World Library  
Flag as Inappropriate
Email this Article

Marginal likelihood

Article Id: WHEBN0000979771
Reproduction Date:

Title: Marginal likelihood  
Author: World Heritage Encyclopedia
Language: English
Subject: Bayesian inference, Posterior predictive distribution, Empirical Bayes method, Free energy principle, Variational Bayesian methods
Collection: Bayesian Statistics, Probability Theory
Publisher: World Heritage Encyclopedia
Publication
Date:
 

Marginal likelihood

In statistics, a marginal likelihood function, or integrated likelihood, is a likelihood function in which some parameter variables have been marginalized. In the context of Bayesian statistics, it may also be referred to as the evidence or model evidence.

Given a set of independent identically distributed data points \mathbb{X}=(x_1,\ldots,x_n), where x_i \sim p(x_i|\theta) according to some probability distribution parameterized by θ, where θ itself is a random variable described by a distribution, i.e. \theta \sim p(\theta|\alpha), the marginal likelihood in general asks what the probability p(\mathbb{X}|\alpha) is, where θ has been marginalized out (integrated out):

p(\mathbb{X}|\alpha) = \int_\theta p(\mathbb{X}|\theta) \, p(\theta|\alpha)\ \operatorname{d}\!\theta

The above definition is phrased in the context of Bayesian statistics. In classical (frequentist) statistics, the concept of marginal likelihood occurs instead in the context of a joint parameter θ=(ψ,λ), where ψ is the actual parameter of interest, and λ is a non-interesting nuisance parameter. If there exists a probability distribution for λ, it is often desirable to consider the likelihood function only in terms of ψ, by marginalizing out λ:

\mathcal{L}(\psi;\mathbb{X}) = p(\mathbb{X}|\psi) = \int_\lambda p(\mathbb{X}|\psi,\lambda) \, p(\lambda|\psi) \ \operatorname{d}\!\lambda

Unfortunately, marginal likelihoods are generally difficult to compute. Exact solutions are known for a small class of distributions, particularly when the marginalized-out parameter is the conjugate prior of the distribution of the data. In other cases, some kind of numerical integration method is needed, either a general method such as Gaussian integration or a Monte Carlo method, or a method specialized to statistical problems such as the Laplace approximation, Gibbs sampling or the EM algorithm.

It is also possible to apply the above considerations to a single random variable (data point) x, rather than a set of observations. In a Bayesian context, this is equivalent to the prior predictive distribution of a data point.


Contents

  • Applications 1
    • Bayesian model comparison 1.1
  • See also 2
  • References 3

Applications

Bayesian model comparison

In Bayesian model comparison, the marginalized variables are parameters for a particular type of model, and the remaining variable is the identity of the model itself. In this case, the marginalized likelihood is the probability of the data given the model type, not assuming any particular model parameters. Writing θ for the model parameters, the marginal likelihood for the model M is

p(x|M) = \int p(x|\theta, M) \, p(\theta|M) \, \operatorname{d}\!\theta

It is in this context that the term model evidence is normally used. This quantity is important because the posterior odds ratio for a model M1 against another model M2 involves a ratio of marginal likelihoods, the so-called Bayes factor:

\frac{p(M_1|x)}{p(M_2|x)} = \frac{p(M_1)}{p(M_2)} \, \frac{p(x|M_1)}{p(x|M_2)}

which can be stated schematically as

posterior odds = prior odds × Bayes factor

See also

References

  • Charles S. Bos. "A comparison of marginal likelihood computation methods". In W. Härdle and B. Ronz, editors, COMPSTAT 2002: Proceedings in Computational Statistics, pp. 111–117. 2002. (Available as a preprint on the web: [1])
  • The on-line textbook: Information Theory, Inference, and Learning Algorithms, by David J.C. MacKay.
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
 
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
 
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.
 


Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.