Marginal likelihood

In statistics, a marginal likelihood function, or integrated likelihood, is a likelihood function in which some parameter variables have been marginalised. It may also be referred to as evidence, but this usage is somewhat idiosyncratic.

Given a parameter θ=(ψ,λ), where ψ is the parameter of interest, it is often desirable to consider the likelihood function only in terms of ψ. If there exists a probability distribution for λ, sometimes referred to as the nuisance parameter, in terms of ψ, then it may be possible to marginalise or integrate out λ:

$\mathcal{L}(\psi;x) = p(x|\psi) = \int_\Lambda p(x|\psi,\lambda) p(\lambda|\psi) \ \operatorname{d} \lambda$

Unfortunately, marginal likelihoods are generally difficult to compute. Exact solutions are known for a small class of distributions. In general, 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.

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

$p(x|M) = \int p(x|\theta, M) \, p(\theta|M) \, d\theta$

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

• Empirical Bayes methods
• Marginal probability

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.