Conditional Weight of Evidence for Gradient-Boosted Credit Models: Estimation, Additivity and Its Relation to Adverse Action Reasons

Agus Sudjianto; Denis Burakov

stat.MLOct 17, 2026

Conditional Weight of Evidence for Gradient-Boosted Credit Models: Estimation, Additivity and Its Relation to Adverse Action Reasons

Authors: Agus Sudjianto, Denis Burakov

Abstract

Traditional weight of evidence assigns a credit characteristic a value based on its association with default across a population. Credit analysis also requires an assessment of that characteristic in the context of an individual applicant's record. We study conditional weight of evidence for this purpose. It measures the change in default log odds from observing a characteristic after the remaining credit information is known. Its sign identifies whether the characteristic adds adverse or favorable information for that applicant, and its magnitude measures the strength of that information. The same characteristic value can therefore receive different assessments for applicants with different payment histories or other credit attributes. We develop estimators using gradient-boosted trees and evaluate their accuracy in a bureau-style simulation. Regressions of the full model's predicted probabilities recover conditional evidence more accurately than separate models fitted to default outcomes, provided the full model is accurate. They also recover a deliberately constructed file-age effect whose direction varies with utilization and is often missed by marginal weight of evidence. Local class-frequency estimates improve on marginal weight of evidence but remain inaccurate for some weak effects. An application to FICO HELOC data illustrates estimation when the population evidence is unknown. Conditional weight of evidence provides an applicant-specific interpretation of the information in a credit characteristic; its population summary, a conditional information value, splits the joint information in the record across blocks when averaged over orderings. Neither is an adverse action reason, which explains the fitted score difference between the applicant and chosen passing records.

Figures & tables

Block	Fields	Generation given the factors
Revolving	limit, balance, utilization	$\log\text{limit}=\log 14000+0.45m-0.15s+0.4\varepsilon$ ; $\operatorname{logit}\text{util}=-0.7+s+0.3q+0.6\varepsilon$ ; balance $=$ util $\times$ limit
Delinquency	accounts 30+, 60+, 90+ days past due; months since delinquency	$n_{30}\sim\mathrm{Pois}(0.25e^{0.8s+0.25q})$ ; $n_{60}\sim\mathrm{Bin}(n_{30},0.4)$ ; $n_{90}\sim\mathrm{Bin}(n_{60},0.4)$ ; months since $\sim\min(\mathrm{Geom}(0.08),60)$ if $n_{30}>0$
File age	oldest and average trade age (months)	$\log(\text{oldest}-6)=\log 70+0.8m-0.15s+0.35\varepsilon$ ; average $=$ oldest $\times(0.25+0.55\,\mathrm{Beta}(3,2))$
Credit seeking	inquiries and new accounts in 6 months	inquiries $\sim\mathrm{Pois}(e^{0.1+0.4s+0.5q})$ ; new $\sim\mathrm{Bin}(\text{inquiries},0.35)$
Installment	original amount, balance, fraction outstanding	$\log\text{original}=\log 18000+0.4m+0.4\varepsilon$ ; $\operatorname{logit}\text{fraction}=0.1+0.5s+0.5\varepsilon$
Public record	indicator	$\mathbb{P}(\text{record})=\sigma(-3.3+0.8s)$

Table 1: Characteristic blocks of the bureau-style simulation. Each block is generated from the latent factors (stress

s

, maturity

m

, credit seeking

q

) and independent noise.

\mathrm{Pois}

\mathrm{Bin}

and

\mathrm{Geom}

denote Poisson, binomial and geometric draws;

\varepsilon

is standard normal noise drawn separately for each quantity.

Sample	Records	Role
Development	100,000	production model, block-excluded models, geometry models, block classes
Calibration	25,000	Platt recalibration of full and outcome-fitted reduced models; choice of $(\kappa,\lambda)$ on 4,000 records
Evidence	100,000	projection regressions; leaf-local counts; $\hat{\pi}$
Reference	50,000	cutoff (riskiest 20 percent declined) and passing band (0.3 in log odds below the cutoff)
Test	25,000	3,000 random applicants and 400 declined applicants at which evidence and attribution are evaluated

Table 2: Samples of each simulation replicate and their roles.

