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Generalized Linear Models and the IRLS Algorithm: An Introduction to Exponential Family Distributions

This lecture explores generalized linear models (GLMs) and the Iterated Reweighted Least Squares (IRLS) algorithm, focusing on the exponential family of distributions. Key topics include moments and canonical parameter representation, the sufficiency of T(x) regarding parameters, and maximum likelihood estimation (MLE) using moment averages. The session covers the conditional mean in relation to GLMs, the application of canonical link functions that simplify MLE, and the practical implementation of online gradient descent. An illustrative example using logistic regression is included.

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Generalized Linear Models and the IRLS Algorithm: An Introduction to Exponential Family Distributions

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  1. Lecture 8 Generalized Linear Models & Iterated Reweighted Least Squares (IRLS) Algorithm

  2. Exponential Family • Moments & canonical parameters representation for EFD. • Sufficiency: T(x) is all there is to know about parameters. • ML estimation: moments are simply average SS. • Generalized Linear Models for discriminative Supervised L. - p(Y|X) = expFamDistr. - conditional mean = f(z) f = link func. or response func. - z = a’*x (linear) • Canonical link function gives simple MLE problem (linear in x) • Online gradient descent algorithm.

  3. IRLS • Do Newton-Ralphson iterations. • Updates become like solving a weighted least squares problem, with weights changing at each iteration. • example: logistic regression • demo_LogReg

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