The marginal likelihood p(x) = ∫ p(x|θ) π(θ) dθ (the model evidence) is the probability of the observed data x under a statistical model with its parameters θ integrated out against the prior π. Its parameters are the likelihood p(x|θ) with its parameter space, the prior π over θ, and the data x; the persistence mechanism is the integral itself - the expectation E_π[p(x|θ)] - which projects the joint p(x,θ) onto the data, collapsing a family of parameterized models into a single model-level number. It persists because it is the normalizing constant of Bayes's theorem p(θ|x) = p(x|θ)π(θ)/p(x), the numerator of the Bayes factor that compares competing models on the data alone, and the quantity whose decomposition log p(x) = ELBO + KL(q‖p) drives variational inference, and whose numerical difficulty is why Laplace approximation, importance sampling, and MCMC exist. [formal: verisimilitudo marginalis | substrate: mind | horizon: a life | explicit: yes | epoch: 0.70]
Accepted ontology entry
marginal-likelihood
The marginal likelihood p(x) = ∫ p(x|θ) π(θ) dθ (the model evidence) is the probability of the observed data x under a statistical model with its parameters θ integrated out against the prior π. Its parameters are the likelihood p(x|θ) wit…
Definition
Why it is in scope
The human-made probability of the observed data under a model with its parameters integrated out, built to persist as the normalizing constant of Bayes's theorem, the numerator of the Bayes factor, and the quantity whose decomposition drives variational inference.
Names and aliases
- marginal-likelihooden · CANONICAL
Relations from this entry
- likelihoodDERIVED_FROM →
Marginal likelihood p(x) = ∫ p(x|θ)π(θ)dθ is derived from the likelihood function p(x|θ) by integrating over the parameter space weighted by the prior. Likelihood was formalized by Fisher in the 1920s; marginal likelihood (model evidence) came later as the Bayesian extension. The likelihood is the base construct; marginal likelihood is its integrated variant.
- statistical-modelDEPENDS_ON →
Marginal likelihood p(x) = ∫ p(x|θ)π(θ)dθ requires a statistical model p(x|θ) with its parameter space as its core operand. Without the statistical model (the likelihood function and its parameterization), the marginal likelihood integral cannot even be formulated. The model is constitutive to marginal likelihood's definition and computation.
Relations to this entry
- cmsfu7dna00gjqszgtdoqq5pr← DEPENDS_ON
Posterior probability p(theta|x) = p(x|theta)pi(theta)/p(x) requires the marginal likelihood p(x) as its normalizing denominator. Without marginal likelihood, posterior probabilities cannot be computed at all. The removal of marginal likelihood means the posterior stops operating.
- evidence-lower-bound← SERVES
ELBO's designed purpose is to provide a tractable lower bound on log marginal likelihood (model evidence). It exists for the sake of computing model evidence when the integral is intractable. In variational inference, maximizing ELBO is how we approximate log p(x). Servant (ELBO) → master (marginal-likelihood). Law 8d.
- bayes-factor← DEPENDS_ON
Bayes factor IS the ratio of marginal likelihoods. Remove marginal likelihood — you cannot compute a Bayes factor. Present-tense operational necessity: every Bayes factor computation requires evaluating marginal likelihoods under each model.
- evidence-lower-bound← DERIVED_FROM
ELBO was derived as a tractable lower bound ON marginal likelihood. Marginal likelihood p(x) = integral p(x|theta)pi(theta)dtheta was formalized in Bayesian statistics in the 19th century; ELBO (Jordan et al. 1999) was introduced decades later as a computable approximation. The identity log p(x) = ELBO(q) + KL(q||p) shows ELBO = log p(x) - KL: ELBO is literally derived from marginal likelihood (minus KL). Marginal likelihood came first.
Record identity
- Created
- Sep 3, 2026, 3:04 PM UTC
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- bbfee8587cff49c6eb99c8d600fb55b0b5315470209ec4bdc64516170258d361