Estimating the mixing density of a latent mixture model is an important task in signal processing. Nonparametric maximum likelihood estimation is one popular approach to this problem. If the latent variable distribution is assumed to be continuous, then bootstrapping can be used to approximate it. However, traditional bootstrapping requires repeated evaluations on resampled data and is not scalable. In this letter, we construct a generative process to rapidly produce nonparametric maximum likelihood bootstrap estimates. Our method requires only a single evaluation of a novel two-stage optimization algorithm. Simulations and real data analyses demonstrate that our procedure accurately estimates the mixing density with little computational cost even when there are a hundred thousand observations.

Nonparametric varying coefficient (NVC) models are useful for modeling time-varying effects on responses that are measured repeatedly for the same subjects. When the number of covariates is moderate or large, it is desirable to perform variable selection from the varying coefficient functions. However, existing methods for variable selection in NVC models either fail to account for within-subject correlations or require the practitioner to specify a parametric form for the correlation structure. In this paper, we introduce the nonparametric varying coefficient spike-and-slab lasso (NVC-SSL) for Bayesian high dimensional NVC models. Through the introduction of functional random effects, our method allows for flexible modeling of within-subject correlations without needing to specify a parametric covariance function. We further propose several scalable optimization and Markov chain Monte Carlo (MCMC) algorithms. For variable selection, we propose an Expectation Conditional Maximization (ECM) algorithm to rapidly obtain maximum a posteriori (MAP) estimates. Our ECM algorithm scales linearly in the total number of observations $N$ and the number of covariates $p$. For uncertainty quantification, we introduce an approximate MCMC algorithm that also scales linearly in both $N$ and $p$. We demonstrate the scalability, variable selection performance, and inferential capabilities of our method through simulations and a real data application. These algorithms are implemented in the publicly available R package NVCSSL on the Comprehensive R Archive Network.

OBJECTIVE: The purpose of this study was to measure the association between neighborhood deprivation and cesarean delivery following labor induction among people delivering at term (≥37 weeks of gestation). MATERIALS AND METHODS: We conducted a retrospective cohort study of people ≥37 weeks of gestation, with a live, singleton gestation, who underwent labor induction from 2010 to 2017 at Penn Medicine. We excluded people with a prior cesarean delivery and those with missing geocoding information. Our primary exposure was a nationally validated Area Deprivation Index with scores ranging from 1 to 100 (least to most deprived). We used a generalized linear mixed model to calculate the odds of postinduction cesarean delivery among people in 4 equally-spaced levels of neighborhood deprivation. We also conducted a sensitivity analysis with residential mobility. RESULTS: Our cohort contained 8672 people receiving an induction at Penn Medicine. After adjustment for confounders, we found that people living in the most deprived neighborhoods were at a 29% increased risk of post-induction cesarean delivery (adjusted odds ratio = 1.29, 95% confidence interval, 1.05-1.57) compared to the least deprived. In a sensitivity analysis, including residential mobility seemed to magnify the effect sizes of the association between neighborhood deprivation and postinduction cesarean delivery, but this information was only available for a subset of people. CONCLUSIONS: People living in neighborhoods with higher deprivation had higher odds of postinduction cesarean delivery compared to people living in less deprived neighborhoods. This work represents an important first step in understanding the impact of disadvantaged neighborhoods on adverse delivery outcomes.