Quantized Prediction Models: Output-Space Regularization for Stable and Interpretable Regression

Quantized Prediction Models: Output-Space Regularization for Stable and Interpretable Regression Carlos R. Ortiz-G´omez Independent Researcher December 2025 DOI: 10.5281/zenodo.17785949 Abstract We introduce Quantized Prediction Models (QPMs), a regression framework that constrains predictions to a finite set of output levels. Unlike traditional parameter-based regularization, QPMs regularize directly in the output space through supervised quantization, creating a functional bottleneck that suppresses noise, improves stability, and produces discrete, monotone predictors suited for governance-sensitive domains. QPMs employ an optimizer-driven binning procedure that concentrates resolution around highstakes decision thresholds while maintaining a globally coherent, interpretable score ladder. To address distributional drift, we propose an Anchored Calibration update that adjusts bin representatives while keeping bin edges fixed, thereby guaranteeing zero rank-order churn. Theoretically, QPMs reduce local sensitivity, decrease variance, and preserve monotonicity under anchored updates. Empirically, they match the predictive performance of standard gradient boosting models while delivering dramatically improved stability, offering a practical, governancealigned alternative for applications in which stability is as critical as accuracy. 1 Introduction Modern regression models such as gradient-boosted machines and neural networks deliver state-of-theart predictive performance on tabular data. However, these models often exhibit output jitter: small perturbations in the input can produce unpredictable, non-monotonic fluctuations in the output. In high-stakes domains such as credit decisioning, these fluctuations are operationally costly. They can cause rank-order drift across retraining cycles, instability in score ladders, and inconsistent treatment strategies—all of which degrade the reliability of downstream decisions. This challenge is increasingly observed across production ML systems in regulated decision environments. Although modern models improve headline metrics such as AUC or KS, their continuous outputs can be sensitive to distributional drift, dataset revisions, or minor perturbations in feature construction. As a result, practitioners frequently encounter a tension between the expressive power of flexible ML models and the stability, monotonicity, and governance requirements that traditional scorecards provide by construction. The result is a structural misalignment: ML practitioners optimize for predictive accuracy, whereas risk practitioners optimize for output stability and replicability across time. Existing regularization methods—such as L1/L2 penalties, dropout, learning-rate control, or treedepth limits—constrain parameters or model complexity, but do not address the core issue: unconstrained continuous outputs amplify noise and propagate instability into operational systems. What is missing is a mechanism that regularizes the prediction space directly. Relation to Existing Work. Our formulation builds on a long lineage of scorecard development methods used in credit risk (e.g., Siddiqi, 2012) and supervised discretization approaches such as optimal binning. QPMs do not replace these traditions; rather, they formalize supervised binning as an explicit output-space regularizer and introduce mechanisms— optimizer-driven non-uniform binning and Anchored Calibration—that provide guarantees on stability and monotone updating under drift. 