Abstract

Most Machine Learning generative models assume the Multivariate Gaussian (Normal) distribution. However, practically all real-world data is non-Gaussian and very often exhibits a number of undesirable features, such as heavy tails, skewness, and heterogeneity, especially in the public health and socioeconomic domains. Because of this intrinsic mismatch, extreme risks, such as tail risk, have always been underestimated, compromising classification accuracy. This paper constructs a maximum likelihood estimation framework for the Alternative Multivariate Exponential Power Distribution (AMEPD) in order to close this critical gap beyond the conventional models (Gaussian and the classical Multivariate Exponential Power Distribution, MEPD). An adaptive, shape-dependent scaling coefficient, c(q), allows the AMEPD - a generalisation of the Normal distribution - to capture a broader range of non-Gaussian behaviours. One of the major obstacles limiting the use of AMEPD is the intricacy of parameter estimation arising from the highly nonlinear log-likelihood function. We propose a robust quasi-Newton optimization framework using the limited-memory BFGS algorithm in conjunction with a Cholesky reparameterization (Σ = LL^⊤) to ensure positive-definite scale matrix Σ and robust, numerically stable convergence in high dimensions. The framework is, therefore, applied empirically to the 2022 Ghana Demographic and Health Survey (DHS) dataset in order to classify childbirth outcomes into safe or unsafe. Using both Newton-Raphson and L-BFGS optimisation techniques, a comprehensive comparison of AMEPD and MEPD under both Linear Discriminant Analysis-LDA (equal covariance) and Quadratic Discriminant Analysis-QDA (unequal covariance) architectures was conducted. The results clearly manifest the superiority of the AMEPD framework: the best classification performance was recorded for an AMEPD-QDA (Newton), with 85.4% correct classification rate and an AUC-ROC of 0.892, outperforming the Gaussian baseline by a large margin (+6.3%). Such better performance by AMEPD can be attributed to its adaptive parameter c(q), which models heterogeneity successfully; more specifically, the safe class exhibited an almost Gaussian distribution (q ̂_1=1.82), while the class of unsafe delivery was heavy-tailed (q ̂_2=0.68). Furthermore, QDA models consistently outperformed LDA by 1.5–3.2%, which is an indication that heteroscedasticity does indeed exist in maternal health data. Clinically, an important 82.7% recall for unsafe deliveries was attained by the AMEPD. The discriminant weight of the number of ANC visits was the highest, 0.342, among the predictors. Thus, the most accurate and robust machine learning framework for distinguishing between delivery outcomes is a combination of AMEPD distributional flexibility and a QDA covariance structure that is optimized using either the L-BFGS or Newton method. This approach accelerates the progress toward achieving SDG 3.1 through the early identification of high-risk pregnancies by health programs based on risk scores developed from AMEPD.