  Bayesian optimization offers a sample-efficient framework for navigating the
exploration-exploitation trade-off in the vast design space of biological
sequences. Whereas it is possible to optimize the various properties of
interest jointly using a multi-objective acquisition function, such as the
expected hypervolume improvement (EHVI), this approach does not account for
objectives with a hierarchical dependency structure. We consider a common use
case where some regions of the Pareto frontier are prioritized over others
according to a specified $\textit{partial ordering}$ in the objectives. For
instance, when designing antibodies, we would like to maximize the binding
affinity to a target antigen only if it can be expressed in live cell culture
-- modeling the experimental dependency in which affinity can only be measured
for antibodies that can be expressed and thus produced in viable quantities. In
general, we may want to confer a partial ordering to the properties such that
each property is optimized conditioned on its parent properties satisfying some
feasibility condition. To this end, we present PropertyDAG, a framework that
operates on top of the traditional multi-objective BO to impose this desired
ordering on the objectives, e.g. expression $\rightarrow$ affinity. We
demonstrate its performance over multiple simulated active learning iterations
on a penicillin production task, toy numerical problem, and a real-world
antibody design task.
