How does Naïve Bayes relate to Bayesian decision theory and optimal classification?

Naïve Bayes implements Bayesian decision theory by choosing the class with maximum posterior probability.

Bayesian decision theory states that the optimal classifier minimizes expected risk by selecting the class with the highest posterior probability P(C|X). Naïve Bayes approximates this posterior using independence assumptions. Despite simplifications, it often approximates the Bayes optimal decision boundary well in high-dimensional sparse domains. The decision rule is argmax_C P(C) Π P(x_i|C).