Probability Distributions.- Linear Models for Regression.- Linear Models for Classification.- Neural Networks.- Kernel Methods.- Sparse Kernel Machines.- Graphical Models.- Mixture Models and EM.- Approximate Inference.- Sampling Methods.- Continuous Latent Variables.- Sequential Data.- Combining Models.

Pattern Recognition and Machine Learning

Here, we describe the third major release of RELION. CPU-based vector acceleration has been added in addition to GPU support, which provides flexibility in use of resources and avoids memory limitations. Reference-free autopicking with Laplacian-of-Gaussian filtering and execution of jobs from python allows non-interactive processing during acquisition, including 2D-classification, de novo model generation and 3D-classification. Per-particle refinement of CTF parameters and correction of estimated beam tilt provides higher resolution reconstructions when particles are at different heights in the ice, and/or coma-free alignment has not been optimal. Ewald sphere curvature correction improves resolution for large particles. We illustrate these developments with publicly available data sets: together with a Bayesian approach to beam-induced motion correction it leads to resolution improvements of 0.2-0.7 A compared to previous RELION versions.

/pdf/new-tools-for-automated-high-resolution-cryo-em-structure-d3sztfqg71.pdf

New tools for automated high-resolution cryo-EM structure determination in RELION-3.

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 article, 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 a member of that family which is close to the target density. 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...

Variational Inference: A Review for Statisticians

Diffraction (X-ray, neutron and electron) and electron cryo-microscopy are powerful methods to determine three-dimensional macromolecular structures, which are required to understand biological processes and to develop new therapeutics against diseases. The overall structure-solution workflow is similar for these techniques, but nuances exist because the properties of the reduced experimental data are different. Software tools for structure determination should therefore be tailored for each method. Phenix is a comprehensive software package for macromolecular structure determination that handles data from any of these techniques. Tasks performed with Phenix include data-quality assessment, map improvement, model building, the validation/rebuilding/refinement cycle and deposition. Each tool caters to the type of experimental data. The design of Phenix emphasizes the automation of procedures, where possible, to minimize repetitive and time-consuming manual tasks, while default parameters are chosen to encourage best practice. A graphical user interface provides access to many command-line features of Phenix and streamlines the transition between programs, project tracking and re-running of previous tasks.

/pdf/macromolecular-structure-determination-using-x-rays-neutrons-7udc75ny0s.pdf

Macromolecular structure determination using X-rays, neutrons and electrons: recent developments in Phenix

Probabilistic Programming allows for automatic Bayesian inference on user-defined probabilistic models. Recent advances in Markov chain Monte Carlo (MCMC) sampling allow inference on increasingly complex models. This class of MCMC, known as Hamiltonian Monte Carlo, requires gradient information which is often not readily available. PyMC3 is a new open source Probabilistic Programming framework written in Python that uses Theano to compute gradients via automatic dierentiation as well as compile probabilistic programs on-the-fly to C for increased speed. Contrary to other Probabilistic Programming languages, PyMC3 allows model specification directly in Python code. The lack of a domain specific language allows for great flexibility and direct interaction with the model. This paper is a tutorial-style introduction to this software package.

Probabilistic Programming in Python using PyMC3

We propose a definition of local resolution for three-dimensional electron cryo-microscopy (cryo-EM) density maps that uses local sinusoidal features. Our algorithm has no free parameters and is applicable to other imaging modalities, including tomography. By evaluating the local resolution of single-particle reconstructions and subtomogram averages for four example data sets, we report variable resolution across a 4- to 40-A range.

/pdf/quantifying-the-local-resolution-of-cryo-em-density-maps-194ktl42w0.pdf

Quantifying the local resolution of cryo-EM density maps.

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.

Probabilistic modeling is iterative. A scientist posits a simple model, fits it to her data, refines it according to her analysis, and repeats. However, fitting complex models to large data is a bottleneck in this process. Deriving algorithms for new models can be both mathematically and computationally challenging, which makes it difficult to efficiently cycle through the steps. To this end, we develop automatic differentiation variational inference (ADVI). Using our method, the scientist only provides a probabilistic model and a dataset, nothing else. ADVI automatically derives an efficient variational inference algorithm, freeing the scientist to refine and explore many models. ADVI supports a broad class of models--no conjugacy assumptions are required. We study ADVI across ten modern probabilistic models and apply it to a dataset with millions of observations. We deploy ADVI as part of Stan, a probabilistic programming system.

/pdf/automatic-differentiation-variational-inference-1ona33tq7g.pdf

Automatic differentiation variational inference

Probabilistic modeling is iterative. A scientist posits a simple model, fits it to her data, refines it according to her analysis, and repeats. However, fitting complex models to large data is a bottleneck in this process. Deriving algorithms for new models can be both mathematically and computationally challenging, which makes it difficult to efficiently cycle through the steps. To this end, we develop automatic differentiation variational inference (ADVI). Using our method, the scientist only provides a probabilistic model and a dataset, nothing else. ADVI automatically derives an efficient variational inference algorithm, freeing the scientist to refine and explore many models. ADVI supports a broad class of models-no conjugacy assumptions are required. We study ADVI across ten different models and apply it to a dataset with millions of observations. ADVI is integrated into Stan, a probabilistic programming system; it is available for immediate use.

Alp Kucukelbir

Papers

Variational Inference: A Review for Statisticians

Quantifying the local resolution of cryo-EM density maps.

Variational Inference: A Review for Statisticians

Automatic differentiation variational inference

Automatic Differentiation Variational Inference