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SKILL.md
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
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
Comparison Table
Scenario
Bayesian
Frequentist
N = 50, 10 parameters
Preferred — priors regularize
MLE may not converge
N = 100,000, 5 parameters
Either works; MLE faster
Preferred — priors irrelevant
Hierarchical / multi-group
Preferred — partial pooling
Mixed effects via ML is comparable
Uncertainty in counterfactuals
Preferred — natural propagation
Delta method or bootstrap
Structural model, large state space
Difficult — MCMC over full model
Preferred — NFXP/MPEC more tractable
Non-standard likelihood
Either — depends on differentiability
GMM often more flexible
Model comparison, non-nested
LOO-CV / WAIC
AIC/BIC, Vuong test
Publication: applied micro top-5
Use sparingly; justify carefully
Standard expectation
Prior Elicitation
The core task: encode genuine prior knowledge without overwhelming the likelihood.
Weakly Informative Defaults
These encode vague knowledge — parameters are unlikely to be astronomically large — without strongly influencing the posterior when data are informative. These follow Gelman et al. (2008, 2017) recommendations:
Parameter type
Recommended prior
Rationale
Location / intercept (raw scale)
Normal(0, 10)
Very diffuse; most applications won't have effects > 10 SDs
Location (standardized predictors)
Normal(0, 2.5)
Rules out extreme effects; standard for logistic regression
Scale / variance
HalfNormal(1) or Exponential(1)
Positive, concentrates near zero but with wide right tail
Log-scale parameters (elasticities)
Normal(0, 1) on log scale
Implies elasticity plausibly between 0.14 and 7.4
Correlation matrices
LKJ(2)
Shrinks toward identity; LKJ(1) is uniform on correlations
Simplex (probability vectors)
Dirichlet(1, ..., 1)
Uniform over simplex
Informative Priors from Calibration Targets
When the literature provides benchmark values, use them as prior means with SD reflecting plausible variation:
Use the methods-explorer agent to find reference parameter values
Set prior mean to the benchmark: e.g., price elasticity of -1.2 from literature
Set prior SD to cover the plausible range: if literature range is [-0.5, -2.0], use Normal(-1.2, 0.4) so that the range is roughly within 2 SDs
Verify with a prior predictive check that implied observables are plausible
# Example: Informative prior on price elasticity from literature
# Literature: price elasticity typically -0.5 to -2.0 (mean around -1.2)
import pymc as pm
with pm.Model() as demand_model:
alpha = pm.Normal("alpha", mu=-1.2, sigma=0.4) # price elasticity
beta_inc = pm.Normal("beta_inc", mu=0, sigma=1.0) # income elasticity
sigma = pm.HalfNormal("sigma", sigma=1.0)
Prior predictive decision rules:
Prior implies impossible values (negative income, market share > 1) → constrain or use transformed parameter
Prior predictive range is > 100x observed data range → prior too diffuse; tighten it
Prior predictive range is narrower than observed variation → prior too tight; loosen it
Prior sensitivity analysis: After estimation, refit with prior SD at 0.5x and 2x. If the posterior mean shifts by more than 0.5 SD, the prior is informative — report sensitivity results.
MCMC Quick Reference
Key thresholds for diagnosing MCMC convergence. For full code and detailed guidance, see references/diagnostics-guide.md.
Diagnostic
Good
Borderline
Bad
Action if Bad
R-hat
< 1.01
1.01–1.05
>= 1.05
Run longer chains; reparametrize
Bulk ESS
> 400
200–400
< 200
Increase draws; reduce autocorrelation
Tail ESS
> 1000
400–1000
< 400
Increase draws; focus on scale parameters
Divergences
0
—
Any
Raise target_accept to 0.9–0.95; non-centered parametrization
BFMI
> 0.3
0.2–0.3
< 0.2
Reparametrize; check prior scales
Convergence checklist (run before every reported result):
For full diagnostic code including trace plots, pair plots, and BFMI computation, see references/diagnostics-guide.md.
Implementation Quick Start
A minimal PyMC example for hierarchical demand. For full Stan, PyMC, NumPyro, and brms examples (including Bayesian IV, reparametrization patterns, and Cholesky covariance), see references/implementation.md.
import pymc as pm
import arviz as az
with pm.Model() as hierarchical_demand:
# Hyperpriors
mu_beta = pm.Normal("mu_beta", mu=-1.0, sigma=0.5)
sigma_beta = pm.HalfNormal("sigma_beta", sigma=0.3)
# Non-centered parametrization — always use this for hierarchical models
beta_raw = pm.Normal("beta_raw", mu=0, sigma=1, shape=n_markets)
beta = pm.Deterministic("beta", mu_beta + sigma_beta * beta_raw)
sigma = pm.HalfNormal("sigma", sigma=1.0)
mu = beta[market_idx] * log_price
log_quantity = pm.Normal("log_quantity", mu=mu, sigma=sigma, observed=Y_obs)
trace = pm.sample(
draws=2000, tune=1000, chains=4,
target_accept=0.9, # raise if divergences appear
random_seed=42, return_inferencedata=True
)
# Always run diagnostics immediately after sampling
print(az.summary(trace, var_names=["beta", "mu_beta", "sigma_beta"]))
Framework selection guide:
Framework
Best for
Stan (cmdstanpy)
Complex custom models, structural work, production code
R users, standard hierarchical families, formula interface
For Bayesian structural models (BLP, dynamic discrete choice, hierarchical DiD) and reparametrization strategies, see references/structural-models.md.
Integration with compound-science
Agents to invoke alongside Bayesian estimation:
numerical-auditor: Review MCMC convergence diagnostics — R-hat, ESS, divergences. Report format should include all five convergence metrics.
econometric-reviewer: Review prior elicitation strategy, sensitivity analysis, and whether priors are consistent with identification. Use for prior predictive checks and moment-matching to literature targets.
methods-explorer: Find literature calibration targets to set informative prior means. Ask for point estimates and uncertainty ranges, not just means.
econometric-reviewer: Verify that reported posterior means, credible intervals, and model comparison statistics match the actual ArviZ/Stan output.
Related skills:
structural-modeling: Frequentist counterpart — NFXP, MPEC, BLP, dynamic discrete choice. Use Bayesian skills on top of the structural model framework when small samples or hierarchical structure warrants it.
causal-inference: For reduced-form causal methods. Bayesian DiD and RD designs follow the same identification logic; the Bayesian layer adds partial pooling and uncertainty propagation.
Extensions to Bayesian context:
empirical-playbook skill (diagnostic-battery.md): Convergence diagnostics (R-hat, ESS, divergences) are a subset of the full diagnostic battery
numerical-auditor agent: Prior predictive simulation is a special case of the Monte Carlo simulation workflow
empirical-playbook skill (sensitivity-analysis.md): Prior sensitivity analysis (vary prior SD 0.5x and 2x) is a natural robustness check for Bayesian models
Common Anti-Patterns
Anti-Pattern
Problem
Better Approach
Reporting results without checking R-hat
Non-convergence masquerades as valid posterior
Always run full diagnostic checklist first
Using uniform priors (flat priors)
Improper or extremely diffuse; creates pathological geometry in hierarchical models
Use weakly informative priors (Normal(0, 2.5) or HalfNormal(1))
Centered parametrization for hierarchical models
Funnel geometry; divergences and low ESS for scale parameters
Non-centered parametrization always for hierarchical models
Using variational inference for final results
Underestimates posterior variance; may miss multimodality
