What is a common application of marginalization in Bayesian inference?

In Bayesian inference, marginalization is a fundamental process used to compute the posterior distribution of model parameters given observed data. This is accomplished by integrating over the prior distribution of the parameters and the likelihood of the observed data. The posterior distribution, which encapsulates our updated beliefs about the parameters after seeing the data, is often obtained through marginalization.

Specifically, when calculating the posterior, we typically want to account for all possible values of the parameters by integrating them out. This allows us to focus on the variable of interest while considering the uncertainty or variability introduced by other parameters in the model. The result is a comprehensive view of the parameter's distribution influenced by both prior knowledge and the observed data.

In contrast, estimating the mean of a population, reducing bias in coefficient estimates, and assessing model fit through residual analysis involve different statistical methodologies that do not directly involve the marginalization technique inherent to Bayesian approaches. Each of these applications focuses on distinct aspects of statistical analysis rather than the Bayesian process of updating beliefs through marginalization.