An interactive mathematics tool

Gaussian Process Explorer

A few observations. A family of possible functions. Explore how assumptions about smoothness and noise shape what we can infer.

Calculations run in this browser. Your points stay in this page’s memory and clear when you reload.

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Kernel & noise

Length scale 1.00

Larger values make nearby function values more strongly correlated.

Signal standard deviation 1.00

Sets the prior’s vertical scale before seeing observations.

Observation noise standard deviation 0.15

Larger values allow more difference between an observation and the underlying function.

Reset clears all points and restores the default parameters.

Posterior

5 / 40 observations

Click to add a point; click an existing point to remove it. The observation table offers the same removal controls.

Mean of f
95% pointwise interval for f
Observations

The shaded region is mean ± 1.96 standard deviations for the latent function f, at each x. It excludes future observation noise and is not a simultaneous band for the entire function.

Observation data & removal controls

Your current observations
Point	x	y	Action

Numerical posterior values

Latent function f at eleven locations; 95% pointwise intervals
x	Mean	Lower	Upper

The model behind the picture

This is a zero-mean Gaussian process with a squared-exponential (RBF) kernel:

k(x, x′) = σ_f² exp(−(x − x′)² / (2ℓ²))

Here ℓ is the length scale and σ_f is the signal standard deviation. Observations follow y = f(x) + ε, with independent Gaussian noise of standard deviation σ_n. The training covariance adds σ_n² to its diagonal. Parameters are chosen by you rather than fitted automatically.

Repeated inputs at zero noise may need a small numerical regularization, which is reported below the plot. For conflicting observations at the same x, a positive noise level is a more appropriate model.

The browser version is self-contained and accepts points directly. The original Python application source remains available separately.