In Bayesian statistics, what is the term for the probability of the observed data after integrating over parameters?

In Bayesian statistics, what is the term for the probability of the observed data after integrating over parameters?

• A. Prior probability.
• B. Posterior probability.
• C. Likelihood.
• D. Marginal likelihood.

Answer: (D)

In Bayesian thinking, the probability of the observed data after averaging over parameter uncertainty is the marginal likelihood, also called the evidence. It is computed by integrating the likelihood over the parameter prior: p(y) = ∫ p(y|θ) p(θ) dθ. This gives the probability of the data under the model, without fixing a specific θ. This differs from the likelihood, which is p(y|θ) as a function of θ for a given data set; the prior, p(θ), is the distribution over parameters before seeing the data; and the posterior, p(θ|y), combines the prior with the data to update beliefs about θ after observing y. The marginal likelihood is key for model comparison via Bayes factors, since it assesses how well a model explains the data overall, after integrating out θ.

In Bayesian thinking, the probability of the observed data after averaging over parameter uncertainty is the marginal likelihood, also called the evidence. It is computed by integrating the likelihood over the parameter prior: p(y) = ∫ p(y|θ) p(θ) dθ. This gives the probability of the data under the model, without fixing a specific θ.

This differs from the likelihood, which is p(y|θ) as a function of θ for a given data set; the prior, p(θ), is the distribution over parameters before seeing the data; and the posterior, p(θ|y), combines the prior with the data to update beliefs about θ after observing y. The marginal likelihood is key for model comparison via Bayes factors, since it assesses how well a model explains the data overall, after integrating out θ.