Bayesian Experimental Design

Objective Functions

Single-objective: Target Tracking

\[ \text{OFV}(\mathbf{x}) = -\sum_{k} w_k \left(\hat{x}_k - x_k^*\right)^2 \]

where \(\hat{x}_k\) is the model prediction and \(x_k^*\) is the target for species \(k\).

Multi-objective: Hypervolume

The multi-objective OFV is the Expected Hypervolume Improvement (EHVI):

\[ \text{qNEHVI}(\mathbf{x}) = \mathbb{E}\left[\text{HVI}\left(\mathbf{f}(\mathbf{x}),\;\mathcal{P},\;\mathbf{r}\right)\right] \]

where \(\mathcal{P}\) is the current Pareto front and \(\mathbf{r}\) is the reference point.

Acquisition Functions

Name	Type	Notes
PI	Single-obj	Probability of Improvement
EI	Single-obj	Expected Improvement
LogEI	Single-obj	Numerically stable log-EI (Ament et al. 2023)
UCB	Single-obj	Upper Confidence Bound, \(\beta\)-parameterised
qEI	Batch	Monte-Carlo EI
qLogEI	Batch	Log-space MC EI
qUCB	Batch	MC UCB
qEHVI	Multi-obj	Expected HV Improvement
qNEHVI	Multi-obj	Noisy EHVI
qNParEGO	Multi-obj	Pareto-EGO scalarisation

All acquisitions are optimised via botorch.optim.optimize_acqf with num_restarts=20, raw_samples=512.

Pareto Optimality

A point \(\mathbf{y}\) is Pareto-optimal if there is no \(\mathbf{y}'\) such that \(y'_i \geq y_i\) for all \(i\) and \(y'_j > y_j\) for at least one \(j\).

The hypervolume indicator quantifies the quality of the Pareto front:

\[ \text{HV}(\mathcal{P},\;\mathbf{r}) = \lambda\left(\bigcup_{\mathbf{y}\in\mathcal{P}} [\mathbf{r},\mathbf{y}]\right) \]

where \(\lambda\) denotes the Lebesgue measure.