	Revolving	Delinq.	File age	Seeking	Install.	Public rec.
Mean attribution, production model, score band	0.42	0.85	$-$ 0.02	0.15	0.03	0.05
Mean attribution, true log odds, score band	0.40	0.86	$-$ 0.03	0.16	0.04	0.05
Mean attribution, production model, proximity	0.46	0.74	0.03	0.15	0.03	0.06
Leading-reason share, production model	0.26	0.57	0.05	0.07	0.00	0.06
Leading-reason share, true log odds	0.23	0.56	0.06	0.08	0.00	0.07
Leading-reason share, proximity references	0.33	0.51	0.01	0.09	0.01	0.05

Table 3: Grouped baseline-Shapley attribution of the score gap for 400 declined applicants per replicate, means over five replicates. Attributions are in estimated default log odds; the mean total gap to the references is 1.48. Leading-reason shares are the fractions of applicants whose largest attribution falls on each block.

	Applicant	Score-band references	Proximity references
Revolving utilization	0.53	0.53	0.27
Revolving limit ($)	30,809	15,936	29,493
Revolving balance ($)	16,229	8,176	8,089
Accounts 30+ days past due	1.00	0.58	1.00
Accounts 60+ days past due	1.00	0.20	0.84
Accounts 90+ days past due	1.00	0.10	0.40

Table 4: A declined applicant from replicate 1 whose leading block changes with the reference policy. Reference columns give means over the 50 references of each policy; the share of references with any account 30+ days past due is 0.54 under the score-band policy and 1.00 under the proximity policy. The lower panel gives block attributions

C^{f}_{j}

of the production model and the applicant’s oracle evidence

W_{j}

and

E_{j}

(logit units).

Figure 1: Mean attribution of the score gap and mean oracle evidence by block for the declined applicants, averaged over five replicates. Left: score attributions against the specified passing references. Right: population evidence allocated by order averaging or measured after all other blocks are known. The panels share units but use different comparison distributions.

Figure 2: Revolving-block conditional evidence for evaluation applicants in replicate 1. Each point is an applicant with its own remaining credit characteristics. Oracle evidence (left) varies less with utilization in the under-48-month group; projection (right) recovers the broad difference between groups with additional dispersion. Zero marks no change in default odds from observing the revolving block. These are cross-applicant patterns, not effects holding the remaining fields fixed.

	Revolving	Delinq.	File age	Seeking	Install.	Public rec.
Oracle sd, pointwise $W_{j}$	0.51	0.78	0.31	0.29	0.10	0.20
Oracle sd, class-level $W^{B}_{j}$	0.46	0.68	0.29	0.26	0.09	0.20
Zero evidence, pointwise target	0.51	0.79	0.31	0.29	0.10	0.20
Direct	0.20	0.21	0.19	0.17	0.15	0.16
Projection	0.17	0.18	0.13	0.13	0.09	0.10
Zero evidence, class-level target	0.47	0.69	0.29	0.26	0.09	0.20

Table 5: Error of conditional evidence estimates against the oracle in the bureau-style simulation: root mean squared error (logit units), mean over five replicates, for about 3,350 combined evaluation applicants per replicate. Direct and projection estimates are compared with the pointwise evidence

W_{j}

and the global and leaf-local estimates with the class-level evidence

W^{B}_{j}

; errors in different panels refer to different targets and are not compared with each other. The first two rows give the standard deviation of the oracle across applicants. Zero-evidence baselines assign zero to every applicant for each target. Replicate standard deviations of the RMSE are at most 0.02 for direct and projection estimates and at most 0.07 for leaf-local estimates.

Figure 3: Leaf-local RMSE against the class-level oracle across kernel exponents. Lines minimize oracle error over the smoothing constant after averaging across replicates; they describe the error surface rather than a deployable selection rule. Stars show mean RMSE at the held-out-likelihood-selected exponent and smoothing constant, placed at the modal selected exponent for each block.