1 Figure 1: Output-Space Regularization in Action. The light blue line shows the continuous, noisy predictions of the latent model (here a GBM) on the UCI Default test set, sorted by the latent QPM score. The orange step function shows the final Quantized Prediction Model (QPM) output. The quantization operator acts as a functional regularizer, suppressing local high-variance fluctuations while preserving the global monotonic risk structure. The goal is not to claim an entirely new paradigm, but to clarify and extend established practices through a unified mathematical and governance-oriented framework. Our Proposal. We introduce Quantized Prediction Models (QPM), a framework that constrains the model output to a finite, ordered set of prediction levels (a “ladder”). As illustrated in Figure 1, this quantization operator acts as an information bottleneck that suppresses noise, stabilizes rank ordering, and yields operationally stable and auditable model outputs. The approach retains the expressive power of flexible function approximators while enforcing the stability properties favored in traditional risk modeling. Our Contributions. This work makes the following contributions: •We formulate supervised regression as an adaptive output quantization problem, showing how discretized predictions act as a principled functional regularizer. •We introduce Anchored Calibration, a post-training update rule that incorporates new information while guaranteeing zero rank-order churn when bin edges remain fixed. •We develop an optimizer-driven binning procedure that allocates local resolution where operational sensitivity is highest. Instead of distributing bins uniformly across the prediction range, QPMs allow practitioners to specify a high-resolution window—typically near decision thresholds or tail regions—while maintaining coarser granularity elsewhere. This “focus region” design preserves global monotonicity and interpretability while providing the fine-grained discrimination needed for policy cutoffs and other narrow-band decisions. •We provide theoretical results showing that QPMs suppress sub-bin variability and exhibit a denoising effect analogous to an autoencoder in output space. •Through empirical studies on public credit datasets, we show that QPMs preserve AUC and KS performance while dramatically reducing prediction drift and stabilizing score ladders across retraining cycles. By directly regularizing the output space, QPMs reconcile the strengths of modern machine learning with the stability, interpretability, and governance requirements of operational risk management. This enables the deployment of high-power models in environments where stability is as critical as accuracy. 2 2 Related Work 2.1 Gradient Boosting and Ensemble Methods Gradient boosting [Friedman,2001] and its modern implementations such as XGBoost [Chen and Guestrin,2016], LightGBM [Ke et al.,2017], and CatBoost [Dorogush et al.,2018] have become the dominant modeling paradigms for tabular data. These models provide excellent predictive performance through stage-wise additive modeling and sophisticated regularization of the parameter space. However, they impose no constraints on the output space of the predictive function. Consequently, their continuous score outputs may fluctuate under distributional shifts or training perturbations, leading to instability in downstream decision systems. Our work differs by introducing a regularization mechanism that directly constrains the model’s outputs, thereby improving stability while preserving the predictive power of boosting-based models. 2.2 Post-Hoc Calibration, Scorecards, and Supervised Discretization Classical credit scoring models rely on logistic regression [Hosmer et al.,2013], often combined with coarse-classing or supervised binning to produce monotone, interpretable risk ladders [Siddiqi,2012]. These scorecards implicitly regularize the shape and resolution of the predictive function by restricting coefficients and discretizing inputs. Several post-hoc calibration methods refine continuous model outputs into better-behaved probability estimates. Platt scaling [Platt,1999] and isotonic regression [Zadrozny and Elkan,2002] enforce monotonicity in logits or scores, but do not modify model structure or constrain the set of possible outputs. Their outputs remain continuous and can therefore exhibit drift or instability across retraining cycles. Supervised discretization tools such as OptBinning [Anderson,2020] provide an optimization-based approach to binning numerical predictors under monotonicity or minimum-support constraints. These approaches learn interpretable partitions for features, not for predictions. QPMs invert this paradigm: they discretize the model output itself. This yields stability benefits that cannot be achieved by input discretization alone. 