Weather forecasting was among the earliest fields to adopt Bayesian ideas at operational scale. The atmosphere is a chaotic system: small errors in initial conditions amplify exponentially, making deterministic prediction impossible beyond roughly two weeks. Bayesian methods address this fundamental limit by treating forecasts as probability distributions rather than single predictions, combining information from multiple models and observation networks to produce calibrated uncertainty estimates.
Bayesian Model Averaging for Weather
Numerical weather prediction (NWP) centers around the world run different models — the European ECMWF, the American GFS, the UK Met Office model — each with distinct physics parameterizations and initial conditions. Bayesian model averaging (BMA) combines these models into a single predictive distribution by weighting each model according to its recent forecast skill.
Where wₖ = posterior model weight for model k (Σ wₖ = 1)
fₖ = deterministic forecast from model k
gₖ = conditional PDF (often Gaussian or gamma, depending on variable)
θₖ = bias and spread parameters estimated from training data
BMA consistently produces better-calibrated probabilistic forecasts than any individual model. For temperature, the component distributions are typically Gaussian; for precipitation, a point mass at zero (for dry forecasts) is mixed with a gamma distribution for positive amounts. The weights and parameters are estimated from a rolling training window using the EM algorithm or full MCMC.
Ensemble Forecasting and Calibration
Ensemble prediction systems run a model multiple times with perturbed initial conditions and physics schemes, producing a spread of forecasts that samples uncertainty. Raw ensembles are often underdispersive — they are too confident — requiring Bayesian calibration. Bayesian ensemble model output statistics (EMOS) fits a predictive distribution whose mean and variance are functions of ensemble statistics, with parameters estimated in a Bayesian framework to avoid overfitting in small training samples.
A forecast system is calibrated if events assigned probability p occur with frequency p over time. Bayesian methods naturally target calibration because they produce genuine posterior predictive distributions. Reliability diagrams — plotting observed frequency against forecast probability — are the standard evaluation tool, and Bayesian post-processing consistently improves the calibration of raw model output.
Data Assimilation
Data assimilation — the process of combining model forecasts with observations to estimate the current atmospheric state — is fundamentally Bayesian. The model forecast provides a prior distribution over the state vector (temperature, pressure, humidity, wind at every grid point), observations provide a likelihood, and the analysis is the posterior. The ensemble Kalman filter (EnKF), widely used in operational weather centers, approximates this Bayesian update using the sample covariance from an ensemble of model runs.
Where x = atmospheric state vector (millions of dimensions)
π(x) = prior from model forecast
p(y | x) = observation likelihood
π(x | y) = analysis (posterior state estimate)
Modern data assimilation systems ingest millions of observations per six-hour cycle — satellite radiances, radiosondes, aircraft reports, surface stations, radar — and produce analyses over state vectors with billions of elements. The computational challenge of performing approximate Bayesian inference at this scale has driven major advances in ensemble methods, localization techniques, and hybrid variational-ensemble approaches.
"The purpose of weather forecasting is not to say what will happen, but to say what might happen and how likely each possibility is. The atmosphere demands honesty about uncertainty." — Tilmann Gneiting, pioneer of probabilistic forecasting
Severe Weather and Extremes
Bayesian methods are particularly valuable for severe weather prediction, where tail probabilities matter most. Bayesian extreme value analysis estimates the return levels of rare events — the 100-year rainfall, the record heat wave — while properly accounting for the short observational record. Bayesian networks model the conditional dependencies among atmospheric variables that lead to tornado outbreaks, derechos, and other compound hazards.