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1 results

  • 10 - Bayesian Methods

  • James Burridge, University of Portsmouth, Nick Tosh, University of Galway
  • Book:
    Inference in Statistical Modelling and Machine Learning
    Print publication:
    30 June 2026, pp 153-191

  • Summary

    This chapter introduces key concepts and methods in Bayesian statistical modelling. The posterior predictive distribution captures both epistemic uncertainty in model parameters and aleatory uncertainty in future outcomes. A Bayesian p-value gives the probability that a statistic computed from data output by a given model will be more extreme than the value of the same statistic computed from observed data. Bayesian p-values close to 0 or 1 suggest the model may be inadequate. Markov chain Monte Carlo is a general-purpose tool for sampling from complex, unnormalised distributions. It produces dependent samples, so the effective sample size is usually smaller than the number of iterations. Informative priors are useful when data leave large uncertainties in parameter values. Empirical Bayes combines information across related datasets by estimating a distribution over parameters using frequentist methods. Hierarchical modelling provides a unified Bayesian framework for handling multiple related datasets, capturing group structure via a hierarchical graph.