Empirical Evidence of Posterior Probability Bias Under SMOTE Oversampling in Imbalanced Credit Card Fraud Detection: A Comparative Study of HistGradient Boosting and Logistic Regression | IJCT Volume 13 – Issue 3 | IJCT-V13I3P123

This study set out to determine whether SMOTE oversampling induces posterior probability bias similar to the under-sampling induced bias demonstrated by Dal Pozzolo, Caelen, Johnson, and Bontempi in their work on undersampling. The answer to this question, supported by clear and unambiguous experimental evidence, is yes. The four conclusions that summarize the findings are given below. First, SMOTE-induced probability bias is confirmed and demonstrated with evidence. When SMOTE was applied as the sole class imbalance intervention to Logistic Regression (LR), false alarms increased by a factor of 165, from 35 to 5,775 across 5 Cross Validation (CV) folds, while ROC-AUC remained virtually unchanged at 0.9806. On the isolated test set, LR generated 1,415 false alarms at the default threshold of 0.5. This precise pattern of spikes in false alarm inflation with stable ranking (ROC-AUC), is the operational fingerprint (evidence) of posterior probability bias as described by authors of the earlier study. This study demonstrates this fingerprint for SMOTE over-sampling on the same dataset where Dal Pozzolo et al. demonstrated it for under-sampling. Thus, class imbalance not only hinders learning minority patterns but also distorts posterior probabilities when resampling methods are applied, whether by under-sampling (Dal Pozzolo et al., 2015) or by over-sampling (the empirical findings in this study). Second, class_weight=’balanced’ is not the source of the bias. An initial experiment applied both SMOTE and class_weight=’balanced’ simultaneously. And then SMOTE was isolated and evidence shows that class_weight=’balanced is not the source of the bias. This confirms that class weighting introduces operationally benign distortions. Third, model expressiveness and SMOTE response are inseparable. HistGradient Boosting responded to SMOTE with improved performance, that is, false alarms decreased and F1 rose by 28.3%. With Logistic Regression’s precision collapsed. Therefore, SMOTE benefits are model-dependent: a classifier must have sufficient capacity to exploit synthetic samples without being miscalibrated by them. Logistic Regression’s linear decision boundary lacks this capacity on this dataset. Fourth, Randomized Search is the preferred optimization strategy. Randomized Search outperformed Exhaustive Grid Search for HistGradient Boosting (CV F1: 0.8293 vs 0.7036) while it spent less time evaluating fewer combinations. This empirically confirming the work of Bergstra and Bengio on this real-world imbalanced Kaggle Credit Card dataset. The most important direction for future work is the formal implementation of Dal Pozzolo et al.’s probability correction formula adapted for SMOTE oversampling. Their formula – p’ = (β × p_s) / (β × p_s − p_s + 1), where β = true_fraud_prior / smote_fraud_rate ≈ 0.00346 – should correct the upward probability shift that this study has identified as the source of Logistic Regression’s false alarm inflation. If this formular is applied successfully, this would dramatically reduce false alarms while preserving recall, and as a result, will provide a practical remedy for the bias demonstrated in this study, and for the first time, formally validate Dal Pozzolo et al.’s correction framework in relation to over-sampling. Additional future directions should explore evaluation on a second dataset to assess generalizability.