\kappa=1

is the unsharpened proximity average.

Evidence records	Revolving	Delinq.	File age	Seeking	Install.	Public rec.
2,500	0.62	0.32	0.41	0.36	0.63	0.06
10,000	0.51	0.19	0.39	0.22	0.32	0.07
30,000	0.36	0.17	0.29	0.16	0.20	0.11
100,000	0.22	0.14	0.23	0.15	0.16	0.08
Global WoE, 100,000	0.92	0.58	0.39	0.62	0.77	0.13
Global WoE, 2,500	1.01	0.66	0.45	0.71	0.98	0.14

Table 6: Leaf-local estimation with evidence samples of different sizes, replicate 1. RMSE against the class-level oracle for the class geometry at the selected

(\kappa,\lambda)

, and for global WoE computed from the same subsample. The last row gives the median default-class effective sample size at 100,000 records.

	Revolving	Delinq.	File age	Seeking	Install.	Public rec.
Coalition evidence $E_{j}$ , oracle mean	0.43	0.62	0.05	0.19	0.12	0.05
Leave-one-block-out $W_{j}$ , oracle mean	0.19	0.47	0.07	0.05	0.00	0.03
Coalition excess $H_{j}$ , oracle mean	0.24	0.15	$-$ 0.02	0.14	0.12	0.01
RMSE of projection estimate of $E_{j}$	0.09	0.18	0.07	0.08	0.05	0.06
Oracle sd of $E_{j}$	0.68	1.14	0.28	0.57	0.35	0.35
$\|W^{b}_{j}-W_{j}\|$ , all test applicants	0.23	0.35	0.00	0.08	0.00	0.05

Table 7: Coalition evidence of the 400 declined applicants per replicate and the effect of the earlier approval policy, means over five replicates (logit units). The coalition excess is

H_{j}=E_{j}-W_{j}

. The absolute booked-minus-through-the-door difference

|W^{b}_{j}-W_{j}|

is computed from the oracle for the test applicants; earlier-policy declines are applicants with booking probability

s(x)<0.5

and current-model declines are applicants with

f(x)>c

. The last three rows give the RMSE of direct estimates fitted on booked accounts only, against the through-the-door and booked oracles, and of direct estimates fitted on all applicants against the through-the-door oracle.

Simulation
	Revolving	Delinq.	File age	Seeking	Install.	Public rec.
Marginal IV, oracle	1.43	1.51	0.07	0.75	0.68	0.12
Conditional IV $\mathrm{IV}_{j}$ , oracle	0.32	0.80	0.16	0.08	0.01	0.05
Coalition IV $\mathrm{IV}^{E}_{j}$ , oracle	0.74	1.04	0.11	0.32	0.24	0.07
Conditional IV $\mathrm{IV}_{j}$ , projection	0.31	0.79	0.14	0.08	0.01	0.04
Coalition IV $\mathrm{IV}^{E}_{j}$ , projection	0.72	1.04	0.10	0.30	0.23	0.06

Table 8: Conditional information value (natural units). Simulation: oracle and projection values on 1,000 random test applicants per replicate, class means weighted by the true default probability; means over 5 replicates, replicate standard deviations at most 0.10 for the oracle values. HELOC: projection estimates over the test records, means over 10 splits. Marginal values use the evidence of the block alone,

V_{x}(\{j\})

, or its projection estimate.

	Delinq.	File age	Trades	Inquiries	Revolving	Install.
Corr(direct, projection)	0.84	0.72	0.50	0.76	0.87	0.35
Corr(projection, leaf-local default geometry)	0.75	0.58	0.14	0.48	0.76	0.04
Corr(leaf-local default geometry, global WoE)	0.93	0.89	0.69	0.92	0.95	0.46
Mean coalition evidence $E_{j}$	0.30	0.15	0.09	0.13	0.37	0.02
Mean leave-one-block-out $W_{j}$	0.30	0.14	0.04	0.13	0.33	0.04
Mean attribution $C^{f}_{j}$	0.37	0.22	0.11	0.18	0.36	0.01

Table 9: FICO HELOC data, ten random splits. Correlations are across test applicants within a split, averaged over splits. Means of

E_{j}

W_{j}

and the attribution

C^{f}_{j}

are over declined applicants. Estimated evidence uses the projection estimator.

