How deeply does Experimentation for Engineers cover lower confidence bound?

Defines the formula, explains k and uncertainty, demonstrates calculation, and embeds LCB in a repeated optimization loop.

Lower confidence bound

As explained in Experimentation for Engineers

A lower confidence bound is an acquisition function that combines a predicted response with a multiple of predictive uncertainty to favor promising or informative evaluations.

A lower confidence bound (LCB) acquisition function scores a candidate as expected response minus a multiple of predictive uncertainty: LCB = expectation − k × uncertainty. Minimizing it balances exploitation—seeking low predicted CPU time—with exploration—sampling uncertain regions. section 6.3.1

Selecting the next parameter

Given parameters [0.5, 0.0] with measurements [1.52, 1.21], fit GPR4, estimate predictions and uncertainties across x̂ values from 0 to 1, and compute lcb = ŷ − 1 × σŷ. The minimum occurs at x̂ = 0.11111111, which becomes the next measurement candidate. section 6.3.1

Use LCB inside an iterative optimization workflow:

Build a surrogate from existing parameter–measurement pairs. section 6.4.2
Minimize LCB, measure the proposed parameter, and add the result to the data. section 6.4.2
Increase k to emphasize uncertainty reduction; decrease it to emphasize predicted performance. No straightforward universal choice exists, so k = 1 is a practical starting point when uncertainty shares the measurement scale. section 6.3.1
Stop when measured CPU time stops changing, using a plot of CPU time against iteration as a check. section 6.4.2

parameters = [0.5, 0.0]
measurements = [1.52, 1.21]
gpr4 = GPR4(parameters, measurements, sigma=.15)
x_hats = np.linspace(0,1,100)
y_hats, sigma_y_hats = zip(*[gpr4.estimate(x_hat) for x_hat in x_hats])
k = 1
lcb = np.array(y_hats) - k*np.array(sigma_y_hats)

Inspect the compact NumPy calculation that forms LCB from surrogate predictions and predictive uncertainties. section 6.3.1

How Experimentation for Engineers covers it

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Related concepts

acquisition functionkind of · An acquisition function is a scoring rule that ranks candidate evaluations by their expected usefulness for selecting the next sample or experiment.
random searchused by · Random search is an optimization technique that explores a search space by evaluating candidate solutions or configurations selected through randomized choices.
surrogate modeluses · A surrogate model is an approximate model that estimates the behavior or outputs of a more complex, costly, or otherwise unavailable model, process, or function.
confidence bounduses · A statistical limit that bounds an estimated quantity with a stated level of confidence or coverage.
upper confidence boundcontrasts with · Upper confidence bound is a decision rule that combines an estimated value with an uncertainty bonus to select actions or candidates optimistically.

This explainer was generated by Manning's knowledge graph from the text of Experimentation for Engineers. The section links point to the parts of the book it draws on.