A worker arranges packets of condoms at a pharmacy in Dhaka. Bangladesh's family planning system is buckling under severe contraceptive shortages. DHAKA - Bangladesh's once-praised family planning system is buckling under severe contraceptive shortages, raising fears of a rise in unplanned pregnancies in one of the world's most densely populated countries. For decades, the South Asian nation was hailed as a success for slashing birth rates through an expansive state-backed family planning program that sent field workers door to door with pills, condoms and advice on birth spacing. But that system is now faltering, with government clinics across the country of 170 million people running out of basic contraceptives after procurement failures and administrative disruption left supplies depleted in nearly a third of districts. In a time of both misinformation and too much information, quality journalism is more crucial than ever.

We prove that conditional diffusion models whose reverse kernels are finite Gaussian mixtures with ReLU-network logits can approximate suitably regular target distributions arbitrarily well in context-averaged conditional KL divergence, up to an irreducible terminal mismatch that typically vanishes with increasing diffusion horizon. A path-space decomposition reduces the output error to this mismatch plus per-step reverse-kernel errors; assuming each reverse kernel factors through a finite-dimensional feature map, each step becomes a static conditional density approximation problem, solved by composing Norets' Gaussian-mixture theory with quantitative ReLU bounds. Under exact terminal matching the resulting neural reverse-kernel class is dense in conditional KL.

We propose stepwise variational inference (VI) with vine copulas: a universal VI procedure that combines vine copulas with a novel stepwise estimation procedure of the variational parameters. Vine copulas consist of a nested sequence of trees built from copulas, where more complex latent dependence can be modeled with increasing number of trees. We propose to estimate the vine copula approximate posterior in a stepwise fashion, tree by tree along the vine structure. Further, we show that the usual backward Kullback-Leibler divergence cannot recover the correct parameters in the vine copula model, thus the evidence lower bound is defined based on the Rényi divergence. Finally, an intuitive stopping criterion for adding further trees to the vine eliminates the need to pre-define a complexity parameter of the variational distribution, as required for most other approaches. Thus, our method interpolates between mean-field VI (MFVI) and full latent dependence. In many applications, in particular sparse Gaussian processes, our method is parsimonious with parameters, while outperforming MFVI.

We develop EigenVI, an eigenvalue-based approach for black-box variational inference (BBVI).

V ariational inference (VI) is a method to approximate the computationally intractable posterior distributions that arise in Bayesian statistics.