2.3 Quantization in Deep Learning Quantization has been widely studied in deep learning, primarily as a means of reducing model size and accelerating inference. Techniques such as weight quantization, activation quantization, and integeronly arithmetic enable deployment of neural networks under resource constraints [Jacob et al.,2018, Banner et al.,2018,Gholami et al.,2021]. In these settings, quantization is applied to the parameters or intermediate activations for computational efficiency. In contrast, we apply quantization to the model output, not to internal representations. Our goal is functional regularization rather than compression: discrete outputs explicitly constrain the predictive range, yielding stability and interpretability benefits absent from typical quantization approaches. 2.4 Monotonicity Constraints and Structured Outputs A parallel line of work seeks to incorporate structural constraints into prediction functions. Isotonic regression [Barlow et al.,1972,de Leeuw,2005] and its scalable variants [Tyree et al.,2011] impose global monotonicity, ensuring that the rank ordering of predictions respects known relationships. Latticebased models [You et al.,2017,Canini et al.,2016] combine piecewise-linear functions with partial monotonicity to produce interpretable models on tabular data. These methods constrain the shape of the regression function but do not restrict the set of attainable outputs. QPMs differ in that they enforce a discrete, ordered set of prediction levels independent of any monotonicity requirement. In this sense, QPMs can be viewed as an output-space generalization of isotonic regression: they support arbitrary interactions in the latent model while guaranteeing stable, monotone, and interpretable outputs. 3 2.5 Information Bottleneck Theory The information bottleneck framework [Tishby et al.,1999,Tishby and Zaslavsky,2015] studies how learning systems compress information while retaining task-relevant structure. Quantized representations naturally reduce mutual information between inputs and internal states, acting as noise filters in both theoretical and empirical settings. By constraining the output of a predictive function to a small set of discrete values, QPMs effectively impose an information bottleneck at the prediction level. This reduces sensitivity to small, functionally irrelevant variations in the input, thereby stabilizing rank ordering and mitigating noise amplification. 2.6 Summary Across these strands of literature, the focus has been on constraining model parameters, internal activations, or functional shape. QPMs address a complementary gap by regularizing the output space directly through quantization. This yields a unique combination of prediction stability, noise attenuation, and governance-friendly interpretability while remaining compatible with the expressive power of modern ML architectures. 3 Methodology Quantized Prediction Models (QPMs) combine the expressive power of continuous regressors with the stability and interpretability of piecewise-constant score ladders. This section formalizes the framework, introduces the binning and optimization mechanisms, and describes how QPMs support driftaware updates with provably zero rank-order churn. 3.1 Framework Overview QPMs decompose the prediction process into two components: a flexible latent function and a supervised quantizer applied to its output. Latent Function. Let F(x) denote a continuous regression function, typically a stage-wise additive model such as gradient boosting, trained on tabular features x∈Rdto approximate a probability of default y∈ {0,1}or a bounded risk score. The mapping F:Rd→Rproduces a latent score that may exhibit fine-grained, noisy variation. Quantization Operator. A QPM prediction is obtained by passing F(x) through a learned quantizer: Q(F(x); θ)=qkif F(x)∈[bk, bk+1), where θ={bk}K+1 k=1 are bin edges and {qk}K k=1 are bin representatives. Throughout this work, qk denotes the empirical conditional mean E[Y|F(x)∈[bk, bk+1)), providing an L2-optimal piecewiseconstant approximation to Ygiven the binning. Thus QPM predictors take the form ˆy(x)=Q(F(x); θ), which constrains model outputs to a finite ladder of ordered prediction levels. Conceptual Objective. At a conceptual level, QPM training seeks a quantized empirical-risk minimizer: min F,θ n X i=1yi−Q(F(xi); θ)2+λΩ(F), where Ω(F) controls model complexity (e.g., depth, shrinkage, or smoothness). The quantizer functions as an explicit information bottleneck, suppressing sub-bin fluctuations of Fand improving stability. In practice, we approximate this joint objective via a two-step procedure: (i) train Fas a continuous booster, then (ii) learn θvia supervised discretization of the latent scores. 