Bayesian Co-Kriging
Also known as: Bayesian cokriging, Bayesian co-regionalization, BCK, Bayesian multivariate kriging
Bayesian Co-Kriging is a multivariate geostatistical method that uses auxiliary spatially correlated variables to improve predictions of a primary variable of interest. By placing Bayesian priors on cross-covariance parameters, it propagates all uncertainty — including parameter uncertainty — into the prediction intervals, yielding fully probabilistic maps with calibrated uncertainty bounds.
Key highlights
- Fully propagates parameter uncertainty into prediction intervals, yielding better-calibrated credible maps than plug-in Co-Kriging.
- Leverages auxiliary variables to reduce prediction error where the primary variable is under-sampled.
- Produces a complete predictive distribution at every unobserved location, enabling direct probabilistic statements (e.g., probability of exceeding a threshold).
- Flexible prior specification allows incorporation of expert knowledge about spatial range or nugget effects.
- Handles missing data in either variable naturally within the Bayesian hierarchical framework.
Intuition
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How it works
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When to use it
Use Bayesian Co-Kriging when you have a primary variable with sparse observations and one or more spatially correlated auxiliary variables with denser sampling; when credible intervals (not just kriging variances) are required for decision-making; and when sample sizes are small enough that parameter uncertainty is non-negligible. It is particularly valuable in environmental monitoring, precision agriculture, and public-health mapping where honest uncertainty quantification affects downstream decisions. Avoid it when auxiliary variables are weakly correlated with the primary (gain is minimal but computational cost is high), when the sample is very large (classical Co-Kriging or INLA-based approaches scale better), or when rapid throughput is needed and Bayesian MCMC is computationally prohibitive.
Strengths & limitations
- Fully propagates parameter uncertainty into prediction intervals, yielding better-calibrated credible maps than plug-in Co-Kriging.
- Leverages auxiliary variables to reduce prediction error where the primary variable is under-sampled.
- Produces a complete predictive distribution at every unobserved location, enabling direct probabilistic statements (e.g., probability of exceeding a threshold).
- Flexible prior specification allows incorporation of expert knowledge about spatial range or nugget effects.
- Handles missing data in either variable naturally within the Bayesian hierarchical framework.
- Computationally intensive: MCMC sampling over multivariate covariance parameters scales poorly to very large datasets (n > 5 000).
- Requires specification of priors for cross-covariance parameters; results can be sensitive to prior choice when data are scarce.
- Cross-covariance structure must be modeled explicitly; misspecification of the co-regionalization model can bias predictions.
- Software implementations are less standardized than classical Co-Kriging, requiring familiarity with Bayesian spatial packages (e.g., spBayes, Stan).
Common pitfalls
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Applications
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Frequently asked
How is Bayesian Co-Kriging different from classical Co-Kriging?
Classical Co-Kriging estimates covariance parameters by maximum likelihood or weighted least squares and then treats them as fixed when predicting. Bayesian Co-Kriging treats those parameters as random variables, samples their posterior via MCMC, and integrates predictions over all plausible parameter values — producing wider but better-calibrated uncertainty intervals.
How many auxiliary variables can be included?
In principle any number, but each additional variable adds a cross-covariance parameter block to the model. In practice two to four auxiliary variables are common; beyond that the model becomes difficult to identify and MCMC convergence slows considerably.
Can Bayesian Co-Kriging handle non-Gaussian primary variables?
Yes, through hierarchical extensions: a latent Gaussian spatial process can be linked to a non-Gaussian observation model (e.g., Poisson for counts, Bernoulli for presence-absence) using a generalized linear model link function within the Bayesian framework.
What software can I use?
The R package spBayes provides dedicated functions for Bayesian Co-Kriging. Stan and NIMBLE can implement custom cross-covariance models. INLA-based approaches via R-INLA offer a fast approximation suitable for larger datasets.
When is classical Co-Kriging sufficient instead of the Bayesian version?
When the dataset is large (hundreds to thousands of observations per variable), parameter uncertainty is relatively small, and computational speed matters. In those settings classical Co-Kriging with resampling-based uncertainty estimates is often adequate.
Sources
- 1.Diggle, P. J., & Ribeiro, P. J. (2007). Model-Based Geostatistics. Springer.ISBN 978-0387329079
- 2.Banerjee, S., Carlin, B. P., & Gelfand, A. E. (2015). Hierarchical Modeling and Analysis for Spatial Data (2nd ed.). CRC Press.ISBN 978-1439819173
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Cite this page
ScholarGate. (2026, June 3). Bayesian Co-Kriging. ScholarGate. https://scholargate.app/spatial-analysis/bayesian-co-kriging