Let = (1,..., s) with i real-valued, X have density p(x), and a prior density

Let θ = (θ1,..., θs) with θi real-valued, X have density pθ(x), and

 a prior density π(θ). Then the 100γ% HPD region is the 100γ% credible region R that has minimum volume.

[Apply the Neyman–Pearson fundamental lemma to the problem of minimizing the volume of R.]