4 Implementation in this Work. All experiments in this paper use the following operational recipe: 1. Train a gradient-boosted regressor Fon (xi, yi) using squared-error loss on yi∈ {0,1}. 2. Learn the quantizer parameters θ={bk, qk}by optimizing bin edges and bin representatives on F(xi) and yi, with support and monotonicity constraints. 3. For model updates under drift, keep bin edges fixed and update both Fand the bin representatives {qk}via residual stacking and Anchored Calibration. No in-the-loop quantization or straight-through estimators are used in this work; we rely entirely on post-hoc output-space regularization. 3.2 Optimizer-Driven Supervised Binning Uniform quantization or plain quantile binning ignore operational importance: risk tails, decision boundaries, or medically relevant thresholds typically require higher resolution than bulk regions. We therefore learn bin edges via an optimizer-driven discretization procedure inspired by supervised binning frameworks such as OptBinning [Anderson,2020] and constrained regression. Supervised Objective. Let L(θ) denote the loss incurred when using quantizer θon latent scores F(xi): L(θ) = n X i=1yi−Q(F(xi); θ)2. We seek to minimize this loss subject to ordered edges and operational constraints: min θL(θ) s.t. b1< b2< . . . < bK+1,and #{i:F(xi)∈[bk, bk+1)}≥mmin, optionally combined with monotonicity or smoothness constraints on {qk}. Resolution Density and Focus Region. To allocate bins where accuracy and policy sensitivity are highest, we introduce an importance weight w(z) over the latent score domain z=F(x) and define a resolution density ρ(z)∝w(z). For example, w(z) =        λtail, z ∈risk tail, λboundary, z near a decision cutoff, λbulk,otherwise, with λtail, λboundary > λbulk. Bin edges are then chosen so that Zbk+1 bk ρ(z)dz =1 K, yielding non-uniform, supervised quantization that concentrates resolution around a practitionerspecified focus region while keeping coarser bins elsewhere. In practice, we instantiate this via a calibrated mapping from Fto PD, then search over (nlow, nmid, nhigh) configurations that satisfy minimum band size and monotonicity constraints. Monotone Center Smoothing. Once preliminary bin representatives qkare computed empirically, we optionally project them onto the space of monotone sequences via isotonic regression. This projection enforces global monotonicity and removes small violations arising from finite-sample noise, yielding governance-compliant ladders that behave like traditional scorecards. 3.3 Drift-Aware Updates with Zero Churn Model recalibration on new data often induces prediction churn: samples near bin boundaries swap rank positions, even when global metrics such as AUC or KS improve. QPMs mitigate this via two mechanisms that restrict updates to smooth, monotone adjustments: a residual stacking step on the latent function and an anchored update of bin representatives. 5 3.3.1 Residual Stacking Instead of retraining a new Ft+1 from scratch, we perform a conservative update on top of the existing model. Let rt(xi)=yi−Ft(xi) be residuals on new data. We fit a shallow learner ht(x) (e.g., a small number of additional boosting stages) to predict rt(x) and update: Ft+1(x)=Ft(x)+η ht(x), with a small learning rate η > 0. This follows the gradient boosting paradigm but in a restricted “fine-tuning” regime, limiting the magnitude of changes in Fand thereby reducing large shifts in the latent scoring surface. 