A marginal likelihood is a likelihood function that has been integrated over the parameter space. In Bayesian statistics, it represents the probability of generating the observed sample for all possible values of the parameters; it can be understood as the probability of the model itself and is therefore often referred to as model evidence or simply evidence.

Due to the integration over the parameter space, the marginal likelihood does not directly depend upon the parameters. If the focus is not on model comparison, the marginal likelihood is simply the normalizing constant that ensures that the posterior is a proper probability. It is related to the partition function in statistical mechanics.

Concept

Given a set of independent identically distributed according to some probability distribution itself is a random variable has been marginalized out (integrated out):

The above definition is phrased in the context of Bayesian statistics, one also has the alternative expression

. The marginal likelihood quantifies the agreement between data and prior in a geometric sense made precise via Hilbert spaces in de Carvalho et al. (2019). In classical (frequentist is a non-interesting nuisance parameter:

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/Metropolis sampling, or the EM algorithm.

, rather than a set of observations. In a Bayesian context, this is equivalent to the prior predictive distribution of a data point.

Applications

Bayesian model comparison

In Bayesian model comparison for the model parameters, the marginal likelihood for the model M is

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, called the Bayes factor:

which can be stated schematically as

1. posterior odds = prior odds × Bayes factor

See also

• Empirical Bayes methods

• Lindley's paradox

• Marginal probability

• Bayesian information criterion

References

Further reading

• Charles S. Bos. "A comparison of marginal likelihood computation methods". In W. Härdle and B. Ronz, editors, COMPSTAT 2002: Proceedings in Computational Statistics, pp.111117. 2002. (Available as a preprint on )

• de Carvalho, Miguel; Page, Garritt; Barney, Bradley (2019). "On the geometry of Bayesian inference". Bayesian Analysis. 14 (4): 1013‒1036. (Available as a preprint on the web: https://www.maths.ed.ac.uk/~mdecarv/papers/decarvalho2018.pdf)

• The on-line textbook: Information Theory, Inference, and Learning Algorithms, by David J.C. MacKay.