Bayesian Geary's C
Also known as: Bayesian Geary C, Bayesian spatial contiguity statistic, Geary's C (Bayesian), Bayesian contiguity ratio
Bayesian Geary's C embeds the classical Geary contiguity ratio within a Bayesian hierarchical framework. Instead of a single point estimate and asymptotic p-value, it produces a posterior distribution over the statistic (or over spatially structured random effects), quantifying uncertainty about spatial autocorrelation while formally incorporating prior knowledge about the spatial process.
Key highlights
- Produces full posterior distributions over spatial autocorrelation parameters, enabling principled uncertainty quantification beyond asymptotic p-values.
- Particularly valuable for small sample sizes where classical inference relies on questionable large-sample approximations.
- Allows formal integration of domain knowledge or historical data through informative priors.
- Focuses on local pair-wise differences, making it sensitive to local clustering patterns that global statistics may miss.
- Naturally embedded in hierarchical models, so additional covariates and random effects can be included in the same estimation.
Intuition
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How it works
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When to use it
Use Bayesian Geary's C when you have areal spatial data and want to assess local positive or negative spatial autocorrelation with full uncertainty quantification, especially in small samples where asymptotic p-values are unreliable, or when you want to incorporate prior information about the spatial process (e.g., from earlier studies or domain knowledge). It is well suited to disease mapping, environmental monitoring, and regional economics with sparse data. Avoid it when you need a quick exploratory screening with large samples and no prior information — classical Geary's C is faster and equally informative in that setting. Also avoid when data are point-referenced (geostatistical) rather than areal; kriging-based Bayesian approaches are more appropriate there.
Strengths & limitations
- Produces full posterior distributions over spatial autocorrelation parameters, enabling principled uncertainty quantification beyond asymptotic p-values.
- Particularly valuable for small sample sizes where classical inference relies on questionable large-sample approximations.
- Allows formal integration of domain knowledge or historical data through informative priors.
- Focuses on local pair-wise differences, making it sensitive to local clustering patterns that global statistics may miss.
- Naturally embedded in hierarchical models, so additional covariates and random effects can be included in the same estimation.
- Computationally expensive: MCMC or INLA is required, making analysis slower than computing classical Geary's C.
- Results depend on prior specification; poorly chosen priors can dominate in small samples and mislead inference.
- Requires careful construction of the spatial weights matrix — results can be sensitive to the chosen contiguity definition.
- Less familiar to practitioners than classical Geary's C or Moran's I, making results harder to communicate in applied reports.
Common pitfalls
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Applications
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Frequently asked
How does Bayesian Geary's C differ from classical Geary's C?
Classical Geary's C returns a single point estimate with an asymptotic p-value based on normality assumptions. The Bayesian version produces a full posterior distribution, quantifies uncertainty without relying on large-sample approximations, and allows incorporation of prior information.
What is the interpretation of Geary's C values?
C near 0 indicates strong positive spatial autocorrelation (neighbors are very similar); C near 1 indicates spatial randomness; C near 2 indicates negative autocorrelation (neighbors are dissimilar). In the Bayesian framework these thresholds become posterior probabilities rather than fixed decisions.
Is Bayesian Geary's C better than Bayesian Moran's I?
Neither is universally better. Geary's C is more sensitive to local spatial clustering because it squares pairwise differences between adjacent units, while Moran's I uses deviations from the global mean and is more sensitive to global trends. Use both when exploring data.
Which software can run Bayesian Geary's C?
Stan (via RStan or PyStan), OpenBUGS/WinBUGS, and R-INLA are the most common platforms. In R, the spdep package computes classical Geary's C; the Bayesian variant requires coding the likelihood and prior in Stan or a CAR model in INLA.
How do I choose priors for the spatial autocorrelation parameter?
A common starting point is a weakly informative prior centered on spatial randomness (C = 1) with moderate variance. If earlier studies suggest strong clustering, an informative prior around C < 1 is appropriate. Always run a prior sensitivity analysis to check how much the posterior shifts with different prior specifications.
Sources
- 1.Geary, R. C. (1954). The contiguity ratio and statistical mapping. The Incorporated Statistician, 5(3), 115–145.
- 2.Banerjee, S., Carlin, B. P., & Gelfand, A. E. (2004). Hierarchical Modeling and Analysis for Spatial Data. Chapman & Hall/CRC.ISBN 978-1584884101
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Cite this page
ScholarGate. (2026, June 3). Bayesian Geary's C. ScholarGate. https://scholargate.app/spatial-analysis/bayesian-gearys-c