3.3.2 Anchored Calibration Given old bin representatives µold kand empirical bin means µobs kcomputed on updated data (using the same bin edges), QPM updates them via: µnew k=α µold k+(1−α)µobs k, α ∈[0,1]. Because bin edges remain fixed, each sample retains its previous bin assignment; only the representative values change. The parameter αcontrols how aggressively the scorecard adapts to the new empirical PDs: α= 1 recovers the original model, while α= 0 overwrites bin values with fully updated empirical means (we optionally project µobs to a monotone sequence prior to anchoring). Zero Rank-Order Churn. Let B(F(x)) denote the bin index associated with latent score F(x). If Fand the bin edges are fixed, then for any two points xi, xj: F(xi)< F(xj) =⇒B(F(xi)) ≤B(F(xj)). Anchored Calibration modifies only the representatives µk; it does not alter For the bin edges used to compute B(F(x)). Therefore bin membership is invariant: Bnew(F(xi))=Bold(F(xi)), and the updated predictions satisfy Qnew(F(xi))=µnew B(F(xi)) ≤µnew B(F(xj)) =Qnew(F(xj)), since the sequence {µnew k}is enforced to be monotone. Thus QPM updates preserve rank ordering exactly when bin edges are held fixed. This property is operationally critical in domains that rely on stable decision rules, such as credit underwriting, fraud detection, and medical triage. 4 Experiments We begin by evaluating Quantized Prediction Models (QPMs) on the UCI Default of Credit Card Clients dataset, a standard benchmark for credit risk modeling. This experiment illustrates three key claims: (i) QPMs match or slightly improve upon GBM accuracy (AUC, KS, MSE), (ii) the quantized outputs form a bank-style score ladder with smooth, monotone PDs, and (iii) QPM fine-tuning with fixed bin edges substantially reduces prediction churn relative to both independent rebinning and standard GBM retrains. 4.1 Experiment 1: UCI Default of Credit Card Clients We first evaluate Quantized Prediction Models on the UCI Default of Credit Card Clients dataset [Yeh and Lien,2009], which contains 30,000 credit card accounts, 25 features, and an empirical default rate of 22.1%. This dataset is a standard public benchmark for probability-of-default (PD) modeling and closely resembles industrial credit portfolios. 6 Backbone Optimization. Backbone predictors in all experiments are trained strictly out of the box. We apply no hyperparameter tuning, no feature selection, and no class-imbalance correction (e.g., weighting, resampling, or synthetic augmentation). As a result, backbone GBMs represent minimally optimized learners whose performance reflects default model behavior rather than a tuned or engineered pipeline. This design is intentional: QPM operates exclusively in the output space, and our objective is to isolate its effect by comparing each QPM model directly to its own backbone predictor. Consequently, the meaningful quantity in our results is the relative change in performance (e.g., ∆AUC), not the nominal magnitude of the backbone AUC itself. We split the data into training and test sets and compare three families of models: 1. a traditional scorecard based on logistic regression with coarse-classed features, 2. a strong Gradient Boosting Machine (GBM), and 3. a QPM constructed on top of the same GBM latent function. Scorecard (Logistic Regression). To represent the “white box” side of credit modeling, we train a standard logistic regression scorecard. Continuous variables are coarsely binned using supervised monotone discretization (e.g., quantile or domain-informed splits), transformed into weight-of-evidence (WOE) encodings, and passed to a logistic regressor. The resulting log-odds are mapped into ten equalfrequency bands (approximately 600 accounts each) to form a ten-grade PD ladder. On the test set, the scorecard attains: AUC = 0.7599,KS = 0.4018,MSE = 0.1389. The band-level PDs increase smoothly from ≈6% to ≈72%, yielding a traditional, decile-based risk ladder. GBM (Continuous). As a high-capacity benchmark, we use an XGBoost regressor with hyperparameters tuned for strong performance on tabular credit data: xgb.XGBRegressor( n_estimators=400, learning_rate=0.03, max_depth=4, subsample=0.7, colsample_bytree=0.7, min_child_weight=10, gamma=2, objective="reg:squarederror", tree_method="hist", random_state=857 ) This model produces continuous PD estimates in (0,1) without any output constraints or explicit score ladder structure. QPM (GBM + Supervised Quantizer). QPM uses the same GBM as a latent