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

The authors (Berk, Western, and Weiss) are to be congratulated for addressing a challenging problem and for providing a variety of comments relevant to it. At the outset, it is important to note that the problem of "apparent populations" is not special to social science. Many fields of science deal with unique datasets, which in no sense are imbedded in randomization-based surveys or experiments, nor which can be viewed as drawn from independent and identically distributed actual replications. Consider cosmology and the origins of the universe, studies of atmospheric pollution, and discussions of the evolution of life on earth (for a fascinating account of William James on precisely this issue of unique datasets, apparent populations, and the evolution of man, see Gould 1988, p. 17). "Could a hypothesized model, with allowance for both its stochastic components and the imperfect estimation of its unknown constants (parameters), have plausibly generated the observed dataset?"-This question is a primitive of science, more fundamental than mathematical statisticians' formalizations of it. Such questions can be legitimately asked of any dataset, whether unique or one of many that was chosen by the scientist according to some randomized design. Addressing such questions with mathematical rigor is more

http://hbanaszak.mjr.uw.edu.pl/TempTxt/Rubin_1995_BayesNeymanAndCalibration.pdf

Bayes, neyman, and calibration

/pdf/re-dealing-with-missing-outcome-data-in-randomized-trials-407cff9ahm.pdf

Re: “Dealing With Missing Outcome Data in Randomized Trials and Observational Studies”

SUMMARY. A statistical model developed from scientific theory may "fail to fit" the available data if the scientific theory is incorrect or if the sample size is too small. The former point is obvious but the latter is more subtle. In the latter case, the hypothesized model may fail to fit in the sense that it is viewed as unnecessarily complicated, and so the investigators settle upon a simpler model that ignores structure hypothesized by scientific theory. We describe a simulation-based approach for determining the sample size that would be required for distinguishing between the simpler model and the hypothesized model assuming the latter is correct. Data are simulated assuming the hypothesized model is correct and compared to posterior predictive replications of the data, which are drawn assuming the simpler model is correct. This is repeated for a number of sample sizes. The Bayesian approach offers two especially nice features for addressing a problem of this type: first, we can average over a variety of plausible values for the parameters of the hypothesized model rather than fixing a single alternative; second, the approach does not require that we restrict attention to a limited class of regular models (e.g., t-tests or linear models). The posterior predictive approach to sample size determination is illustrated using an application of finite mixture models to psychological data.

Sample size determination using posterior predictive distributions

BC Neyman, 1923) or the Bayesian perspective (Rubin, 1978). We fully adhere to this dictum, as our analysis accounts for the randomization at the physician level. We thank B&C for forcing clarification on this important point. If possible deviations from stability at the subunit level are of concern, we can define potential outcomes and principal strata at a coarser level (e.g. physicians), which, as B&C also indicate, would result in multilevel compliance, although then more assumptions would be needed to draw causal inferences

/pdf/rejoinder-to-discussions-on-clustered-encouragement-designs-55oogxdw03.pdf

Rejoinder to discussions on clustered encouragement designs with individual noncompliance: Bayesian inference with randomization, and application to advance directive forms

Donald B. Rubin

Papers

Bayes, neyman, and calibration

Re: “Dealing With Missing Outcome Data in Randomized Trials and Observational Studies”

Sample size determination using posterior predictive distributions

Rejoinder to discussions on clustered encouragement designs with individual noncompliance: Bayesian inference with randomization, and application to advance directive forms

Procedures with Nonignorable Nonresponse