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Concept
Marginal likelihood
Summary
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 from a prior and is therefore often referred to as model evidence or simply evidence.
Given a set of independent identically distributed data points where according to some probability distribution parameterized by , where itself is a random variable described by a distribution, i.e. the marginal likelihood in general asks what the probability is, where has been marginalized out (integrated out):
The above definition is phrased in the context of Bayesian statistics in which case is called prior density and is the likelihood. The marginal likelihood quantifies the agreement between data and prior in a geometric sense made precise in de Carvalho et al. (2019). In classical (frequentist) statistics, the concept of marginal likelihood occurs instead in the context of a joint parameter , where is the actual parameter of interest, and is a non-interesting nuisance parameter. If there exists a probability distribution for , it is often desirable to consider the likelihood function only in terms of , by marginalizing out :
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.
It is also possible to apply the above considerations to a single random variable (data point) , rather than a set of observations. In a Bayesian context, this is equivalent to the prior predictive distribution of a data point.
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.
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