What key concept is pivotal in carrying out marginalization in Bayesian inference?

Marginalization in Bayesian inference is fundamentally about integrating over specific dimensions of a joint distribution to obtain the marginal probabilities of interest. This process is crucial when dealing with multivariate distributions, where you want to focus on a subset of variables while accounting for the uncertainty and interactions with the other variables.

The concept of integrating over multiple dimensions allows you to sum or integrate out certain variables to simplify the analysis and interpretation of the resulting distributions. For example, if you have a joint distribution of two variables but are only interested in one variable, marginalization enables you to compute the distribution of that variable alone by integrating out the other variable.

This capability is an essential part of Bayesian analysis, as it helps in making probabilistic statements about specific parameters or predictions while acknowledging the influence of other variables. Thus, the ability to integrate over multiple dimensions is a pivotal component of carrying out effective marginalization in Bayesian inference.