marginal-likelihood-integral-definition
IN premise — summaries/2026/08/24/wiki-Bayesian_inference-chunk-2.md
Created 2026-08-25T02:58:43+00:00
The marginal likelihood (evidence) is defined as p(X|α) = ∫ p(X|θ) p(θ|α) dθ, integrating out the parameter θ over the prior.
Summary
This defines how the system scores how well a model explains the observed data overall: instead of plugging in one specific parameter value, it averages the data's likelihood across every possible parameter value, weighted by how plausible each one is under the prior. That number is what lets the system compare competing models and update its confidence in the hyperparameters, since it captures the total predictive power of the model without needing to commit to a single setting.