  Hypervolume improvement (HVI) is commonly employed in multi-objective
Bayesian optimization algorithms to define acquisition functions due to its
Pareto-compliant property. Rather than focusing on specific statistical moments
of HVI, this work aims to provide the exact expression of HVI's probability
distribution for bi-objective problems. Considering a bi-variate Gaussian
random variable resulting from Gaussian process (GP) modeling, we derive the
probability distribution of its hypervolume improvement via a cell
partition-based method. Our exact expression is superior in numerical accuracy
and computation efficiency compared to the Monte Carlo approximation of HVI's
distribution. Utilizing this distribution, we propose a novel acquisition
function - $\varepsilon$-probability of hypervolume improvement
($\varepsilon$-PoHVI). Experimentally, we show that on many widely-applied
bi-objective test problems, $\varepsilon$-PoHVI significantly outperforms other
related acquisition functions, e.g., $\varepsilon$-PoI, and expected
hypervolume improvement, when the GP model exhibits a large the prediction
uncertainty.
