde{X}_b^{(j)}))^2]\]

This is equivalent to the Sobol sensitivity index formulation. The key difference from CPI is the order of averaging.

Jacobian Transformation to X-space

Both CPI and SCPI compute importance in the disentangled Z-space. To attribute importance to the original features \(X_l\), we use the Jacobian of the decoder transformation \(T^{-1}: Z \to X\):

\[\phi_{X,l} = \sum_{k=1}^{d} H_{lk}^2 \cdot \phi_{Z,k}\]

where \(H = \frac{\partial X}{\partial Z}\) is the Jacobian matrix evaluated at the data points. This correctly accounts for how changes in each latent dimension Z_k affect each original feature X_l.

For linear transformations (as in OTExplainer), this reduces to \(\phi_X = H^T H \phi_Z\) where \(H = L\) is the Cholesky factor. For normalizing flows, the Jacobian varies with position and is computed via automatic differentiation.

When to Use FlowExplainer

Complex non-linear dependencies between features
Non-Gaussian data distributions
When OT assumptions are too restrictive
When you have sufficient data (>500 samples) to train the flow

Shared Disentanglement Diagnostics

All disentangled explainers (OT, EOT, Flow) report two common diagnostics:

Latent independence: median pairwise distance correlation in latent space.
Distribution fidelity: MMD between original data and reconstructed data.

Both are “lower is better” metrics and are reported with qualitative labels (GOOD, MODERATE, POOR) using shared thresholds.

Relationship to Other Methods

Method	Comparison to DFI
SHAP KernelExplainer	Model-agnostic like DFI, but uses sampling-based Shapley estimation. DFI uses OT-based counterfactuals.
SHAP TreeExplainer	Exact Shapley for trees. DFI works with any model.
LIME	Local linear approximation. DFI considers global distribution.
Permutation Importance	Breaks feature dependencies. DFI preserves correlation structure.
Integrated Gradients	Requires gradients. DFI is gradient-free.

Confidence Intervals

A key advantage of DFI is built-in uncertainty quantification. The conf_int() method provides:

Standard errors computed across samples
Confidence intervals using normal approximation
P-values for testing \(H_0: \phi_j = 0\) or \(H_0: \phi_j \leq \delta\)
Variance floor methods for stable inference with small effects
Score: the estimated feature importance (mean UEIF)
Multiple Testing Correction: control of False Discovery Rate (FDR) or Family-Wise Error Rate (FWER) via multitest_method.

This enables statistical feature selection: identify features that are significantly different from zero or a practical threshold.

Multiple Testing Correction

When testing hundreds of features simultaneously, the probability of obtaining false positives (Type I errors) increases. conf_int() supports multiple testing corrections using the statsmodels library.

By setting multitest_method, you can choose from various correction methods:

FWER Control: 'bonferroni', 'holm', 'sidak', etc.
FDR Control: 'fdr_bh' (Benjamini-Hochberg), 'fdr_by' (Benjamini-Yekutieli).

When a correction is applied, conf_int() returns adjusted p-values as pvalue_adj, and the reject_null decision is updated to reflect the specified alpha (e.g., FDR < 0.05).

Group-Level Importance

In many applications features naturally belong to groups (e.g., genomic regions, sensor categories, feature families). The conf_int() method supports a groups argument that aggregates per-sample UEIFs across features within each group and reports group-level importance with proper uncertainty.

Given a group \(S_g \subseteq \{1, \ldots, d\}\), the group importance is:

\[\phi_g = \frac{1}{n} \sum_{i=1}^{n} \sum_{j \in S_g} \text{UEIF}_{ij}\]

The standard error is computed from the per-sample grouped UEIFs \(u_i = \sum_{j \in S_g} \text{UEIF}_{ij}\):

\[\text{SE}_g = \frac{\sigma(u)}{\sqrt{n}} + \frac{c}{\sqrt{n}\; z_{1-\alpha/2}}\]

where \(c\) is a small finite-sample correction constant (default 0.1) that prevents anti-conservative z-scores when the raw SE is very small.

An optional null-thresholding step (threshold_null=True) zeros out per-feature UEIFs with negative mean before aggregation, preventing estimation noise from artificially deflating group importance.

Previously, this was handled by a separate group_importance() method, which is now deprecated in favor of the more flexible conf_int(groups=...) API.