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 ${\displaystyle \mathbb {X} =(x_{1},\ldots ,x_{n}),}$ where ${\displaystyle x_{i}\sim p(x_{i}|\theta )}$ according to some probability distribution parameterized by θ, where θ itself is a random variable described by a distribution, i.e. ${\displaystyle \theta \sim p(\theta |\alpha ),}$ the marginal likelihood in general asks what the probability ${\displaystyle p(\mathbb {X} |\alpha )}$ is, where θ has been marginalized out (integrated out):

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