SummaryThe genetic architecture of autism spectrum disorder involves the interplay of common and rare variation and their impact on hundreds of genes. Using exome sequencing, analysis of rare coding variation in 3,871 autism cases and 9,937 ancestry-matched or parental controls implicates 22 autosomal genes at a false discovery rate (FDR) < 0.05, and a set of 107 autosomal genes strongly enriched for those likely to affect risk (FDR < 0.30). These 107 genes, which show unusual evolutionary constraint against mutations, incur de novo loss-of-function mutations in over 5% of autistic subjects. Many of the genes implicated encode proteins for synaptic, transcriptional, and chromatin remodeling pathways. These include voltage-gated ion channels regulating propagation of action potentials, pacemaking, and excitability-transcription coupling, as well as histone-modifying enzymes and chromatin remodelers, prominently histone post-translational modifications involving lysine methylation/demethylation.

Autism spectrum disorders (ASD) are believed to have genetic and environmental origins, yet in only a modest fraction of individuals can specific causes be identified1,2. To identify further genetic risk factors, we assess the role of de novo mutations in ASD by sequencing the exomes of ASD cases and their parents (n= 175 trios). Fewer than half of the cases (46.3%) carry a missense or nonsense de novo variant and the overall rate of mutation is only modestly higher than the expected rate. In contrast, there is significantly enriched connectivity among the proteins encoded by genes harboring de novo missense or nonsense mutations, and excess connectivity to prior ASD genes of major effect, suggesting a subset of observed events are relevant to ASD risk. The small increase in rate of de novo events, when taken together with the connections among the proteins themselves and to ASD, are consistent with an important but limited role for de novo point mutations, similar to that documented for de novo copy number variants. Genetic models incorporating these data suggest that the majority of observed de novo events are unconnected to ASD, those that do confer risk are distributed across many genes and are incompletely penetrant (i.e., not necessarily causal). Our results support polygenic models in which spontaneous coding mutations in any of a large number of genes increases risk by 5 to 20-fold. Despite the challenge posed by such models, results from de novo events and a large parallel case-control study provide strong evidence in favor of CHD8 and KATNAL2 as genuine autism risk factors.

Cosmological inference becomes increasingly difficult when complex data-generating processes cannot be modeled by simple probability distributions. With the everincreasing size of data sets in cosmology, there is increasing burden placed on adequate modeling; systematic errors in the model will dominate where previously these were swamped by statistical errors. For example, Gaussian distributions are an insufficient representation for errors in quantities like photometric redshifts. Likewise, it can be difficult to quantify analytically the distribution of errors that are introduced in complex fitting codes. Without a simple form for these distributions, it becomes difficult to accurately construct a likelihood function for the data as a function of parameters of interest. Approximate Bayesian computation (ABC) provides a means of probing the posterior distribution when direct calculation of a sufficiently accurate likelihood is intractable. ABC allows one to bypass direct calculation of the likelihood but instead relies upon the ability to simulate the forward process that generated the data. These simulations can naturally incorporate priors placed on nuisance parameters, and hence these can be marginalized in a natural way. We present and discuss ABC methods in the context of supernova cosmology using data from the SDSS-II Supernova Survey. Assuming a flat cosmology and constant dark energy equation of state we demonstrate that ABC can recover an accurate posterior distribution. Finally we show that ABC can still produce an accurate posterior distribution when we contaminate the sample with Type IIP supernovae.

We present a semi‐supervised method for photometric supernova typing. Our approach is to first use the non‐linear dimension reduction technique diffusion map to detect structure in a data base of supernova light curves and subsequently employ random forest classification on a spectroscopically confirmed training set to learn a model that can predict the type of each newly observed supernova. We demonstrate that this is an effective method for supernova typing. As supernova numbers increase, our semi‐supervised method efficiently utilizes this information to improve classification, a property not enjoyed by template‐based methods. Applied to supernova data simulated by Kessler et al. to mimic those of the Dark Energy Survey, our methods achieve (cross‐validated) 95 per cent Type Ia purity and 87 per cent Type Ia efficiency on the spectroscopic sample, but only 50 per cent Type Ia purity and 50 per cent efficiency on the photometric sample due to their spectroscopic follow‐up strategy. To improve the performance on the photometric sample, we search for better spectroscopic follow‐up procedures by studying the sensitivity of our machine‐learned supernova classification on the specific strategy used to obtain training sets. With a fixed amount of spectroscopic follow‐up time, we find that, despite collecting data on a smaller number of supernovae, deeper magnitude‐limited spectroscopic surveys are better for producing training sets. For supernova Ia (II‐P) typing, we obtain a 44 per cent (1 per cent) increase in purity to 72 per cent (87 per cent) and 30 per cent (162 per cent) increase in efficiency to 65 per cent (84 per cent) of the sample using a 25th (24.5th) magnitude‐limited survey instead of the shallower spectroscopic sample used in the original simulations. When redshift information is available, we incorporate it into our analysis using a novel method of altering the diffusion map representation of the supernovae. Incorporating host redshifts leads to a 5 per cent improvement in Type Ia purity and 13 per cent improvement in Type Ia efficiency.

Dimension-reduction techniques can greatly improve statistical inference in astronomy. A standard approach is to use Principal Components Analysis (PCA). In this work we apply a recently-developed technique, diffusion maps, to astronomical spectra for data parameterization and dimensionality reduction, and develop a robust, eigenmodebased framework for regression. We show how our framework provides a computationally efficient means by which to predict redshifts of galaxies, and thus could inform more expensive redshift estimators such as template cross-correlation. It also provides a natural means by which to identify outliers (e.g., misclassified spectra, spectra with anomalous features). We analyze 3835 SDSS spectra and show how our framework yields a more than 95% reduction in dimensionality. Finally, we show that the prediction error of the diffusion map-based regression approach is markedly smaller than that of a similar approach based on PCA, clearly demonstrating the superiority of diffusion maps over PCA for this regression task.