---
title: "On the Expectation-Maximization (EM) algorithm and regression models"
url: https://daily.dev/posts/on-the-expectation-maximization-em-algorithm-and-regression-models-s0tbinrws
source_url: https://www.r-bloggers.com/2026/08/on-the-expectation-maximization-em-algorithm-and-regression-models
type: article
source: "R-bloggers"
published: 2026-08-26T05:15:42.408Z
updated: 2026-08-26T05:16:05.927Z
tags: ["r", "statistics", "regression-analysis"]
reading_time: 7
upvotes: 0
comments: 0
language: en
---

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# On the Expectation-Maximization (EM) algorithm and regression models

**[R-bloggers](https://daily.dev/sources/rbloggers)** · 7 min read · 0 upvotes · 0 comments

## Summary

A mathematical walkthrough derives linear regression and Poisson regression as special cases of the Expectation-Maximization (EM) algorithm. For linear regression, the E-step collapses trivially and the M-step reproduces the closed-form OLS solution in a single iteration. For Poisson regression, a second-order Taylor expansion yields a weighted least-squares Q-function, and the M-step becomes the Iteratively Reweighted Least Squares (IRLS) update used in generalized linear models, requiring multiple iterations to converge since the closed-form solution doesn't exist. References to Casella and Berger, and McCullagh and Nelder are given for further reading.

## Full article

daily.dev links to this article rather than hosting it. Read it at the original source: <https://www.r-bloggers.com/2026/08/on-the-expectation-maximization-em-algorithm-and-regression-models>

## Questions this post answers

### Why does the EM algorithm converge in a single iteration for linear regression?

The M-step yields the global maximum of the Q-function in closed form, and that maximum does not depend on the current parameter estimate. Because ordinary least squares already has a closed-form solution β=(XᵀX)⁻¹Xᵀy, plugging in any starting value produces the same result after one M-step, since the Q-function's maximizer is independent of β(t).

_daily.dev surfaces deep dives like this for developers building statistical or ML models with regression._

### How does the EM algorithm derive the IRLS update used for Poisson regression?

A second-order Taylor expansion of the exponential mean function around the current linear predictor turns the Poisson log-likelihood into a weighted least-squares Q-function, with weights equal to the current fitted means μ(t) and working response ỹ(t)=η(t)+(yi−μ(t))/μ(t). Maximizing this gives β(t+1)=(XᵀW(t)X)⁻¹XᵀW(t)ỹ(t), the standard Iteratively Reweighted Least Squares update for GLMs.

_engineers implementing GLM solvers can find similar derivations curated on daily.dev._

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Tags: [#r](https://daily.dev/tags/r), [#statistics](https://daily.dev/tags/statistics), [#regression-analysis](https://daily.dev/tags/regression-analysis)

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