# Proof: Marginal likelihood is a definite integral of joint likelihood

**Index:**The Book of Statistical Proofs ▷ General Theorems ▷ Bayesian statistics ▷ Probabilistic modeling ▷ Marginal likelihood is integral of joint likelihood

**Theorem:** In a full probability model $m$ describing measured data $y$ using model parameters $\theta$, the marginal likelihood is the integral of the joint likelihood across the parameter space $\Theta$

and related to likelihood function and prior distribution as follows:

\[\label{eq:ml-lf} p(y|m) = \int_{\Theta} p(y|\theta,m) \, p(\theta|m) \, \mathrm{d}\theta \; .\]**Proof:** In a full probability model, the marginal likelihood is defined as the marginal probability of the data $y$, given only the model $m$:

Using the law of marginal probabililty, this can be obtained by integrating the joint likelihood function over the entire parameter space:

\[\label{eq:ml-jl-qed} p(y|m) = \int_{\Theta} p(y,\theta|m) \, \mathrm{d}\theta \; .\]Applying the law of conditional probability, the integrand can also be written as the product of likelihood function and prior density:

\[\label{eq:ml-lf-qed} p(y|m) = \int_{\Theta} p(y|\theta,m) \, p(\theta|m) \, \mathrm{d}\theta \; .\]**∎**

**Sources:**

**Metadata:**ID: P91 | shortcut: ml-jl | author: JoramSoch | date: 2020-05-05, 04:59.