Bayesian Ordinary Kriging
Also known as: Bayesian kriging, BOK, geostatistical Bayesian interpolation, Bayesian spatial prediction
Bayesian Ordinary Kriging is a geostatistical interpolation method that combines classical ordinary kriging with a Bayesian framework to jointly estimate the spatial covariance parameters and produce predictions at unsampled locations. Unlike plug-in kriging, it propagates uncertainty about variogram parameters through to the predictive distribution, yielding more honest uncertainty quantification.
Read the full method
Sign in with a free account to read this section.
Method map
The neighbourhood of related methods — select a node to explore.
When to use it
Use Bayesian Ordinary Kriging when you have georeferenced continuous measurements at point locations and need spatial predictions at unsampled sites with reliable uncertainty bounds. It is especially valuable when the sample is small or moderate (fewer than a few hundred points) and variogram parameter estimates would be highly uncertain under classical fitting. It is also appropriate when a prior on spatial structure is available from earlier studies. Avoid it when the dataset is very large (thousands of points) and computational cost from MCMC becomes prohibitive, or when the stationarity assumption (constant mean, stationary covariance) is clearly violated — in that case consider Bayesian Universal Kriging or non-stationary alternatives. Do not use it as a drop-in replacement for ordinary kriging without checking whether the stationarity assumption is reasonable.
Strengths & limitations
- Propagates variogram parameter uncertainty through to predictions, producing better-calibrated prediction intervals than plug-in ordinary kriging.
- Allows incorporation of prior knowledge about spatial range or variance from earlier surveys or domain expertise.
- Provides a full predictive distribution at each location, enabling probabilistic exceedance mapping (e.g., probability that pollution exceeds a threshold).
- Coherent Bayesian framework makes it straightforward to compare competing spatial covariance models via marginal likelihood (Bayes factors).
- Performs well with sparse data where classical variogram fitting is unreliable.
- Computationally intensive: MCMC sampling scales poorly with large datasets due to repeated matrix inversions of the n-by-n covariance matrix.
- Sensitivity to prior choice can matter substantially with small samples; poorly specified priors on the range parameter can mislead inference.
- Assumes second-order stationarity (constant mean and covariance depending only on distance), which may be violated in heterogeneous landscapes.
- More complex to implement and communicate than classical kriging; requires expertise in Bayesian computation.
Frequently asked
How does Bayesian Ordinary Kriging differ from ordinary kriging?
Classic ordinary kriging estimates variogram parameters (range, sill, nugget) by fitting the experimental variogram and then treats them as fixed and known. Bayesian Ordinary Kriging places probability distributions on these parameters and integrates over them during prediction. This produces wider, more honest prediction intervals, especially when the sample is small.
What priors should I use for the variogram parameters?
A common choice is a log-normal or half-Cauchy prior on the range phi and a half-normal or inverse-gamma prior on sigma-squared. When informative priors are available from earlier surveys of the same region, use them — they can substantially sharpen inference with sparse data. Always perform a sensitivity analysis by trying two or three alternative prior specifications.
When should I choose Bayesian Universal Kriging instead?
If the spatial mean is not constant — for example, elevation increases systematically from coast to mountains — add covariates and use Bayesian Universal Kriging (also called model-based geostatistics with a linear mean function). Ordinary Kriging assumes only a single unknown constant mean across the domain.
How do I handle the computational cost with large datasets?
Standard MCMC with full covariance matrices is O(n^3) per iteration and becomes impractical beyond a few thousand observations. Approximate methods such as nearest-neighbour Gaussian processes (NNGP), fixed-rank kriging, or SPDE/INLA approaches can scale Bayesian spatial models to much larger datasets while retaining most of the inferential benefits.
How do I validate the predictions?
Use leave-one-out or k-fold cross-validation: withhold each observation in turn, predict it from the rest, and compare the predictive interval coverage to the nominal level (e.g., 90% of observations should fall inside the 90% predictive interval). Poor coverage indicates mis-specified covariance structure or priors.
Sources
- Diggle, P. J., & Ribeiro, P. J. (2007). Model-Based Geostatistics. Springer. ISBN: 978-0387329079
- Handcock, M. S., & Stein, M. L. (1993). A Bayesian analysis of kriging. Technometrics, 35(4), 403-410. DOI: 10.1080/00401706.1993.10485354 ↗
How to cite this page
ScholarGate. (2026, June 3). Bayesian Ordinary Kriging. ScholarGate. https://scholargate.app/en/spatial-analysis/bayesian-ordinary-kriging
Which method?
Set this method beside its closest kin and read them side by side — the library lays the books on the table; the choice is yours.
- Bayesian Co-KrigingSpatial analysis↔ compare
- Bayesian KrigingSpatial analysis↔ compare
- Bayesian Universal KrigingSpatial analysis↔ compare
- Ordinary KrigingSpatial analysis↔ compare
- Spatial AutocorrelationSpatial analysis↔ compare