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724724# Discussion and conclusions
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726726- ** Semiparametric marginal estimation (Bernstein–Dirichlet)**
727- - ** Advantages:** smooth joint PDF/CDF estimation • monotonic CDF by construction • boundary correction
727+ - ** Advantages:** smooth joint PDF/CDF estimation • monotonic CDF by construction
728728 - ** Limitations:** * sensitive* to prior choice • heavy tails • disjoint supports • narrow multimodal densities
729729- ** Bayesian nonparanormal network architecture**
730- - ** Advantages:** outperformed baselines in expected LOO-CV predictive accuracy • adaptive multi-scale spatial dependency via SPDE hyperparameters • semiparametric marginal modeling • spectral governance of spatial dependence resolves limitations of static adjacency-based models
730+ - ** Advantages:** outperformed baselines in expected LOO-CV predictive accuracy • semiparametric marginal modeling • spectral governance of spatial dependence resolves limitations of static adjacency-based models
731731 - ** Limitations:** Cholesky $\mathcal{O}(d^2) \text{--} \mathcal{O}(d^{3/2})$ • no tail dependence • Bernstein–Dirichlet limitations
732732- ** Innovations and broader implications**
733733 - ** Copula‐based Besag's pseudolikelihood:** partition-free MRF inference
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746746- Dynamic Bernstein-Dirichlet univariate models
747747- Conditional Bernstein-Dirichlet models
748748- Integration with machine learning architectures (Bayesian deep learning)
749- - Multivariate and cross‑variable extremes ( copula families)
749+ - Other copula families
750750- Scalable inference for large‑scale networks
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