Quick answer: a credit-underwriting model usually observes repayment outcomes only for applicants who were approved. Rejected applicants never received the loan, so the bank does not observe whether they would have repaid under the proposed terms. That creates sample-selection bias: the labelled development sample is partly produced by the previous lending policy. Reject inference is the family of methods that tries to reason about the missing outcomes of rejected applicants using their observed characteristics, modelling assumptions, external performance data or semi-supervised techniques. The central limitation is fundamental: a rejected applicant’s repayment outcome is a counterfactual, not a hidden fact waiting to be read from the database.

The model learns from the people the old policy allowed it to observe.

Page role: the missing-label problem inside credit underwriting

Bukit Timah Tutor already explains credit-scoring algorithms and bank model validation. This article owns a narrower question: what happens when the model’s repayment labels exist mainly for applicants selected by a prior approval policy?

The OCC has studied this exact issue. Its working paper Specification and Informational Issues in Credit Scoring explains that models built solely from extended loans can be affected by selectivity bias and examines reject inference as a possible response.

1. Write the selection problem explicitly

For applicant i, let:

X_i = observed application features;
A_i = 1 if approved, 0 if rejected;
Y_i = repayment/default outcome that would be observed if credit were granted.

The development dataset often contains:

(X_i, A_i, Y_i) when A_i=1

but only:

(X_i, A_i) when A_i=0.

The missing Y values are not usually missing at random. Approval depends on credit characteristics related to Y. The missingness mechanism and the target are therefore connected.

2. Why accepted-only training can distort the learned relationship

Suppose an old policy rejected nearly everyone below a credit score of 600. The bank now wants to build a model for applicants from 500 to 800. Historical loan performance contains many observations from 650–800 but very few from 500–600.

A logistic model can still draw a line through the accepted data and extrapolate downward, but the lower-score region is weakly identified by realised repayment evidence. The model may look smooth because the equation is smooth—not because the bank has observed the relevant borrowers.

This is the first falsifier of naive confidence: good in-sample fit inside the accepted region does not prove good calibration outside the historical approval region.

3. The old policy becomes part of the training data

Imagine approval was historically:

A = 1 if old_score(X) ≥ cutoff, subject to other policy rules.

The labelled sample is then conditional on passing that rule:

P(Y|X, A=1)

while the model may be intended to estimate:

P(Y|X)

for a wider applicant population. Those are not automatically the same object.

Selection can be especially strong when the old score uses variables also used by the new score. The model then learns in a region already filtered by a related model.

4. Reject inference cannot observe the counterfactual

If an applicant was rejected in 2024, there is no literal repayment history for the 2024 loan that was never originated. A reject-inference method therefore constructs or estimates information under assumptions.

This distinguishes three categories:

Observed fact: the applicant’s features and rejection decision.
External evidence: later credit-bureau or other legally usable performance information, where available and appropriate.
Inferred counterfactual: what the applicant would have done under the ungranted loan.

A strong model keeps those categories separate. Inference should never be relabelled as observation.

5. Method 1 — simple accepted-only modelling

The baseline is to train only on accepted applicants with observed outcomes. This method is transparent and does not invent labels.

Its weakness is coverage. It works best when the intended future approval population overlaps strongly with the historical accepted population. It becomes fragile when the bank plans to expand materially into previously rejected segments.

This baseline remains valuable because every reject-inference method should prove it improves something beyond accepted-only modelling rather than merely producing a more complicated scorecard.

6. Method 2 — augmentation by inverse acceptance probability

One idea is to estimate the probability that an applicant would be accepted under the historical policy:

p_i = P(A=1|X_i).

Accepted observations can then receive weights related to 1/p_i, so observations from regions rarely accepted receive more influence.

This resembles inverse-probability weighting in missing-data and causal-inference settings. The crucial assumption is overlap: there must be some accepted observations in the region whose outcomes we want to infer. If acceptance probability is effectively zero, the weight becomes unstable and the data contain little empirical information about repayment there.

Weighting cannot manufacture overlap where the old policy allowed none.

7. Method 3 — parceling or pseudo-labelling rejected applicants

Parceling groups rejected applicants into score bands and assigns estimated proportions of “good” and “bad” outcomes, often using accepted-applicant performance plus an assumption that rejected applicants are worse within each band.

For example, if an accepted score band has a 5% observed default rate, an analyst might assume rejected applicants in that band would have a higher default rate and distribute pseudo-labels accordingly.

The method is simple. Its weakness is also simple: the result can be driven by the chosen uplift assumption. If the analyst assumes rejects are 1.5× worse rather than 2× worse, the inferred labels change even though no new repayment evidence arrived.

Parceling should therefore be treated as a sensitivity framework, not a machine for converting unknown outcomes into facts.

