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I'm working with a Bayesian Optimization problem where the accuracy of sampled data varies based on sampling cost (low-uncertainty data is expensive, high-uncertainty data is cheap).

What kinds of approaches are there for quantifying this trade-off, and making decisions about what is the best balance between sampling cost and data accuracy? And what is the name/term for this kind of analysis? I'm not familiar with this topic so I don't know what to search for.

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This idea of using multiple fidelities(low-uncertainty vs high-uncertainty) data within Bayesian optimization is an area of active research. There are multiple ways of incorporating this information. If you are aware of basic bayesian optimization terminology, it can be incorporated in following ways:

  1. To pick the next point for evaluation during acquisition function optimization.
  2. By modeling directly in the surrogate model(gaussian process).

Some related literature is below:

https://papers.nips.cc/paper/6118-gaussian-process-bandit-optimisation-with-multi-fidelity-evaluations.pdf https://arxiv.org/pdf/1807.01774.pdf

Both of them has source code available too.

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