Unobserved variables. Models and misunderstandings (Q358116)

The unifying theme of this book is that information about unobserved variables in statistical problems is properly conveyed by their posterior distributions. The first two chapters lay the foundations, by defining the notations and setting out conventions and basic results. Specifically, Chapter 1 deals with the background of models, continuous and categorical variables, and describes the seven problems with which the book deals in separate chapters. Chapter 2, on measurement, estimation and prediction, stresses the importance of measurements in statistics. Chapter 3 introduces mixtures and provides an introduction to the key idea of latent variables. Basically, it treats two important questions: how to identify whether or not a given distribution could be a mixture and, if so, to estimate the components. Chapters 4-6 represent a progression from a very simple type of latent variable problems to the full generality of factor analysis and ramifications. Chapter 4 studies models for ability: the normal logistic, fixed effects (Rasch) and mixed effects models. Chapter 5 treats a general latent variable model and Chapter 6 treats the prediction of latent variables. Chapters 7 and 8 deal with topics which pervade many branches of statistics: identifiability and categorical variables, respectively. In Chapter 7 examples are given for factor models, the g factor model of Spearman and the Thomson bonds model applied to ability and intelligence testing. Linear structural equation models are also dealt with. Chapter 8 analyses the role of categorical variables and discusses unordered categories, random effects and ordered categories. Chapter 9 describes models for time series, mainly regression-type and autoregressive models. Problems of imputation and specification of missing data are discussed and the EM algorithm is briefly introduced. This is further discussed in Chapter 10, which treats missing data. This type of data is common in social applications and is probably the most obvious example of unobserved variables. Chapter 11 is about social measurements. Latent variables are often necessary and provide adequate models for social and physical situations. Examples are given on labor wastage and heritability. Chapter 12 treats the Bayesian approach and computational methods. In this approach, parameters are treated as random variables and this seems to put them on the same footing as other unobserved variables. MCMC methods are described, in particular the Gibbs sampler. Finally Chapter 13 summarizes the contents of the book, focusing on the key idea of posterior distributions of unobservable variables.