Gaussian Processes

A Gaussian process is a distribution over functions; it gives the stack calibrated surrogates with built-in error bars, ideal for scarce fusion data.

A prior over functions

A Gaussian process (GP) places a prior directly on functions: any finite set of points is jointly Gaussian, with correlations set by a kernel. Conditioning on observations yields a posterior mean (the surrogate prediction) and posterior variance (its uncertainty) in closed form. GPs are the stack's default surrogate where data is scarce and honest error bars are essential.

GP: f ~ GP( m(x), k(x,x') )

Posterior at test points X* given data (X, y):
  mean = K*  (K + sigma^2 I)^-1 y
  cov  = K** - K* (K + sigma^2 I)^-1 K*'

  K   = k(X, X)   (train-train)
  K*  = k(X*, X)  (test-train)
  common kernel: k(x,x') = s^2 exp(-||x-x'||^2 / 2 l^2)  (RBF)

Kernels encode assumptions

The kernel is where prior knowledge lives: smoothness (RBF), roughness (Matern), periodicity, or length scales that differ per input. Hyperparameters - length scale, signal and noise variance - are learned by maximizing the marginal likelihood, which trades data fit against smoothness automatically, guarding against overfit on small datasets.

# GP marginal likelihood for hyperparameter fit (schematic)
def nll(params, X, y):
    K = kernel(X, X, params) + params.noise*I
    L = cholesky(K)
    a = cho_solve(L, y)
    return 0.5*y@a + sum(log(diag(L))) + 0.5*n*log(2*pi)
params = minimize(nll, init, args=(X, y))   # maximize evidence

Where GPs serve the machines

GPs surrogate expensive maps - a scalar stability margin versus operating parameters, a confinement metric versus profiles - with uncertainty that grows away from data. That growing variance is exactly what scenario optimization needs to avoid overconfident extrapolation, and it is central to the burner, where the GP's error bars balloon in the 166-830x regime rather than pretending to knowledge that does not exist.

• Closed-form posterior mean and variance: calibrated by construction.
• Kernel encodes smoothness/length scales as explicit priors.
• Hyperparameters fit by marginal likelihood: overfit-resistant.
• Variance grows off-data: honest extrapolation, key for the burner.

The cubic cost of the exact GP limits dataset size, so the stack uses sparse/inducing-point approximations for larger problems - trading a little fidelity for scale while keeping the calibrated uncertainty that motivates using a GP at all.