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1 results

  • 9 - Bayesian Linear Regression, Priors, and Regularisation

  • James Burridge, University of Portsmouth, Nick Tosh, University of Galway
  • Book:
    Inference in Statistical Modelling and Machine Learning
    Print publication:
    30 June 2026, pp 133-152

  • Summary

    This chapter introduces simple and multiple Bayesian linear regression models, in which parameters are treated as latent random variables. Thanks to their simplicity, these models yield closed-form posteriors. With flat priors, the posterior closely resembles the frequentist sampling distribution. We also explore the use of shrinkage priors to penalise model complexity and reduce overfitting. A Gaussian prior on the coefficients leads to ridge regression, where the MAP estimate corresponds to L2-regularised least squares. A Laplace prior yields lasso regression, based on L1 regularisation. Both are examples of regularisation techniques, but they behave differently: ridge regression shrinks all coefficients toward zero, while lasso tends to set some exactly to zero, producing a sparse model.