8. Method 4 — fuzzy augmentation

Instead of assigning a rejected applicant one hard pseudo-label, fuzzy augmentation can assign fractional membership to good and bad outcomes based on a modelled probability.

If a reject is assigned estimated default probability 0.30, the training procedure can represent that observation with weighted contributions to both outcome classes rather than declaring it definitely good or bad.

This preserves uncertainty better than hard parceling, but the underlying probability is still model-generated. If the initial model is biased outside the accepted region, fuzzy augmentation can reproduce the same bias more smoothly.

9. Method 5 — mixture models and the EM algorithm

A more formal approach treats rejected-applicant outcomes as latent variables. Mixture models can use the Expectation-Maximisation (EM) algorithm:

E-step: estimate probabilities of latent good/bad outcomes for rejects using current parameters.
M-step: update model parameters using observed accepted outcomes plus expected contributions from rejected cases.
Repeat until the chosen convergence criterion is reached.

Academic reject-inference research has explored mixture-model and semi-supervised approaches of this kind. The benefit is principled treatment of missing labels inside one statistical model. The limitation remains identification: EM can optimise a likelihood under assumptions, but it cannot prove that those assumptions describe the unobserved counterfactual population.

10. Method 6 — external performance information

Where law, consent, contracts and data-governance rules permit, a lender may have later external information about rejected applicants—for example, subsequent credit performance elsewhere. That information can be more valuable than synthetic pseudo-labels because it is an observed outcome, though not necessarily the outcome of the exact rejected product.

External outcomes still require caution:

the applicant may receive a different loan amount or interest rate elsewhere;
the other lender may use different servicing or line-management policies;
macroeconomic conditions may change before the later observation;
credit-bureau reporting can be incomplete or delayed.

Observed elsewhere is better than invented, but it is not identical to “what would have happened under our declined contract.”

11. Policy drift changes the labelled population over time

Suppose the bank lowers its cutoff and begins accepting more marginal applicants. The new approvals gradually generate labels in a region that used to be unobserved. Model performance can then change for two reasons:

the economic relationship between X and default changed;
the bank changed which X values it allows into the labelled sample.

This is policy-induced dataset shift. A monitoring system should record historical decision policy alongside model version and applicant outcome so analysts can distinguish changing world from changing selection rule.

12. The acceptance policy and model form a feedback loop

The deployed model influences approvals. Approvals determine which outcomes become observable. Those outcomes train the next model. The next model influences future approvals.

The feedback loop is:

model → policy → selected borrowers → observed outcomes → next model.

A stable-looking score can therefore become self-confirming. If a segment is rarely approved, the bank receives little direct evidence capable of contradicting its old belief about that segment.

This is why reject inference belongs in model governance, not only model development.

13. Selection bias is not the same thing as unfair discrimination

Selection bias is a statistical property: labelled observations are not representative of the target population because selection depends on variables related to the outcome. Fair-lending and discrimination analysis are legal and policy questions involving protected characteristics, proxies, treatment, impact and applicable jurisdictional rules.

The two can interact. Non-representative data can worsen model performance for groups with less historical coverage, and opaque models can make errors harder to diagnose. But a statistical selection correction does not by itself establish legal compliance or fairness.

Federal Reserve remarks on automated lending have highlighted risks from incomplete or nonrepresentative training data and the need to understand how model features inform credit decisions. See Federal Reserve remarks on fair lending and technology.

14. Explainability still matters when the statistical target is uncertain

A reject-inference model can be numerically sophisticated while obscuring why an applicant’s estimated risk moved. Credit-decision systems often need to preserve reason codes, variable lineage and the ability to explain adverse-action decisions under applicable law.

The CFPB has publicly emphasised that model complexity does not excuse creditors from giving accurate and specific reasons for adverse actions where those requirements apply. See its 2023 guidance on credit denials using complex algorithms. Regulatory policy can change, so institutions must verify current law rather than relying on a static educational article.

15. Validation requires tests outside the comfortable accepted sample

A model can backtest beautifully on accepted borrowers and still fail exactly where the new policy plans to expand. Validation should therefore segment performance by:

historical acceptance-probability bands;
distance from old policy cutoffs;
new versus established product populations;
score ranges with sparse historical approvals;
policy version;
time since deployment.

The April 2026 revised interagency model-risk guidance emphasises risk-based validation, monitoring, data and model limitations. See FDIC: Agencies Revise the Interagency Model Risk Management Guidance.

16. A worked miniature example

Suppose 100,000 people apply for credit:

60,000 are approved;
40,000 are rejected;
6,000 of the approved accounts later default.