function F(x) and learns a non-uniform output quantizer on top. We first train the GBM to convergence, then apply the optimizer-driven binning procedure from Section 3with: focus pd = (0.05,0.20), nlow,max = 6, nmid,max = 6, nhigh,max = 6, min band size = 300,prior π0= ˆptrain,PD cap = 0.80. The algorithm allocates bins to low-, mid-, and high-risk regions, merges small bins to respect the minimum band size, and computes bin-level PD centers with Laplace/Bayesian smoothing and a PD cap of 0.80. On this dataset the chosen configuration is: nlow = 5, nmid = 6, nhigh = 6, K = 14 total bins, 7 Table 1: Predictive performance on the UCI Default dataset. QPM matches or slightly improves GBM accuracy while outperforming a traditional scorecard. Values are point estimates; brackets report bootstrap 95% confidence intervals for GBM and QPM. Model / Output MSE AUC KS Scorecard (LR, 10 bands) 0.1389 0.7599 0.4018 GBM (continuous) 0.1359 0.7727 0.4203 QPM (latent F(x)) 0.1359 0.7773 0.4289 QPM (quantized PD) 0.1359 0.7750 0.4225 with empirical PD centers (smoothed) ranging from 2.46% to 80%: (0.0246,0.0298,0.0590,0.0914,0.1296,0.1472,0.1675,0.1948,0.2806,0.4192,0.5265,0.6269,0.7742,0.8000). Most of the resolution is concentrated in the operationally critical low-to-mid PD region (5%,20%), with coarser bins in the very high risk tail. 4.1.1 Predictive Performance Table 1summarizes out-of-sample performance for the three models. For QPM we report both the latent GBM score F(x) (after training with the QPM objective) and the final quantized PD output. Bootstrap analysis over the test set yields: AUCGBM = 0.7727 [0.7574,0.7881], AUCQPM (quantized) = 0.7752 [0.7599,0.7902], KSGBM = 0.4236 [0.3961,0.4532], KSQPM (quantized) = 0.4231 [0.3935,0.4516]. The confidence intervals substantially overlap, indicating that QPM and GBM are statistically indistinguishable in AUC and KS on this dataset, while both improve upon the logistic scorecard. 4.1.2 Bank-Style Score Ladder Table 2reports the complete QPM bank-style score ladder obtained on the UCI Default test set. The quantizer produces K= 14 ordered grades G01–G14, each defined by a latent-score interval [Fmin, Fmax) with an associated posterior PD center and empirical performance statistics. On the test set, the PD centers rise smoothly from 2.46% to 80%, with strictly monotone progression across all grades and well-behaved lift profiles. The non-uniform bin widths concentrate resolution in the low-to-mid PD region (5%–20%), aligning with operational decision boundaries such as acquisition cutoffs and early collections triggers. Qualitatively, the resulting ladder resembles a traditional credit scorecard: it segments the portfolio into interpretable, risk-aligned strata while preserving the stability and monotonicity properties expected in regulated environments. At the same time, the latent GBM provides richer predictive structure than a handcrafted logistic regression model, yielding a hybrid architecture that is both powerful and governance-friendly. 4.1.3 Churn Under Model Updates We next examine how QPM behaves under model updates, comparing: 1. QPM A: Independent Retrains with Re-optimized Bins. Train QPM twice on different temporal slices and re-optimize the quantizer each time. 2. QPM B: Fine-Tune with Fixed Bin Edges (Ours). Fix the bin edges learned in the first fit and update only the latent GBM and bin means on the second slice. 8 Table 2: QPM bank-style scorecard on the UCI Default test set. Each grade corresponds to a nonuniform bin in latent score space, with monotone PD centers produced by the optimized quantizer. Grade Band Fmin Fmax PD #Acct %Portf #Def DefR Odds Lift Cum%Acct Cum%Def G01 0 -0.0147 0.0479 0.0246 222 0.0370 12 0.0541 17.50 0.2444 0.0370 0.0090 G02 1 0.0479 0.0635 0.0298 318 0.0530 13 0.0409 23.46 0.1848 0.0900 0.0188 G03 2 0.0635 0.0889 0.0590 665 0.1108 42 0.0632 14.83 0.2856 0.2008 0.0505 G04 3 0.0889 0.1142 0.0914 841 0.1402 84 0.0999 9.01 0.4516 0.3410 0.1138 G05 4 0.1142 0.1395 0.1296 700 0.1167 94 0.1343 6.45 0.6072 0.4577 0.1846 G06 5 0.1395 0.1648 0.1472 579 0.0965 89 0.1537 5.51 0.6950 0.5542 0.2517 G07 6 0.1648 0.1901 0.1675 479 0.0798 89 0.1858 4.38 0.8401 0.6340 0.3188 G08 7 0.1901 0.2154 0.1948 368 0.0613 63 0.1712 4.84 0.7741 0.6953 0.3662 G09 8 0.2154 0.3303 0.2806 735 0.1225 210 0.2857 2.50 1.2919 0.8178 0.5245 G10 9 0.3303 0.4452 0.4192 313 0.0522 119 0.3802 1.63 1.7190 0.8700 0.6142 G11 10 0.4452 0.5601 0.5265 145 0.0242 75 0.5172 0.93 2.3387 0.8942 0.6707 G12 11 0.5601 0.6751 0.6269 259 0.0432 159 0.6139 0.63 2.7757 0.9373 0.7905 G13 12 0.6751 0.7900 0.7742 326 0.0543 238 0.7301 0.37 3.3010 0.9917 0.9699 G14 13 0.7900 0.9049 0.8000 50 0.0083 40 0.8000 0.25 3.6172 1.0000 1.0000 3. GBM vs QPM: Head-to-Head Churn. Compare stochastic GBM retrains to QPM fine-tunes on the same splits. 