Varying-coefficient mixture of experts model for dynamic heterogeneous populations: application to mouse cortical development

As cells differentiate, gene-gene associations may change. Because the composition of cell subtypes also shifts with development, it is challenging to establish whether those changes reflect real changes in gene regulation within one or more subpopulations or a spurious one due to the shifting composition. Only formal inference can separate them. We propose a Varying-Coefficient Mixture-of-Experts (VCMoE) model in which the coefficients of both the gating functions and the expert components vary smoothly with the index variable, so that a subgroup-specific association can be inferred without being confounded by the concurrent shift in subgroup composition. We develop a label-consistent EM algorithm and establish the identifiability and asymptotic theory that make the resulting inference valid. These guarantees support simultaneous confidence bands, constructed using asymptotic and bootstrap methods, together with a generalized likelihood ratio test for constant coefficient functions. Simulations demonstrate accurate estimation and satisfactory coverage. Applied to single-nucleus RNA-sequencing data from embryonic mouse cortex, VCMoE finds that the repression of Bcl11b by Satb2 within the upper layer is not yet in place when those neurons first appear and becomes established over the course of differentiation, a dynamic that an ordinary mixture-of-experts model neglected. A package for implementing the method is available.