Maximum likelihood or restricted maximum likelihood (REML) estimates of the parameters in linear mixed-effects models can be determined using the lmer function in the lme4 package for R. As for most model-fitting functions in R, the model is described in an lmer call by a formula, in this case including both fixed- and random-effects terms. The formula and data together determine a numerical representation of the model from which the profiled deviance or the profiled REML criterion can be evaluated as a function of some of the model parameters. The appropriate criterion is optimized, using one of the constrained optimization functions in R, to provide the parameter estimates. We describe the structure of the model, the steps in evaluating the profiled deviance or REML criterion, and the structure of classes or types that represents such a model. Sufficient detail is included to allow specialization of these structures by users who wish to write functions to fit specialized linear mixed models, such as models incorporating pedigrees or smoothing splines, that are not easily expressible in the formula language used by lmer.

/pdf/fitting-linear-mixed-effects-models-using-lme4-4zzrpoy9lh.pdf

Fitting Linear Mixed-Effects Models Using lme4

https://rss.onlinelibrary.wiley.com/doi/pdf/10.1111/j.2517-6161.1977.tb01600.x

Maximum likelihood from incomplete data via the EM algorithm

Deep learning is a form of machine learning that enables computers to learn from experience and understand the world in terms of a hierarchy of concepts. Because the computer gathers knowledge from experience, there is no need for a human computer operator to formally specify all the knowledge that the computer needs. The hierarchy of concepts allows the computer to learn complicated concepts by building them out of simpler ones; a graph of these hierarchies would be many layers deep. This book introduces a broad range of topics in deep learning. The text offers mathematical and conceptual background, covering relevant concepts in linear algebra, probability theory and information theory, numerical computation, and machine learning. It describes deep learning techniques used by practitioners in industry, including deep feedforward networks, regularization, optimization algorithms, convolutional networks, sequence modeling, and practical methodology; and it surveys such applications as natural language processing, speech recognition, computer vision, online recommendation systems, bioinformatics, and videogames. Finally, the book offers research perspectives, covering such theoretical topics as linear factor models, autoencoders, representation learning, structured probabilistic models, Monte Carlo methods, the partition function, approximate inference, and deep generative models. Deep Learning can be used by undergraduate or graduate students planning careers in either industry or research, and by software engineers who want to begin using deep learning in their products or platforms. A website offers supplementary material for both readers and instructors.

Deep Learning

Meta-Analysis in Clinical Trials*

We describe latent Dirichlet allocation (LDA), a generative probabilistic model for collections of discrete data such as text corpora. LDA is a three-level hierarchical Bayesian model, in which each item of a collection is modeled as a finite mixture over an underlying set of topics. Each topic is, in turn, modeled as an infinite mixture over an underlying set of topic probabilities. In the context of text modeling, the topic probabilities provide an explicit representation of a document. We present efficient approximate inference techniques based on variational methods and an EM algorithm for empirical Bayes parameter estimation. We report results in document modeling, text classification, and collaborative filtering, comparing to a mixture of unigrams model and the probabilistic LSI model.

/pdf/latent-dirichlet-allocation-4gmscb06en.pdf

Latent dirichlet allocation

A common reaction among applied statisticians is that the Bayesian statistician's energies in an applied problem must be directed at the a priori elicitation of one model specification from which an optimal design and all inferences follow automatically by applying Bayes's theorem to calculate conditional distributions of unknowns given knowns. I feel, however, that the applied Bayesian statistician's tool-kit should be more extensive and include tools that may be usefully labeled frequency calculations. Three types of Bayesianly justifiable and relevant frequency calculations are presented using examples to convey their use for the applied statistician.

https://projecteuclid.org/download/pdf_1/euclid.aos/1176346785

Bayesianly Justifiable and Relevant Frequency Calculations for the Applied Statistician

Multiple imputation for non-response replaces each missing value by two or more plausible values. The values can be chosen to represent both uncertainty about the reasons for non-response and uncertainty about which values to impute assuming the reasons for non-response are known. This paper provides an overview of methods for creating and analysing multiply-imputed data sets, and illustrates the dramatic improvements possible when using multiple rather than single imputation. A major application of multiple imputation to public-use files from the 1970 census is discussed, and several exploratory studies related to health care that have used multiple imputation are described.

Multiple imputation in health-care databases: an overview and some applications.

Most questions in social and biomedical sciences are causal in nature: what would happen to individuals, or to groups, if part of their environment were changed? In this groundbreaking text, two world-renowned experts present statistical methods for studying such questions. This book starts with the notion of potential outcomes, each corresponding to the outcome that would be realized if a subject were exposed to a particular treatment or regime. In this approach, causal effects are comparisons of such potential outcomes. The fundamental problem of causal inference is that we can only observe one of the potential outcomes for a particular subject. The authors discuss how randomized experiments allow us to assess causal effects and then turn to observational studies. They lay out the assumptions needed for causal inference and describe the leading analysis methods, including, matching, propensity-score methods, and instrumental variables. Many detailed applications are included, with special focus on practical aspects for the empirical researcher.

Causal Inference for Statistics, Social, and Biomedical Sciences: An Introduction

When assignment to treatment group is made solely on the basis of the value of a covariate, X, effort should be concentrated on estimating the conditional expectations of the dependent variable Y g...

Donald B. Rubin

Papers

Bayesianly Justifiable and Relevant Frequency Calculations for the Applied Statistician

Multiple imputation in health-care databases: an overview and some applications.

Causal Inference for Statistics, Social, and Biomedical Sciences: An Introduction

Assignment to Treatment Group on the Basis of a Covariate

Discussion of "Randomization Analysis of Experimental Data in the Fisher Randomization Test" by D. Basu