Mathematics Applications to Detect Fetal Syndromes

Abstract

This research addresses the detection of fetal syndromes as a high-dimensional, non-linear binary classification problem, we mathematically formulate and empirically evaluate three classes of models: probabilistic classifiers based on Bayesian inference with multivariate Gaussian assumptions, geometric classifiers such as Support Vector Machines with non-linear kernels, and deep learning models based on multi-layer neural networks, the study's central hypothesis posits that the complex, synergistic interplay between sonographic and biochemical markers can only be captured by models with high representational capacity. Using a large clinical dataset, we demonstrate the hierarchical superiority of a Deep Neural Network (DNN), which achieved a test set Area Under the Curve (AUC) of 0.982 and a Matthews Correlation Coefficient (MCC) of 0.869, through the application of SHapley Additive exPlanations (SHAP), we deconstruct the model to reveal that higher-order interaction effects account for approximately 20% of its predictive power. Furthermore, by employing a Bayesian Neural Network (BNN), we introduce a framework for quantifying predictive uncertainty, decomposing it into its aleatoric and epistemic components, the results show that the BNN can reliably flag atypical patient profiles by exhibiting high epistemic uncertainty, a critical feature for clinical safety, this work concludes that the problem's underlying geometry is that of complex, intertwined manifolds, and that models capable of learning these structures while quantifying their own uncertainty represent the next frontier in prenatal diagnostics.