Posterior mean maximizer

As explained in Bayesian Optimization in Action

A posterior mean maximizer is the input that maximizes the predictive posterior mean of an objective function.

In multifidelity optimization, the posterior mean maximizer is the input with the highest predicted objective value under the current Gaussian-process belief. It answers: if optimization stopped now, which point should be recommended? section 9.4

Use the posterior mean maximizer to:

• Measure progress when the high-fidelity incumbent is unavailable or misleading. section 9.4
• Distinguish optimization scenarios by evaluating the objective at the current best posterior-mean location. section 9.4
• Implement the recommendation by optimizing a high-fidelity posterior-mean acquisition function. section 9.4

Figure 9.13 Measuring performance with the high-fidelity incumbent is inappropriate in multifidelity optimization. On the left, the high-fidelity incumbent is roughly 0.8, while we haven’t discovered the objective’s optimum. On the right, even though we are close to locating the objective’s optimum, there’s no high-fidelity query to record the incumbent value with.

Compare why high-fidelity incumbent values can misrepresent progress and why posterior-mean recommendations provide a more useful metric. section 9.4

How Bayesian Optimization in Action covers it

How deep it goes

Defines the metric, explains its intuition, and gives a BoTorch implementation using posterior-mean optimization.

Examples

Uses contrasting multifidelity scenarios, Figure 9.13, and BoTorch helper policies for posterior-mean optimization.

What it assumes

Assumes Bayesian optimization, Gaussian-process beliefs, acquisition optimization, and multifidelity models.

What you can do after

Apply it via BoTorch by optimizing PosteriorMean with FixedFeatureAcquisitionFunction at the high fidelity.

How it connects

Connects Gaussian processes, Bayesian optimization, acquisition functions, high-fidelity incumbents, and multifidelity optimization.

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This explainer was generated by Manning's knowledge graph from the text of Bayesian Optimization in Action. The section links point to the parts of the book it draws on.