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Many two-level nested simulation applications involve the conditional expectation of some response variable, where the expected response is the quantity of interest, and the expectation is with respect to the inner-level random variables, conditioned on the outer-level random variables. The latter typically represent random risk factors, and risk can be quantified by estimating the probability density function (pdf) or cumulative distribution function (cdf) of the conditional expectation. Much prior work has considered a naïve estimator that uses the empirical distribution of the sample averages across the inner-level replicates. This results in a biased estimator, because the distribution of the sample averages is over-dispersed relative to the distribution of the conditional expectation when the number of inner-level replicates is finite. Whereas most prior work has focused on allocating the numbers of outer- and inner-level replicates to balance the bias/variance tradeoff, we develop a bias-corrected pdf estimator. Our approach is based on the concept of density deconvolution, which is widely used to estimate densities with noisy observations but has not previously been considered for nested simulation problems. For a fixed computational budget, the bias-corrected deconvolution estimator allows more outer-level and fewer inner-level replicates to be used, which substantially improves the efficiency of the nested simulation. 

Abstract Embryonic development is largely conserved among mammals. However, certain genes show divergent functions. By generating a transcriptional atlas containing >30,000 cells from post-implantation non-human primate embryos, we uncover that ISL1 , a gene with a well-established role in cardiogenesis, controls a gene regulatory network in primate amnion. CRISPR/Cas9-targeting of ISL1 results in non-human primate embryos which do not yield viable offspring, demonstrating that ISL1 is critically required in primate embryogenesis. On a cellular level, mutant ISL1 embryos display a failure in mesoderm formation due to reduced BMP4 signaling from the amnion. Via loss of function and rescue studies in human embryonic stem cells we confirm a similar role of ISL1 in human in vitro derived amnion. This study highlights the importance of the amnion as a signaling center during primate mesoderm formation and demonstrates the potential of in vitro primate model systems to dissect the genetics of early human embryonic development.