BAM: Building Auditable Tabular Prediction Models with Bivariate Gaussians

In tabular data prediction, generalized additive models (GAMs) and their variants with pairwise interactions (GA²M) are often regarded as "glass-box" models, because each pairwise shape function can be plotted as a contour map. But reality is harsh: the shape functions learned by methods such as EBM and NODE-GAM are often overly complex—irregular contours and severe local fluctuations make them effectively impossible for practitioners to interpret, let alone validate with domain knowledge.

When "Interpretability" Becomes a Mere Formality: The Predicament of Complex Shape Functions

GA²M's interpretability relies on visualizing shape functions, but visualization does not equal comprehensibility. If a contour map is full of jagged edges and isolated islands, even if it can be drawn, no one can extract reliable business rules from it. The paper's authors point out that this requirement for "auditability" demands that the shape functions themselves be simple enough, not merely that they can be plotted after the fact.

This raises a key question: does simplifying shape functions sacrifice predictive accuracy? The authors give a negative answer—provided the right feature pairs are chosen. As long as the correct pairwise interactions are captured, the simplified model can fully match the accuracy of standard GAMs.

GAM (generalized additive model) represents the predicted value as the sum of univariate shape functions for each feature, in the form (g(\mathbb{E}[y]) = \beta_0 + \sum_j f_j(x_j)), where each (f_j) can be visualized individually. GA²M adds pairwise interaction terms (f_{jk}(x_j, x_k)) on top of this to capture synergy effects between features. EBM (Explainable Boosting Machine) is an efficient implementation of GA²M that uses boosted trees to learn shape functions and then converts them into lookup tables; NODE-GAM uses neural networks to learn shape functions. Although both types of methods can plot each component, due to the flexibility of trees or neural networks, the learned shape functions may contain numerous high-frequency fluctuations and isolated regions, making it difficult to extract concise rules from the visualized graphs.

BAM's Design: Every Interaction Is an Asymmetric Bivariate Gaussian

Based on this insight, the authors propose Bivariate Additive Models (BAM). BAM belongs to the GA²M family, but its core constraint is that every pairwise shape function is restricted to a simple parametric form, specifically instantiated as an asymmetric bivariate Gaussian distribution.

The direct benefit of this design is that each component can be visualized as a unimodal contour map. Unimodal means the shape function has only one "peak," allowing practitioners to clearly judge how a feature pair jointly affects predictions and to compare it against domain knowledge. More importantly, the selection of feature pairs is done automatically during training, requiring no manual exhaustive combination search.

Compared with the complex shape functions produced by models such as EBM, each component of BAM is visually far simpler, greatly reducing audit costs.

The asymmetric bivariate Gaussian distribution is a generalization of the standard bivariate Gaussian distribution, allowing the two variables to have different variances and a general covariance structure. Its density function takes the form (\exp(-\frac{1}{2}(\mathbf{x}-\boldsymbol{\mu})^T \boldsymbol{\Sigma}^{-1} (\mathbf{x}-\boldsymbol{\mu}))), where (\boldsymbol{\Sigma}) is an asymmetric covariance matrix (in practice typically parameterized as (\sigma_1, \sigma_2, \rho)). This distribution exhibits a unimodal, elliptical contour on the two-dimensional plane, with the peak position and shape determined by the mean vector and covariance matrix. Using it as a pairwise shape function means each interaction term can only express a single "hill"-shaped smooth surface, thereby naturally excluding hard-to-audit patterns such as multiple peaks and jagged edges.

Experimental Performance: The Trade-off Between Accuracy and Auditability

The paper conducts experiments on six tabular benchmark datasets, and the results show that BAM's predictive accuracy is comparable to standard GAMs and close to the level of GA²M. This means that constraining shape functions to simple parametric forms does not lead to significant accuracy loss.

The authors also demonstrate the trade-off between accuracy and the number of components that need to be inspected: more components usually yield higher accuracy, but the audit burden also increases. This trade-off provides practitioners with a clear adjustment knob—model complexity can be decided based on audit resources.

The six tabular benchmark datasets typically come from public repositories such as OpenML or UCI, covering binary classification and regression tasks, with feature dimensions ranging from dozens to hundreds. These benchmarks are widely used to evaluate the performance of GAM-type models; for example, the original EBM paper used a similar collection of datasets. On these benchmarks, BAM achieves accuracy comparable to standard GAMs (such as models containing only univariate terms), indicating that the introduction of pairwise interaction terms does not lose information due to parametric constraints; at the same time, its accuracy is close to GA²M, showing that simple parametric forms are sufficient to capture most important interaction effects. This result provides empirical support for the argument that "constraining shape functions does not sacrifice accuracy."

Implications for Tabular Prediction Practice

BAM's value lies in redefining the standard of interpretability: not "can be plotted," but "can be audited." In scenarios such as financial risk control and medical diagnosis where model decisions need to be traceable, this strong auditability is particularly important. Practitioners can quickly check whether the shape of each bivariate Gaussian aligns with business intuition, without having to search for a needle in a haystack of complex contour maps.

Of course, BAM's limitations are also obvious: the unimodal assumption of the asymmetric bivariate Gaussian may fail to capture certain real complex interaction patterns. The paper does not avoid this constraint but presents it as a design choice—trading expressive power for auditability. For tabular prediction tasks that pursue interpretability, this is a compromise worth noting.

One-line Summary

BAM constrains pairwise interactions to asymmetric bivariate Gaussians, enabling auditable tabular prediction.

Paragraph Summary

In tabular prediction, although GA²M models can visualize pairwise shape functions, the functions learned by EBM and NODE-GAM are overly complex, rendering interpretability a mere formality. The paper proposes Bivariate Additive Models (BAM), which restrict each pairwise shape function to an asymmetric bivariate Gaussian distribution, making each component present a unimodal contour map that practitioners can easily audit. Feature pairs are selected automatically during training without manual specification. On six tabular benchmarks, BAM's accuracy is comparable to standard GAMs and close to GA²M, demonstrating that simplifying shape functions does not significantly sacrifice accuracy. The model also offers a trade-off adjustment between accuracy and the number of components to inspect, providing a new option for scenarios requiring strong auditability such as financial risk control and medical diagnosis.