bayesian-estimation

Files (4)

• SKILL.md
• diagnostics-guide.md · references/
• implementation.md · references/
• structural-models.md · references/

SKILL.md (first lines)

# 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

### 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:

1. Use the `methods-explorer` agent to find reference parameter values
2. Set prior mean to the benchmark: e.g., price elasticity of -1.2 from literature
3. 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
4. Verify with a prior predictive check that implied observables are plausible

```python
# 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):**

- [ ] R-hat < 1.01 for all parameters
- [ ] Bulk ESS > 400 for all parameters
- [ ] Tail ESS > 400 (ideally > 1000 for reported credible intervals)

From James-Traina/compound-science, MIT.