Conditioning updates a probability distribution after fixing or observing one or more variables. In a joint multivariate normal distribution, observing one value produces a posterior belief about all related variables. Positive correlation can shift a variable’s mean and reduce its uncertainty; the observed variable becomes fixed. section 2.2.2
Conditioning on B = 4
Suppose A, B, and C represent uncertain prices. Conditioning their joint Gaussian distribution on B = 4 updates A because A and B are positively correlated, while B has no remaining uncertainty. Conditioning on C = 4 leaves other marginal distributions unchanged when C is unrelated to them. section 2.2.2
Bayes’ theorem gives the Gaussian update in closed form, producing exact posterior means and variances. Gaussian processes use the same idea to learn from observations, becoming more confident near observed data and more uncertain farther away. section 2.2.2 section 2.6
