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Form involving a probability density function

Suppose Ω is a measurable subset of the real line and f (x) is a non-negative function such that

In probabilistic language, f is a probability density function.

Then Jensen's inequality becomes the following statement about convex integrals:

If g is any real-valued measurable function and φ {\textstyle \varphi } is convex over the range of g, then

If g (x) = x, then this form of the inequality reduces to a commonly used special case:

This is applied in Variational Bayesian methods.