Bayesian Weighted Least Squares (Bayesian WLS)
Bayesian Weighted Least Squares · Also known as: Bayesian weighted regression, BWLS, Bayesian heteroscedastic regression, weighted Bayesian linear regression
Bayesian Weighted Least Squares combines the classical WLS weighting scheme — which downweights observations with high error variance — with Bayesian prior distributions over the regression coefficients and error variance. The result is a posterior distribution that reflects both the data likelihood and prior beliefs, providing full uncertainty quantification in heteroscedastic settings.
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 WLS when you have heteroscedastic regression data and want principled uncertainty quantification beyond point estimates and confidence intervals. It is particularly useful with small samples, where classical WLS standard errors can be unreliable, or when meaningful prior information is available. Choose it over Bayesian OLS when error variances differ across observations and weights can be specified or estimated. Avoid it when weights are completely unknown and cannot be approximated, or when the sample is large enough that the Bayesian-classical distinction is negligible and computational cost matters; in those cases classical WLS with robust standard errors is simpler.
Strengths & limitations
- Provides full posterior distributions over coefficients, yielding credible intervals and predictive distributions rather than point estimates alone.
- Handles heteroscedasticity structurally through observation-specific weights, improving efficiency over unweighted Bayesian OLS.
- Allows incorporation of prior information, which stabilises estimates and aids identification in small samples.
- Posterior predictive checks offer natural, coherent model criticism without separate diagnostic tests.
- Shrinkage from informative priors reduces overfitting in high-dimensional or collinear settings.
- Requires specification of weights (inverse variances); misspecified weights can bias results more than in classical WLS.
- Prior choice is consequential in small samples; poorly calibrated priors can dominate the likelihood.
- Computationally more demanding than classical WLS, especially when MCMC is needed for non-conjugate models.
- Interpretation of posterior credible intervals differs philosophically from frequentist confidence intervals, which may complicate communication.
Frequently asked
How do I choose the weights in Bayesian WLS?
Weights should reflect the inverse of each observation's error variance. In practice they may come from known sampling variances (e.g., in meta-analysis), from a first-stage variance model, or from theoretical considerations. Sensitivity analysis over plausible weight specifications is always recommended.
What is the difference between Bayesian WLS and Bayesian GLS?
WLS uses a diagonal weight matrix, assuming errors are uncorrelated across observations but have different variances. GLS uses a full covariance matrix, also handling correlations (e.g., autocorrelation or spatial dependence). If your errors are both heteroscedastic and correlated, Bayesian GLS is the more general choice.
Do I need MCMC or can I use an analytical posterior?
With conjugate Normal or Normal-Inverse-Gamma priors the posterior is available in closed form, so MCMC is not needed. Non-conjugate priors or variance-model extensions (e.g., a hierarchical weight model) require MCMC or variational inference.
How does Bayesian WLS differ from robust regression?
Robust regression (e.g., M-estimation or Bayesian robust regression with heavy-tailed likelihoods) downweights outliers automatically based on residual size. Bayesian WLS uses pre-specified or estimated weights reflecting known heteroscedasticity, not residual magnitude. They address different problems and can be combined.
When does the prior matter most?
The prior is most influential with small samples, many predictors relative to observations, or when the likelihood is flat (weak identification). With large, information-rich datasets the likelihood dominates and results converge to the classical WLS estimates regardless of the prior.
Sources
- Zellner, A. (1971). An Introduction to Bayesian Inference in Econometrics. Wiley, New York. ISBN: 978-0471169376
- Koop, G. (2003). Bayesian Econometrics. Wiley, Chichester. ISBN: 978-0470845677
How to cite this page
ScholarGate. (2026, June 3). Bayesian Weighted Least Squares. ScholarGate. https://scholargate.app/en/econometrics/bayesian-wls
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 Fixed Effects ModelEconometrics↔ compare
- Bayesian OLSEconometrics↔ compare
- Bayesian Random Effects ModelEconometrics↔ compare
- Robust WLSEconometrics↔ compare