The observed default rate among accepted applicants is 10%. It would be incorrect to say the whole applicant population has a 10% default rate. The rejected 40,000 have no outcome under the proposed loan.

Now suppose the bank wants to approve 10,000 additional applicants next year from a score region historically rejected. Three models estimate their default rate at 14%, 18% and 25% depending on reject-inference assumptions.

The disagreement is not noise to average away. It identifies an epistemic weak point. The decision system should carry that uncertainty into pricing, limits, capital, monitoring and the design of any controlled expansion rather than pretending one estimate is observed truth.

17. Alternatives to reject inference

Stay within the historical support. Use the model only where accepted-outcome data are dense enough.
Use external observed performance. Where lawful and comparable, supplement missing internal labels.
Controlled policy expansion. Expand cautiously into adjacent populations and monitor real outcomes under appropriate governance.
Multiple models. Compare accepted-only, weighted, semi-supervised and structural approaches rather than selecting one method by habit.
Uncertainty bands. Report ranges where identification is weak instead of forcing a point estimate.

No alternative removes the counterfactual problem completely. The question is which design creates the most correctable evidence with the least unsupported inference.

18. The reject-inference algorithmic pipeline

Define the target outcome and performance window.
Reconstruct historical approval policies and cutoffs.
Measure accepted/rejected coverage in feature space.
Identify regions with weak or zero overlap.
Build an accepted-only baseline.
Estimate acceptance propensity where appropriate.
Test alternative reject-inference methods.
Keep inferred labels distinct from observed labels.
Use external outcomes where lawful and comparable.
Validate calibration and ranking by policy/coverage segment.
Run sensitivity tests over key reject assumptions.
Document uncertainty for unsupported regions.
Monitor newly approved marginal populations.
Update the model when real outcomes replace inferred ones.

19. Failure modes

Observed/inferred collapse. Pseudo-labels are stored as if they were actual repayment outcomes.
No-overlap extrapolation. Weighting is used in regions the old policy never approved.
Self-confirming model. The same score creates selection and is then validated on its selected borrowers.
Parceling certainty. A chosen reject bad-rate uplift is treated as empirical fact.
Policy-history loss. Analysts cannot reconstruct which cutoff generated each training cohort.
External-data equivalence. Performance on a different product is treated as the exact counterfactual of the declined loan.
Complexity substitution. Semi-supervised or EM methods replace rather than expose identification assumptions.
Global metric comfort. AUC or accuracy is strong overall while performance is weak near the expansion boundary.

20. Diagnostics and falsifiers

What fraction of the target population lies outside strong historical approval support?
How does predicted default change across reject-inference methods?
Which rejected regions have effectively zero accepted analogues?
Does the model remain calibrated for newly approved applicants near the old cutoff?
Can every training label be identified as observed or inferred?
What happens if the assumed reject bad-rate uplift is halved or doubled?
Does an accepted-only challenger outperform the reject-inference model out of time?
Which evidence would cause the bank to abandon the reject-inference adjustment?

Suppose someone claims, “Reject inference gives us the default rate of applicants we declined.” A falsifier is simply the causal structure of the problem: the declined contract was never originated, so its repayment outcome was never observed. Reject inference estimates under assumptions; it does not recover a missing historical fact.

21. Verification and update triggers

preserve policy version and decision reason with every application;
compare inferred outcomes with later observed outcomes in newly opened regions;
monitor overlap and propensity-weight stability;
recalibrate after material cutoff or underwriting-rule changes;
test model drift separately from selection-policy drift;
independently review explainability and data lineage;
retire pseudo-label assumptions as real evidence accumulates;
re-verify current regulatory and consumer-protection requirements before deployment changes.

Connections across the finance-and-banking algorithms lane

Credit scoring — the prediction model whose labelled sample may be selected.
Revolving-credit utilisation and EAD — approval changes which utilisation paths become observable.
Delinquency and cure — downstream performance labels used to validate underwriting.
Model validation — the governance owner for limitations, benchmarks and monitoring.

Research anchors

The deeper lesson

Reject inference is a lesson in the geometry of evidence. The bank knows what happened to people it approved. It knows some features of people it rejected. It does not know the repayment outcome of a loan that never existed. Statistical methods can bridge part of that gap only by adding assumptions, external evidence or new observed outcomes. A strong underwriting system therefore does not hide the missing region behind a smooth score. It marks where evidence becomes extrapolation, monitors the populations created by policy, and replaces inference with reality whenever new outcomes arrive.

Educational note: This article explains public statistical and banking-model concepts. It is not lending advice, legal advice, fair-lending compliance advice or a recommendation about any individual applicant.

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How Credit Underwriting Models Learn from Accepted Applicants: Reject Inference, Selection Bias, Missing Labels, Policy Drift and Validation