Solved – In the most basic sense, what is marginal likelihood

In Bayesian statistics, the marginal likelihood $$m(x) = int_Theta f(x|theta)pi(theta),text dtheta$$ where

1. $x$ is the sample
2. $f(x|theta)$ is the sampling density, which is proportional to the model likelihood
3. $pi(theta)$ is the prior density

is a misnomer in that

1. it is not a likelihood function [as a function of the parameter], since the parameter is integrated out (i.e., the likelihood function is averaged against the prior measure),
2. it is a density in the observations, the predictive density of the sample,
3. it is not defined up to a multiplicative constant,
4. it does not solely depend on sufficient statistics

Other names for $m(x)$ are evidence, prior predictive, partition function. It has however several important roles:

1. this is the normalising constant of the posterior distribution$$pi(theta|x) = dfrac{f(x|theta)pi(theta)}{m(x)}$$
2. in model comparison, this is the contribution of the data to the posterior probability of the associated model and the numerator or denominator in the Bayes factor.
3. it is a measure of goodness-of-fit (of a model to the data $x$), in that $2log m(x)$ is asymptotically the BIC (Bayesian information criterion) of Schwarz (1978).

See also

Normalizing constant in Bayes theorem

Normalizing constant irrelevant in Bayes theorem?

Intuition of Bayesian normalizing constant