transformed-optimizers-advanced

Skill: Transformed-Output GP Optimizers

Use this skill when measurements are guaranteed positive (y > 0) or bounded (y ∈ [0, 1]). A plain GP doesn't know about these constraints and will happily predict negative intensities or probabilities outside [0, 1]. The transformed optimizers fit the GP in an unconstrained "link" space (log or logit) and push the Gaussian posterior back through the inverse link via evaluate_posterior(x), so the original-scale predictions and credible intervals are guaranteed to respect the constraint.

When to use which

Observation type	Optimizer	Example domains
Strictly positive, y > 0	LogGPOptimizer	Intensities, rates, concentrations, fluxes, lifetimes
Bounded, y ∈ [a, b]	LogitGPOptimizer(..., range=(a, b))	Yields ([0, 100]%), probabilities, contrasts, transmittance, normalized intensities
Unconstrained / can be negative	plain GPOptimizer	Phase shifts, demeaned signals, temperatures in °C

Both transformed classes are drop-in replacements for GPOptimizer in the single-task scalar case — the constructor, train, ask, tell, optimize, kernel / mean / noise hooks, and pickling are inherited unchanged. The transform is invisible to the rest of the workflow.

Quick start

import numpy as np
from gpcam import LogGPOptimizer

# y_data must be > 0 — LogGPOptimizer raises ValueError otherwise
gpo = LogGPOptimizer(x_data, y_data)
gpo.train(hyperparameter_bounds=hp_bounds)

post = gpo.evaluate_posterior(x_grid)
# post["median"]                  -> exact (= exp of the latent GP mean)
# post["mean"]                    -> closed-form lognormal mean
# post["lower"], post["upper"]    -> exact 95% credible band; both > 0

from gpcam import LogitGPOptimizer

# y_data must be in [0, 1]; boundary values are clipped to [eps, 1-eps] with a warning
gpo = LogitGPOptimizer(x_data, y_data, eps=1e-6, n_samples=10000)
gpo.train(hyperparameter_bounds=hp_bounds)

post = gpo.evaluate_posterior(x_grid)
# Same dict shape; mean/std are Monte-Carlo (logistic-normal has no closed form).
# All entries lie strictly inside (0, 1).

For observations bounded in an arbitrary closed interval [a, b] (yield in [0, 100]%, transmittance in [0, 1], an angle in [0, 90], …), pass range=(a, b). The data is linearly rescaled to [0, 1] before the logit transform, posterior outputs are mapped back to [a, b], and predictions / credible bands stay strictly inside (a, b):

gpo = LogitGPOptimizer(x_data, y_data, range=(0.0, 100.0))   # yield in [0, 100]%
post = gpo.evaluate_posterior(x_grid)
# post["median"], post["lower"], post["upper"], post["samples"]  -- all in (0, 100).

evaluate_posterior — the original-scale accessor

The inherited posterior_mean(x) / posterior_covariance(x) operate in the transformed (latent) space. Use evaluate_posterior(x) whenever you want the posterior on the original scale:

post = gpo.evaluate_posterior(x, level=0.95)
# returns: {"median", "mean", "std", "lower", "upper", "level"}

Because both links are monotone increasing, the median and the credible interval transform exactly. The mean/std are exact closed forms for LogGPOptimizer (lognormal) and Monte-Carlo estimates for LogitGPOptimizer.

Raw posterior samples (for histograms or custom quantities)

Pass return_samples=True to also get an array of original-scale posterior draws of shape (n_points, n_samples):

post = gpo.evaluate_posterior(x_query, return_samples=True, n_samples=8000)
samples = post["samples"]            # shape (len(x_query), 8000)

# Histogram at the first query point:
import matplotlib.pyplot as plt
plt.hist(samples[0], bins=50, density=True)

Distributions are: Gaussian for plain GPOptimizer, lognormal for LogGPOptimizer, logistic-normal for LogitGPOptimizer. Use samples to compute expectations of arbitrary functions: np.mean(g(samples), axis=1).

LogitGPOptimizer knobs

• range (default (0.0, 1.0)): (lower, upper) bounds of the observation domain. Data is linearly rescaled to [0, 1] before the logit transform; predictions are mapped back to [lower, upper]. Raises ValueError if lower >= upper.
• eps (default 1e-6): clipping margin (in the rescaled [0, 1] space) so logit(0) / logit(1) don't blow up. Increase (e.g. 1e-4) for noisy boundary data at the cost of small bias; decrease for cleaner data.
• n_samples (default 10000): MC sample count for the closed-form-less mean / std (also the default for return_samples=True when no explicit n_samples is passed to evaluate_posterior). Higher is more accurate but slower.

Acquisition functions on transformed data

ask() and the inherited acquisition functions all run on the transformed GP. Both log and logit are monotone increasing, so ranking acquisitions ("variance", "ucb", "lcb", "maximum", "minimum") still pick the same locations as on the original scale.

Two cases that need care:

• "target probability": pass already-transformed bounds. For LogGPOptimizer, use args={"a": np.log(a), "b": np.log(b)}. For LogitGPOptimizer, use args={"a": scipy.special.logit(a), "b": scipy.special.logit(b)}.
• "expected improvement" / "probability of improvement": these compare against the best observed transformed value, which is also the best original value, so semantics carry over. The score magnitude differs from a plain GP fit but the ranking is consistent.

What get_data() returns

get_data()["y data"] is the transformed y stored on the GP. The transformed optimizers add a "original y data" key containing the inverse-transformed values for convenience:

data = gpo.get_data()
data["y data"]            # log(y) or logit(y)  -- in the GP's modeling space
data["original y data"]   # y  -- the observations as you provided them

Noise variances

When you provide noise_variances (observation noise) at construction or via tell(), they are transformed by the delta method (var_z ≈ (g'(y))² · var_y) before reaching the latent GP. Provide noise in the original scale; the optimizer handles the transform.

Pickling

LogGPOptimizer and LogitGPOptimizer pickle the same way as GPOptimizer. LogitGPOptimizer's extra attributes (eps, n_samples) survive the round-trip.

Example notebook

See examples/GPOptimizer_LogAndLogit.ipynb for a side-by-side visual tour — latent-space GP, back-transformed band, plain-GP comparison that violates the constraint, and posterior-sample histograms (lognormal vs. logistic-normal).