Marginal likelihood

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 data points ${\displaystyle \mathbf{X}=(x_1,\dots ,x_n),}$ where ${\displaystyle x_i\sim p(x|\theta )}$ according to some probability distribution parameterized by ${\displaystyle \theta }$, where ${\displaystyle \theta }$ itself is a random variable described by a distribution, i.e. ${\displaystyle \theta \sim p(\theta \mid \alpha ),}$ the marginal likelihood in general asks what the probability ${\displaystyle p(\mathbf{X}\mid \alpha )}$ is, where ${\displaystyle \theta }$ has been marginalized out (integrated out):

${\displaystyle p(\mathbf{X}\mid \alpha )={\int }_{\theta }p(\mathbf{X}\mid \theta )p(\theta \mid \alpha )\mathrm{d}\theta }$

The above definition is phrased in the context of Bayesian statistics in which case ${\displaystyle p(\theta \mid \alpha )}$ is called prior density and ${\displaystyle p(\mathbf{X}\mid \theta )}$ is the likelihood. Recognizing that the marginal likelihood is the normalizing constant of the Bayesian posterior density ${\displaystyle p(\theta \mid \mathbf{X},\alpha )}$, one also has the alternative expression

${\displaystyle p(\mathbf{X}\mid \alpha )=\frac{p(\mathbf{X}\mid \theta ,\alpha )p(\theta \mid \alpha )}{p(\theta \mid \mathbf{X},\alpha )}}$

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

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

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 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) ${\displaystyle x}$, rather than a set of observations. In a Bayesian context, this is equivalent to the prior predictive distribution of a data point.