Expectation Propagation for Likelihoods Depending on an Inner Product of Two Multivariate Random Variables

We will also use Fubini’s theorem [2, p. 170–171] and Lebesgue’s dominated convergence theorem [2, p. 91]. Fubini’s theorem allows us to change the order of integration in a double integral of a function, when we know that the function is integrable in one of the orders (i.e., the integral of the absolute value of the function is finite in one of the orders). Lebesgue’s dominated convergence theorem states that if |fn(x)| ≤ g(x), where g(x) is integrable, and limn→∞ fn = f pointwise, then limn→∞ ∫ fn(x)dx = ∫ f(x)dx.