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  1. Free, publicly-accessible full text available July 3, 2024
  2. Monte Carlo algorithms, such as Markov chain Monte Carlo (MCMC) and Hamiltonian Monte Carlo (HMC), are routinely used for Bayesian inference; however, these algorithms are prohibitively slow in massive data settings because they require multiple passes through the full data in every iteration. Addressing this problem, we develop a scalable extension of these algorithms using the divide‐and‐conquer (D&C) technique that divides the data into a sufficiently large number of subsets, draws parameters in parallel on the subsets using apoweredlikelihood and produces Monte Carlo draws of the parameter by combining parameter draws obtained from each subset. The combined parameter draws play the role of draws from the original sampling algorithm. Our main contributions are twofold. First, we demonstrate through diverse simulated and real data analyses focusing on generalized linear models (GLMs) that our distributed algorithm delivers comparable results as the current state‐of‐the‐art D&C algorithms in terms of statistical accuracy and computational efficiency. Second, providing theoretical support for our empirical observations, we identify regularity assumptions under which the proposed algorithm leads to asymptotically optimal inference. We also provide illustrative examples focusing on normal linear and logistic regressions where parts of our D&C algorithm are analytically tractable.

     
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  3. Robert McCulloch (Ed.)
  4. null (Ed.)
    Linear mixed-effects models play a fundamental role in statistical methodology. A variety of Markov chain Monte Carlo (MCMC) algorithms exist for fitting these models, but they are inefficient in massive data settings because every iteration of any such MCMC algorithm passes through the full data. Many divide-and-conquer methods have been proposed to solve this problem, but they lack theoretical guarantees, impose restrictive assumptions, or have complex computational algorithms. Our focus is one such method called the Wasserstein Posterior (WASP), which has become popular due to its optimal theoretical properties under general assumptions. Unfortunately, practical implementation of the WASP either requires solving a complex linear program or is limited to one-dimensional parameters. The former method is inefficient and the latter method fails to capture the joint posterior dependence structure of multivariate parameters. We develop a new algorithm for computing the WASP of multivariate parameters that is easy to implement and is useful for computing the WASP in any model where the posterior distribution of parameter belongs to a location-scatter family of probability measures. The algorithm is introduced for linear mixed-effects models with both implementation details and theoretical properties. Our algorithm outperforms the current state-of-the-art method in inference on the functions of the covariance matrix of the random effects across diverse numerical comparisons. 
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  5. The genetic and molecular basis of heterosis has long been studied but without a consensus about mechanism. The opposite effect, inbreeding depression, results from repeated self-pollination and leads to a reduction in vigor. A popular explanation for this reaction is the homozygosis of recessive, slightly deleterious alleles upon inbreeding. However, extensive studies in alfalfa indicated that inbreeding between diploids and autotetraploids was similar despite the fact that homozygosis of alleles would be dramatically different. The availability of tetraploid lines of maize generated directly from various inbred lines provided the opportunity to examine this issue in detail in perfectly matched diploid and tetraploid hybrids and their parallel inbreeding regimes. Identical hybrids at the diploid and tetraploid levels were inbred in triplicate for seven generations. At the conclusion of this regime, F1 hybrids and selected representative generations (S1, S3, S5, S7) were characterized phenotypically in randomized blocks during the same field conditions. Quantitative measures of the multiple generations of inbreeding provided little evidence for a distinction in the decline of vigor between the diploids and the tetraploids. The results suggest that the homozygosis of completely recessive, slightly deleterious alleles is an inadequate hypothesis to explain inbreeding depression in general. 
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