One of the core problems of modern statistics is to approximate difficult-to-compute probability densities. This problem is especially important in Bayesian statistics, which frames all inference about unknown quantities as a calculation involving the posterior density. In this paper, we review variational inference (VI), a method from machine learning that approximates probability densities through optimization. VI has been used in many applications and tends to be faster than classical methods, such as Markov chain Monte Carlo sampling. The idea behind VI is to first posit a family of densities and then to find the member of that family which is close to the target. Closeness is measured by Kullback-Leibler divergence. We review the ideas behind mean-field variational inference, discuss the special case of VI applied to exponential family models, present a full example with a Bayesian mixture of Gaussians, and derive a variant that uses stochastic optimization to scale up to massive data. We discuss modern research in VI and highlight important open problems. VI is powerful, but it is not yet well understood. Our hope in writing this paper is to catalyze statistical research on this class of algorithms.

Many of the classification algorithms developed in the machine learning literature, including the support vector machine and boosting, can be viewed as minimum contrast methods that minimize a convex surrogate of the 0-1 loss function. The convexity makes these algorithms computationally efficient. The use of a surrogate, however, has statistical consequences that must be balanced against the computational virtues of convexity. To study these issues, we provide a general quantitative relationship between the risk as assessed using the 0-1 loss and the risk as assessed using any nonnegative surrogate loss function. We show that this relationship gives nontrivial upper bounds on excess risk under the weakest possible condition on the loss function: that it satisfy a pointwise form of Fisher consistency for classification. The relationship is based on a simple variational transformation of the loss function that is easy to compute in many applications. We also present a refined version of this result in the case of low noise. Finally, we present applications of our results to the estimation of convergence rates in the general setting of function classes that are scaled convex hulls of a finite-dimensional base class, with a variety of commonly used loss functions.

Materials and Methods 1. Generation of sequencing targets. Sequencing targets were amplified from genomic DNA using Platinum Taq HiFi (Invitrogen) following the manufacturer's recommendations, with primers

Type I collagen, the predominant protein of vertebrates, polymerizes with type III and V collagens and non-collagenous molecules into large cable-like fibrils, yet how the fibril interacts with cells and other binding partners remains poorly understood. To help reveal insights into the collagen structure-function relationship, a data base was assembled including hundreds of type I collagen ligand binding sites and mutations on a twodimensional model of the fibril. Visual examination of the distribution of functional sites, and statistical analysis of mutation distributions on the fibril suggest it is organized into two domains. The "cell interaction domain" is proposed to regulate dynamic aspects of collagen biology, including integrin-mediated cell interactions and fibril remodeling. The "matrix interaction domain" may assume a structural role, mediating collagen cross-linking, proteoglycan interactions, and tissue mineralization. Molecular modeling was used to superimpose the positions of functional sites and mutations from the two-dimensional fibril map onto a three-dimensional x-ray diffraction structure of the collagen microfibril in situ, indicating the existence of domains in the native fibril. Sequence searches revealed that major fibril domain elements are conserved in type I collagens through evolution and in the type II/XI collagen fibril predominant in cartilage. Moreover, the fibril domain model provides potential insights into the genotype-phenotype relationship for several classes of human connective tissue diseases, mechanisms of integrin clustering by fibrils, the polarity of fibril assembly, heterotypic fibril function, and connective tissue pathology in diabetes and aging.Type I collagen is the most abundant protein in humans and other vertebrates, comprising much of the fibrous extracellular matrix scaffold of bones, tendons, skin, and many other tissues (1-4). In general, type I collagen and its binding partners are proposed to provide mechanical strength and form to tissues. Collagenous scaffolds are laid down and remodeled by cells and are also a predominant substrate for cell interactions, migration, and differentiation. Consequently, various debilitating human diseases are associated with type I collagen mutations, including osteogenesis imperfecta (OI, 2 brittle bone disease), Ehlers Danlos syndrome, vascular disorders, and others (3, 5). Type I collagen is also employed in human medicine as hemostatic sponges and implants to repair wounds and in tissue engineering applications as scaffolds (6).Type I collagen is synthesized in the endoplasmic reticulum as ␣1 and ␣2 procollagen chains, each encoded by separate genes that are translated into proteins somewhat longer than 1000 amino acid residues (3, 7). Nucleation domains on the C-terminal propeptide promote the polymerization of two ␣1 and one ␣2 chains into the procollagen triple helical monomer (Fig. 1, A and B). The triple helical domain of procollagen is composed of contiguous glycine-X-Y tri-peptide repeats, with the obligate glyci...