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2350 results in Statistical theory and methods

Page 45 of 118

Graphical Models for Categorical Data

Graphical Models for Categorical Data
  • Alberto Roverato
  • Published online:
    16 June 2017
    Print publication:
    24 August 2017
  • For advanced students of network data science, this compact account covers both well-established methodology and the theory of models recently introduced in the graphical model literature. It focuses on the discrete case where all variables involved are categorical and, in this context, it achieves a unified presentation of classical and recent results.
Topics at the Frontier of Statistics and Network Analysis

Topics at the Frontier of Statistics and Network Analysis
  • (Re)Visiting the Foundations
  • Eric D. Kolaczyk
  • Published online:
    06 June 2017
    Print publication:
    10 August 2017
  • This snapshot of the current frontier of statistics and network analysis focuses on the foundational topics of modeling, sampling, and design. Primarily for graduate students and researchers in statistics and closely related fields, emphasis is not only on what has been done, but on what remains to be done.

  • 13 - Objective Bayes Inference and MCMC

  • from Part II - Early Computer-Age Methods
  • Bradley Efron, Stanford University, California, Trevor Hastie, Stanford University, California
  • Book:
    Computer Age Statistical Inference
    Published online:
    05 July 2016
    Print publication:
    21 July 2016, pp 233-263

  • Summary

    From its very beginnings, Bayesian inference exerted a powerful influence on statistical thinking. The notion of a single coherent methodology employing only the rules of probability to go from assumption to conclusion was and is immensely attractive. For 200 years, however, two impediments stood between Bayesian theory's philosophical attraction and its practical application.

  • In the absence of relevant past experience, the choice of a prior distribution introduces an unwanted subjective element into scientific inference.

  • Bayes’ rule (3.5) looks simple enough, but carrying out the numerical calculation of a posterior distribution often involves intricate higherdimensional integrals.

    • The two impediments fit neatly into the dichotomy of Chapter 1, the first being inferential and the second algorithmic.

    A renewed cycle of Bayesian enthusiasm took hold in the 1960s, at first concerned mainly with coherent inference. Building on work by Bruno de Finetti and L. J. Savage, a principled theory of subjective probability was constructed: the Bayesian statistician, by the careful elicitation of prior knowledge, utility, and belief, arrives at the correct subjective prior distribution for the problem at hand. Subjective Bayesianism is particularly appropriate for individual decision making, say for the business executive trying to choose the best investment in the face of uncertain information.

    It is less appropriate for scientific inference, where the sometimes skeptical world of science puts a premium on objectivity. An answer came from the school of objective Bayes inference. Following the approach of Laplace and Jeffreys, as discussed in Section 3.2, their goal was to fashion objective, or “uninformative,” prior distributions that in some sense were unbiased in their effects upon the data analysis.

    In what came as a surprise to the Bayes community, the objective school has been the most successful in bringing Bayesian ideas to bear on scientific data analysis. Of the 24 articles in the December 2014 issue of the Annals of Applied Statistics, 8 employed Bayesian analysis, predominantly based on objective priors.

    This is where electronic computation enters the story. Commencing in the 1980s, dramatic steps forward were made in the numerical calculation of high-dimensional Bayes posterior distributions. Markov chain Monte Carlo (MCMC) is the generic name for modern posterior computation algorithms.

  • 16 - Sparse Modeling and the Lasso

  • from Part III - Twenty-First-Century Topics
  • Bradley Efron, Stanford University, California, Trevor Hastie, Stanford University, California
  • Book:
    Computer Age Statistical Inference
    Published online:
    05 July 2016
    Print publication:
    21 July 2016, pp 298-323

  • Summary

    The amount of data we are faced with keeps growing. From around the late 1990s we started to see wide data sets, where the number of variables far exceeds the number of observations. This was largely due to our increasing ability to measure a large amount of information automatically. In genomics, for example, we can use a high-throughput experiment to automatically measure the expression of tens of thousands of genes in a sample in a short amount of time. Similarly, sequencing equipment allows us to genotype millions of SNPs (single-nucleotide polymorphisms) cheaply and quickly. In document retrieval and modeling, we represent a document by the presence or count of each word in the dictionary. This easily leads to a feature vector with 20,000 components, one for each distinct vocabulary word, although most would be zero for a small document. If we move to bi-grams or higher, the feature space gets really large.

    In even more modest situations, we can be faced with hundreds of variables. If these variables are to be predictors in a regression or logistic regression model, we probably do not want to use them all. It is likely that a subset will do the job well, and including all the redundant variables will degrade our fit. Hence we are often interested in identifying a good subset of variables. Note also that in these wide-data situations, even linear models are over-parametrized, so some form of reduction or regularization is essential.

    In this chapter we will discuss some of the popular methods for model selection, starting with the time-tested and worthy forward-stepwise approach. We then look at the lasso, a popular modern method that does selection and shrinkage via convex optimization. The LARs algorithm ties these two approaches together, and leads to methods that can deliver paths of solutions.

    Finally, we discuss some connections with other modern big-and widedata approaches, and mention some extensions.

    Forward Stepwise Regression

    Stepwise procedures have been around for a very long time. They were originally devised in times when data sets were quite modest in size, in particular in terms of the number of variables. Originally thought of as the poor cousins of “best-subset” selection, they had the advantage of being much cheaper to compute (and in fact possible to compute for large p).We will review best-subset regression first.

  • 17 - Random Forests and Boosting

  • from Part III - Twenty-First-Century Topics
  • Bradley Efron, Stanford University, California, Trevor Hastie, Stanford University, California
  • Book:
    Computer Age Statistical Inference
    Published online:
    05 July 2016
    Print publication:
    21 July 2016, pp 324-350

  • Summary

    In the modern world we are often faced with enormous data sets, both in terms of the number of observations n and in terms of the number of variables p. This is of course good news—we have always said the more data we have, the better predictive models we can build. Well, we are there now—we have tons of data, and must figure out how to use it.

    Although we can scale up our software to fit the collection of linear and generalized linear models to these behemoths, they are often too modest and can fall way short in terms of predictive power. A need arose for some general purpose tools that could scale well to these bigger problems, and exploit the large amount of data by fitting a much richer class of functions, almost automatically. Random forests and boosting are two relatively recent innovations that fit the bill, and have become very popular as “out-thebox” learning algorithms that enjoy good predictive performance. Random forests are somewhat more automatic than boosting, but can also suffer a small performance hit as a consequence.

    These two methods have something in common: they both represent the fitted model by a sum of regression trees. We discuss trees in some detail in Chapter 8. A single regression tree is typically a rather weak prediction model; it is rather amazing that an ensemble of trees leads to the state of the art in black-box predictors!

    We can broadly describe both these methods very simply.

    Random forest Grow many deep regression trees to randomized versions of the training data, and average them. Here “randomized” is a wideranging term, and includes bootstrap sampling and/or subsampling of the observations, as well as subsampling of the variables.

    Boosting Repeatedly grow shallow trees to the residuals, and hence build up an additive model consisting of a sum of trees.

    The basic mechanism in random forests is variance reduction by averaging. Each deep tree has a high variance, and the averaging brings the variance down. In boosting the basic mechanism is bias reduction, although different flavors include some variance reduction as well. Both methods inherit all the good attributes of trees, most notable of which is variable selection.

Page 45 of 118