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2327 results in Pattern Recognition and Machine Learning

Page 26 of 117

  • 50 - Least-Squares Problems

  • Ali H. Sayed, École Polytechnique Fédérale de Lausanne
  • Book:
    Inference and Learning from Data
    Published online:
    24 February 2023
    Print publication:
    22 December 2022, pp 2165-2220

  • Summary

    We studied in Chapters 29 and 30 the mean‐square error (MSE) criterion in some detail, and applied it to the problem of inferring an unknown (or hidden) variable x from the observation of another variable x when {x, y} are related by means of a linear regression model or a state‐space model.

  • 28 - Bayesian Inference

  • Ali H. Sayed, École Polytechnique Fédérale de Lausanne
  • Book:
    Inference and Learning from Data
    Published online:
    17 March 2023
    Print publication:
    22 December 2022, pp 1092-1120

  • Summary

    The mean-square-error (MSE) criterion (27.17) is one notable example of the Bayesian approach to statistical inference. In the Bayesian approach, both the unknown quantity, x, and the observation, y, are treated as random variables and an estimator x^ for x is sought by minimizing the expected value of some loss function denoted by Q(x,x^). In the previous chapter, we focused exclusively on the quadratic loss Q(x,x^)=(x−x^)2 for scalar x. In this chapter, we consider more general loss functions, which will lead to other types of inference solutions such as the mean-absolute error (MAE) and the maximum a-posteriori (MAP) estimators. We will also derive the famed Bayes classifier as a special case when the realizations for x are limited to the discrete values x∈{±1}.

Exponential Families in Theory and Practice
  • Bradley Efron
  • Published online:
    25 November 2022
    Print publication:
    15 December 2022
  • During the past half-century, exponential families have attained a position at the center of parametric statistical inference. Theoretical advances have been matched, and more than matched, in the world of applications, where logistic regression by itself has become the go-to methodology in medical statistics, computer-based prediction algorithms, and the social sciences. This book is based on a one-semester graduate course for first year Ph.D. and advanced master's students. After presenting the basic structure of univariate and multivariate exponential families, their application to generalized linear models including logistic and Poisson regression is described in detail, emphasizing geometrical ideas, computational practice, and the analogy with ordinary linear regression. Connections are made with a variety of current statistical methodologies: missing data, survival analysis and proportional hazards, false discovery rates, bootstrapping, and empirical Bayes analysis. The book connects exponential family theory with its applications in a way that doesn't require advanced mathematical preparation.

Page 26 of 117