Bayesian Estimation

Bayesian Estimation

Reference for Bayesian estimation in quantitative social science: from prior elicitation to MCMC implementation to posterior reporting. Covers the full workflow of specifying a Bayesian model, running inference, diagnosing convergence, and communicating results — with applications to structural models, hierarchical designs, and small-sample settings.

When to Use This Skill

Use when the user is:

• Specifying priors and setting up a Bayesian model in Stan, PyMC, NumPyro, brms, or rstanarm
• Running MCMC and diagnosing R-hat, ESS, divergences, or trace plots
• Implementing hierarchical (multilevel) models with partial pooling
• Adding Bayesian inference to a structural model (BLP, dynamic discrete choice, DSGE)
• Reporting credible intervals, posterior predictive checks, or model comparison statistics
• Eliciting priors from calibration targets or literature benchmarks
• Debugging sampling pathologies: divergences, low acceptance rates, poor mixing

Skip when:

• The model is large-N and well-identified (frequentist MLE/GMM is more efficient and faster)
• The task is pure structural estimation without Bayesian components (use structural-modeling skill)
• The user needs classical causal inference (use causal-inference skill)

When to Use Bayesian Estimation

Advantages

Small samples: Priors act as regularization. With N < 100 observations and several parameters, MLE can overfit or fail to converge. A weakly informative prior is often equivalent to several additional observations of prior knowledge.

Hierarchical structure: When data have natural groupings (markets, countries, firms), Bayesian partial pooling is more efficient than either pooling all groups (ignores variation) or fitting each group separately (ignores shared structure). Random effects Bayesian models borrow strength across groups.

Uncertainty propagation: The posterior is a full distribution. Downstream quantities (elasticities, welfare changes, counterfactuals) inherit full uncertainty without a delta method approximation.

Natural model comparison: Posterior predictive checks and LOO-CV are cleaner than frequentist test-based model selection, especially for non-nested models.

Constrained parameters: Parameters with domain restrictions (discount factors in [0,1], positive variances, simplex probability vectors) are handled naturally via transformed parameter blocks and appropriate priors.

When to Prefer Frequentist

• Large N (N > 10,000): Priors become irrelevant as the likelihood dominates; MLE or GMM is faster
• Simple models where MLE is efficient (OLS, logit, Poisson): No benefit from Bayesian approach
• Publication venues requiring classical standard errors and p-values (some applied micro journals)
• Models requiring solving equilibrium constraints (NFXP/MPEC) at each draw — computationally prohibitive