Gaussian Processes in Astronomy

Why This Matters

Astronomical data are short, irregularly sampled, and contaminated by physical nuisance signals: stellar rotation, granulation, instrumental drift, atmospheric turbulence. Most physical signals of interest (planetary transits at $\sim 100$ ppm, radial-velocity wobbles of $\sim 1$ m/s) sit below the amplitude of these correlated nuisances. Treating noise as i.i.d. Gaussian biases parameter estimates and inflates false-positive rates.

Gaussian processes give a principled framework for jointly modeling a physical signal and a stationary stochastic noise process. The kernel encodes prior beliefs about the timescale and smoothness of the nuisance; the posterior returns parameter estimates with calibrated uncertainty. This matters for the yes/no decisions that drive follow-up time on Keck or JWST.

The cubic cost $O(N^3)$ of standard GPs forced the field to develop scalable inference for one-dimensional time series. The celerite algorithm reduces inference to $O(N)$ for a class of mixture-of-exponential kernels, which made GPs the default tool for Kepler, K2, TESS, and ground-based RV pipelines.

Core Ideas

Stellar variability and transit detection. Kepler photometry shows correlated brightness variations from rotation-modulated starspots, p-mode oscillations, and granulation. A quasi-periodic kernel $k(\tau) = A \exp(-\tau^2 / 2\ell^2) \cos(2\pi \tau / P)$ models rotation while leaving room for a transit dip parameterized by a Mandel-Agol model. Marginal likelihood discriminates planet vs. stellar artifact. The celerite kernel (Foreman-Mackey et al. 2017, AJ 154) writes the covariance as a sum of damped harmonic oscillators, yielding a semiseparable matrix with linear-time Cholesky.

Exoplanet radial velocities. Stellar surface activity injects RV signals at the rotation period and its harmonics, often comparable to or larger than the planetary signal. The standard practice is a joint model: Keplerian orbits for the planets, a GP with quasi-periodic kernel for activity, with hyperparameters sampled by HMC or nested sampling. This was central to confirmations of Proxima b and several TESS systems.

Cosmological field reconstruction. GPs serve as nonparametric priors for fields where the underlying function is smooth but otherwise unknown: reconstructing the Hubble parameter $H(z)$ from supernova distance moduli, mapping the dark-energy equation of state $w(z)$, or inferring weak-lensing convergence maps. The kernel choice (squared-exponential, Matern $\nu = 3/2$, $\nu = 5/2$) controls smoothness assumptions and propagates into the posterior on cosmological parameters. The model-selection question (which kernel) is itself addressed by the marginal likelihood.

Scalable kernels beyond celerite. For two-dimensional fields and higher-dimensional inputs, structured kernel interpolation (KISS-GP) and inducing-point methods (SVGP) extend GPs to $N \sim 10^5$ or larger. These power applications in galaxy survey systematics modeling and 21-cm signal extraction.

Common Confusions

Watch Out

GP residuals are not white

Adding a GP noise model does not whiten the residuals after subtracting the posterior mean. The posterior mean already absorbs the correlated component, so residuals against the full posterior predictive should look white. Plotting residuals against the planet-only model and complaining about correlations is a category error.

References

Related Topics

• Gaussian Processes for ML
• Gaussian Processes Regression
• Bayesian State Estimation
• Time Series Forecasting Basics
• Kalman Filter