# integration – Deterministic Approximation of Products of Integrals

Suppose that I am interested in approximating the product of two integrals. For simplicity, let $$P$$ be a probability density function, let $$f, g$$ be two $$P$$-integrable functions, define

begin{align} phi = int P(x) f(x) , dx \ gamma = int P(x) g(x) , dx, end{align}

and say that I am interested in approximating the product $$phi cdot gamma$$.

I am interested in how a numerical analyst would approach this problem. A naive solution to this would be to do e.g. Gaussian quadrature with $$P$$ as the base measure, use the GQ nodes to form approximations of $$phi, gamma$$, and then multiply the approximations together. Can / does one ever do fancier things than this in practice?

I ask because I come from a statistics / applied probability background, where one often places a premium on having unbiased estimates of integrals. As such, taking the product of two estimates which use the same set of function evaluations will generally induce a statistical bias, and so there are simple and popular techniques which allow for this bias to be removed. I am curious about whether similar approaches are considered in numerical analysis.

To compress the question further: from the point of view of numerical analysis, are there better ways of approximating a product of integrals than taking the product of approximations to each of the integrals?

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