What is marginal likelihood (evidence) in Bayes' theorem?

What is marginal likelihood (evidence) in Bayes' theorem?

• A. The probability of the observed data, p(data).
• B. The prior probability of the hypothesis.
• C. The posterior probability of the hypothesis given the data.
• D. The probability of the parameters given the data.

Answer: (A)

In Bayes' theorem, marginal likelihood (evidence) is the probability of the observed data under the model, found by averaging the data likelihood over the parameter prior. Formally, p(data) = ∫ p(data|θ) p(θ) dθ (or a sum if θ is discrete). This quantity acts as the normalizing constant in Bayes' rule, since p(θ|data) = p(data|θ) p(θ) / p(data). It tells you how plausible the data are when you account for all plausible parameter values weighted by their prior probabilities. It’s not the prior p(θ) (which is before seeing data), nor the posterior p(θ|data) (which updates beliefs after observing data), nor the probability of the parameters given the data (which is the posterior).

In Bayes' theorem, marginal likelihood (evidence) is the probability of the observed data under the model, found by averaging the data likelihood over the parameter prior. Formally, p(data) = ∫ p(data|θ) p(θ) dθ (or a sum if θ is discrete). This quantity acts as the normalizing constant in Bayes' rule, since p(θ|data) = p(data|θ) p(θ) / p(data). It tells you how plausible the data are when you account for all plausible parameter values weighted by their prior probabilities. It’s not the prior p(θ) (which is before seeing data), nor the posterior p(θ|data) (which updates beliefs after observing data), nor the probability of the parameters given the data (which is the posterior).