4. QPM C: Calibration-Only Anchored Updates. Apply Anchored Calibration with fully frozen bin edges and latent model, using new data only to shift bin centers. We measure churn both in terms of percentile ranks and in terms of band assignments. For percentile churn we compute the absolute change in each account’s percentile rank between version 1 and version 2. For band churn we assign accounts to global quantile bands (5 equal-sized bands) or to native QPM bands and track how many accounts change band or jump by two or more bands. Independent Rebin vs Fixed Edges. When QPM is fully re-fit with a fresh quantizer (scenario A), the native 8-band representation (for the churn experiment) exhibits: •31.4% of accounts change native band at least once, •2.8% move by two or more native bands. Under fine-tuning with fixed edges (scenario B), the same native bands yield: •7.0% of accounts change band, •0.3% move by two or more bands. Thus, holding bin edges fixed reduces native-band churn by more than a factor of four compared to independent rebinning, while preserving predictive performance. GBM vs QPM: Continuous vs Quantized PD. For a fair comparison to a fully continuous model, we consider 5 global percentile bands applied to both a stochastic GBM retrain and the QPM fine-tune. Between version 1 and version 2 we observe: •GBM retrain (continuous PD): mean absolute percentile shift 0.0332; band churn 14.6%; 0.2% move ≥2 bands. •QPM fine-tune (quantized PD): mean absolute percentile shift 0.0392; band churn 17.7%; 0.3% move ≥2 bands. On this dataset, QPM fine-tuning exhibits churn of the same order of magnitude as GBM retraining when both are summarized into coarse global bands. The small differences fall within the range one would expect from stochasticity and binning effects. The key advantage of QPM lies not in uniformly dominating GBM on this metric, but in the additional structural guarantees it can provide through Anchored Calibration. 9 John Platt. Probabilistic outputs for support vector machines and comparisons to regularized likelihood methods. In Advances in Large Margin Classifiers, pages 61–74. MIT Press, 1999. Naeem Siddiqi. Credit Risk Scorecards: Developing and Implementing Intelligent Credit Scoring. Wiley, 2012. Naftali Tishby and Noga Zaslavsky. Deep learning and the information bottleneck principle. 2015 IEEE Information Theory Workshop (ITW), pages 1–5, 2015. Naftali Tishby, Fernando C Pereira, and William Bialek. The information bottleneck method. Proceedings of the 37th Annual Allerton Conference on Communication, Control, and Computing, pages 368–377, 1999. Stephen Tyree, Kilian Q Weinberger, Kunal Agrawal, Joel Paykin, and B Rouhani. Parallel boosted regression trees for web search ranking. In Proceedings of the 20th International Conference on World Wide Web, pages 387–396, 2011. I-Cheng Yeh and Che-hui Lien. The comparisons of data mining techniques for the predictive accuracy of probability of default of credit card clients. Expert Systems with Applications, 36(2):2473–2480, 2009. doi: 10.1016/j.eswa.2007.12.020. Dataset available at the UCI Machine Learning Repository: https://archive.ics.uci.edu/dataset/350/default+of+credit+card+clients. Seungil You, Dacheng Ding, Kevin Canini, Johannes Pfeifer, and Mayank Gupta. Deep lattice networks and partial monotonic functions. In Advances in Neural Information Processing Systems, volume 30, 2017. Bianca Zadrozny and Charles Elkan. Transforming classifier scores into accurate multiclass probability estimates. In Proceedings of the 8th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pages 694–699, 2002. 16