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Stable Global Weighting of Flow Mixtures using Simplex Exponential Moving Average

Normalising flows provide a powerful variational family for approximate inference, yet individual architectures often fail to generalise across heterogeneous posterior geometries. We revisit mixture-based flow formulations and introduce \emph{AMF\mbox{-}VI\mbox{-}sEMA}, a two-stage framework featuring a \emph{stable global weighting} mechanism based on a \emph{Simplex Exponential Moving Average} (sEMA) update. In Stage~1, a heterogeneous set of experts (\textsc{RealNVP}, \textsc{MAF}, \textsc{RBIG}) are trained independently to specialise in distinct structural regimes. In Stage~2, expert parameters are frozen and global mixture weights are learned through a temperature-controlled softmax of average log-likelihoods, followed by a smooth EMA update on the probability simplex. This design produces a tractable, data-agnostic gating mechanism (without per-sample gating or gradient backpropagation through weights) that adaptively reallocates capacity while avoiding component collapse. We evaluate the framework on ten posterior benchmarks: six canonical 2D synthetic families (Banana, X-Shaped, Bimodal, Multimodal, Two-moons, Rings) and four real/low-dimensional Bayesian targets (BLR, BPR, Weibull, Real-GMM2), with stronger baselines (\textsc{NICE}, \textsc{ResFlow}, and EM-Mixing). Comprehensive evaluation covers NLL, KL divergence, Wasserstein-2 distance, and MMD, together with diagnostics of mixture dynamics, hyperparameter sensitivity, and cross-seed robustness. Empirically, \emph{AMF\mbox{-}VI\mbox{-}sEMA} achieves consistent NLL improvements over its predecessor \emph{AMF\mbox{-}VI} and avoids the catastrophic transport failures of single-flow baselines, while maintaining stable weight trajectories ($N_{\mathrm{eff}}{>}1.4$ on all datasets) with minimal computational overhead.

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