$\texttt{PRISM}$:A 3D Probabilistic Neural Representation for Interpretable Shape Modeling

Understanding how anatomical shapes evolve in response to developmental covariates-and quantifying their spatially varying uncertainties-is critical in healthcare research. Existing approaches typically rely on global time-warping formulations that ignore spatially heterogeneous dynamics. We introduce PRISM, a novel framework that bridges implicit neural representations with uncertainty-aware statistical shape analysis. PRISM models the conditional distribution of shapes given covariates, providing spatially continuous estimates of both the population mean and covariate-dependent uncertainty at arbitrary locations. A key theoretical contribution is a closed-form Fisher Information metric that enables efficient, analytically tractable local temporal uncertainty quantification via automatic differentiation. Experiments on three synthetic datasets and one clinical dataset demonstrate PRISM's strong performance across diverse tasks-from modeling shape evolution to personalized shape prediction and anomaly detection-within a unified framework, while providing interpretable and clinically meaningful uncertainty estimates. The code is publicly available at https://github.com/uncbiag/PRISM.