Marginal likelihood

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.^[1]

Concept

Given a set of

independent identically distributed

data points

\mathbf {X} =(x_{1},\ldots ,x_{n}),

where

x_{i}\sim p(x|\theta )

according to some probability distribution parameterized by

\theta

, where

\theta

itself is a random variable described by a distribution, i.e.

\theta \sim p(\theta \mid \alpha ),

the marginal likelihood in general asks what the probability

p(\mathbf {X} \mid \alpha )

is, where

\theta

has been marginalized out (integrated out):

p(\mathbf {X} \mid \alpha )=\int _{\theta }p(\mathbf {X} \mid \theta )\,p(\theta \mid \alpha )\ \operatorname {d} \!\theta

The above definition is phrased in the context of Bayesian statistics in which case $p(\theta \mid \alpha )$ is called prior density and $p(\mathbf {X} \mid \theta )$ is the likelihood. The marginal likelihood quantifies the agreement between data and prior in a geometric sense made precise^[

frequentist

) statistics, the concept of marginal likelihood occurs instead in the context of a joint parameter $\theta =(\psi ,\lambda )$ , where $\psi$ is the actual parameter of interest, and $\lambda$ is a non-interesting nuisance parameter. If there exists a probability distribution for $\lambda$ ^{[dubious – discuss]}, it is often desirable to consider the likelihood function only in terms of $\psi$ , by marginalizing out $\lambda$ :

${\mathcal {L}}(\psi ;\mathbf {X} )=p(\mathbf {X} \mid \psi )=\int _{\lambda }p(\mathbf {X} \mid \lambda ,\psi )\,p(\lambda \mid \psi )\ \operatorname {d} \!\lambda$

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
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.

Applications

Bayesian model comparison

In
Bayesian model comparison
, the marginalized variables $\theta$ are parameters for a particular type of model, and the remaining variable $M$ 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 $\theta$ for the model parameters, the marginal likelihood for the model M is

$p(\mathbf {X} \mid M)=\int p(\mathbf {X} \mid \theta ,M)\,p(\theta \mid M)\,\operatorname {d} \!\theta$

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, called the Bayes factor:

${\frac {p(M_{1}\mid \mathbf {X} )}{p(M_{2}\mid \mathbf {X} )}}={\frac {p(M_{1})}{p(M_{2})}}\,{\frac {p(\mathbf {X} \mid M_{1})}{p(\mathbf {X} \mid M_{2})}}$

which can be stated schematically as

posterior odds = prior odds × Bayes factor

See also

Empirical Bayes methods

Lindley's paradox

Marginal probability

Bayesian information criterion

This article includes a
improve this article by introducing more precise citations. (July 2010) (Learn how and when to remove this template message
)

References

doi:10.1007/3-540-28820-1_2
.

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. 111–117. 2002. (Available as a preprint on SSRN 332860
)

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: [1])

Lambert, Ben (2018). "The devil is in the denominator". A Student's Guide to Bayesian Statistics. Sage. pp. 109–120.
ISBN 978-1-4739-1636-4
.

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

Retrieved from "https://en.wikipedia.org/w/index.php?title=Marginal_likelihood&oldid=1212036231"

[1] :10.1007/3-540-28820-1_2
.

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