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

	Simulation	HELOC
Platt slope $b$	1.00 to 1.03	0.81 to 1.18
Calibration slope before Platt	1.00 to 1.02	0.91 to 1.08
Calibration slope	0.97 to 1.02	0.87 to 1.14
Calibration intercept	$-$ 0.03 to 0.02	$-$ 0.19 to 0.08
Expected calibration error (deciles)	0.002 to 0.005	0.021 to 0.047
Mean predicted probability	0.087 to 0.092	0.484 to 0.528

Appendix

Table 10: Calibration of the production model on the test sample: ranges over five simulation replicates and ten HELOC splits. The calibration slope and intercept are from a logistic regression of the outcome on the model’s log odds (intercept with slope fixed at one). The lower panel compares the simulation model with the true log odds, for all test applicants and for applicants above the cutoff.

Block	Estimator	$\mathbb{E}[e_{F}^{2}]$	$\mathbb{E}[e_{-j}^{2}]$	$2\mathbb{E}[e_{F}e_{-j}]$	Evidence MSE
Revolving	Direct	0.0477	0.0405	0.0490	0.0392
	Projection	0.0477	0.0297	0.0490	0.0284
Delinquency	Direct	0.0477	0.0375	0.0412	0.0440
	Projection	0.0477	0.0303	0.0439	0.0341
File age	Direct	0.0477	0.0476	0.0602	0.0351
	Projection	0.0477	0.0361	0.0662	0.0177

Appendix

Table 11: Error in the evidence difference, averaged over five replicates on the combined evaluation sample. The first two terms are squared full and reduced log-odds errors; subtracting twice their cross-product gives the evidence MSE. Values are in squared log-odds units.

	Revolving	Delinq.	File age	Seeking	Install.	Public rec.
Direct, combined	0.198	0.210	0.187	0.165	0.153	0.159
Direct, sampled declines	0.236	0.278	0.232	0.195	0.190	0.195
Projection, combined	0.168	0.185	0.133	0.128	0.093	0.104
Projection, sampled declines	0.190	0.263	0.169	0.158	0.108	0.143

Appendix

Table 12: Pointwise conditional-evidence RMSE among the 400 sampled declines per replicate, compared with the combined evaluation sample. Entries are means of replicate RMSE in logit units. The decline sample is selected by the original production model and cutoff.

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Leaf Values as Coordinates: Exact Contrastive Explanation for Gradient-Boosted Ensembles

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Independent Researcher

Counterfactual Explanations Algorithmic Recourse Gradient-Boosted Decision Trees Tabular ML

Open paper AskPDF arXiv

A gradient-boosted ensemble predicts by summing one leaf value per tree. Read those values as coordinates rather than as intermediate results, and every instance becomes a point in R^M on which the model acts linearly: the score is the sum of the coordinates. This small change of view makes contrastive explanation exact. The difference between two instances is a vector that is identically zero wherever they share a leaf, so the gap between a rejected applicant and an accepted one is carried by a handful of coordinates, each traceable to a real split in a real tree. Nothing is fitted, sampled, or assumed additive in features -- the additivity is already there, in the right space. We build a recourse method on this representation and evaluate it on five tabular datasets under repeated cross-validation. Its recommendation reconstructs the model's own decision to 6.2 x 10^-15, so an auditor can re-check the arithmetic without the model. On the credit datasets it is Pareto-non-dominated on effort against realism. And when recommendations are restricted to changes the subject could actually make -- not their age, not a settled delinquency -- it retains 58% of its validity where the strongest baseline retains 41%, a distinction the standard evaluation cannot see because it never asks whether a recommendation can be carried out.

Abstract

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