Author: José Mauricio Gómez Julián
Date: 2026-07-06

# The Empirical-Bayes Workflow in gdpar

### Operational recipe: `gdpar_eb()` end-to-end, four path regimes, EB-vs-FB comparison and troubleshooting (Sub-phase 8.6.E)

#### José Mauricio Gómez Julián

#### 2026-07-06

# 1. What this vignette covers

This is an operational recipe. It walks you through `gdpar_eb()` end to end across the four path regimes canonized in Sub-phase 8.6 of gdpar:

Regime | \(K\) | \(p\) | Stan template pair
--- | --- | --- | ---
Base | 1 | 1 | `amm_eb_marginal.stan` + `amm_eb_conditional.stan`
Path A | 1 | >1 | `amm_eb_marginal_multi.stan` + `amm_eb_conditional_multi.stan`
Path B | >1 | 1 | `amm_eb_marginal_K.stan` + `amm_eb_conditional_K.stan`
Path C | >1 | >1 | `amm_eb_marginal_KxP.stan` + `amm_eb_conditional_KxP.stan`

It then shows how to compare an EB fit against a Fully-Bayes (FB) fit on the same data via `gdpar_compare_eb_fb()`, how to read the numerical diagnostics that surface when the marginal Hessian is poorly conditioned, and when to give up on EB and fall back to FB.

The theoretical canonization of EB-vs-FB — the asymptotic equivalence (Theorem 7A and its multivariate extension Theorem 7A*), the higher-order coverage discrepancy (Proposition 7B scalar / 7B* matricial / 7B* tensorial), the finite-sample compound decision bound (Theorem 7C / 7C* compound multi-slot), and the four discrepancy conditions of Proposition 7D — lives in the canonical vignettes `vignette("v07_eb_vs_fb")` (scalar) and `vignette("v07b_eb_multivariate")` (multivariate extension). Read those if you want to know why a given choice was made; read this one if you want to do it.

The chunks below default to `eval = FALSE` because they compile Stan models and take several minutes per fit; re-enable evaluation on a per-chunk basis or globally via `knitr::opts_chunk$set(eval = TRUE)` if you want to reproduce the runs locally.

# 2. Setup

    library(gdpar)
    # These two are Suggests; gdpar_eb() and gdpar_compare_eb_fb() require
    # them at runtime.
    library(cmdstanr)
    library(posterior)

We will work with a synthetic dataset of size `n = 150` on a single continuous outcome with one covariate, then enrich it to multivariate and multi-slot variants as we walk through the four regimes.

    set.seed(20260526L)
    n  <- 150L
    df <- data.frame(x = stats::rnorm(n))
    df$y_scalar <- 1.0 + 0.4 * df$x + stats::rnorm(n, sd = 0.3)
    # Multivariate (p = 2) outcome for Path A.
    df$y_p2 <- cbind(
      1.0 + 0.4 * df$x + stats::rnorm(n, sd = 0.3),
      -0.5 + 0.2 * df$x + stats::rnorm(n, sd = 0.4)
    )
    # Same dataset is fine for Path B (K > 1, p = 1) and Path C (K > 1,
    # p > 1) by reusing y_scalar / y_p2 with a multi-slot family below.

# 3. Minimal `gdpar_eb()` call (K = 1 + p = 1)

The base regime mirrors the canonical `gdpar()` signature; the only new arguments are `eb_correction = TRUE` (default; applies the Proposition 7B scalar inflation to the conditional credible intervals) and `laplace_control = list(...)` (controls the multi-start Laplace maximizer of v07 Section 11.1 step (i)).

    fit_eb <- gdpar_eb(
      formula        = y_scalar ~ x,
      family         = gdpar_family("gaussian"),
      amm            = amm_spec(a = ~ x),
      data           = df,
      iter_warmup    = 500L,
      iter_sampling  = 500L,
      chains         = 2L,
      refresh        = 0L,
      seed           = 1L,
      laplace_control = list(multi_start_M = 5L)
    )
    print(fit_eb)

The output reports the EB plug-in point estimate \(\widehat\theta_{\text{ref}}^{\text{EB}}\) (from the marginal Laplace), its marginal standard error from the Laplace covariance, the marginal Hessian condition number \(\kappa(H)\), the multi-start dispersion across the `multi_start_M = 5` independent inits, and the conditional HMC convergence diagnostics (\(\widehat R\), ESS, divergences).

`summary(fit_eb)` returns a tidy table of EB credible intervals with the Proposition 7B scalar inflation applied:

    summary(fit_eb)

# 4. Path A (K = 1, p > 1): multivariate outcome

For a \(p\)-dimensional outcome, the amm spec uses `dimwise()` (or a plain `list` of length \(p\)) to declare the per-coordinate components, and the family is promoted to a `gdpar_family_multi` of dimension \(p\). The Stan template pair `amm_eb_marginal_multi.stan` + `amm_eb_conditional_multi.stan` (canonized in Sub-phase 8.6.C under decision D34) is dispatched automatically:

    fit_eb_A <- gdpar_eb(
      formula        = y_p2 ~ x,
      family         = gdpar_family_multi("gaussian", p = 2L),
      amm            = amm_spec(p = 2L, dims = dimwise(a = ~ x)),
      data           = df,
      iter_warmup    = 500L,
      iter_sampling  = 500L,
      chains         = 2L,
      refresh        = 0L,
      seed           = 2L
    )
    print(fit_eb_A)

The corresponding correction is matricial, \(C^*_{g,\alpha} \in \mathbb{R}^{p\times p}\) (Proposition 7B* of v07b Section 5.1). It reduces algebraically to the scalar Proposition 7B at \(p = 1\), so the upgrade from base to Path A is transparent.

# 5. Path B (K > 1, p = 1): multi-parametric family

For a multi-slot distributional regression (e.g. modelling both `mu` and `sigma` of a Gaussian K=2, or `mu` and `phi` of a Negative Binomial K=2), the amm input is either a named list of `amm_spec` (one per slot) or a `gdpar_formula_set` via `gdpar_bf(...)`. The Stan template pair `amm_eb_marginal_K.stan` + `amm_eb_conditional_K.stan` is dispatched automatically:

    fs <- gdpar_bf(y_scalar ~ a(x), sigma ~ a(x))
    fit_eb_B <- gdpar_eb(
      formula        = fs,
      family         = gdpar_family("gaussian"),
      data           = df,
      iter_warmup    = 500L,
      iter_sampling  = 500L,
      chains         = 2L,
      refresh        = 0L,
      seed           = 3L,
      skip_id_check  = TRUE
    )
    print(fit_eb_B)

Coverage of stan_ids per Sub-phase 8.6.C decision D33 (relaxed): `{1, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13}` — Gaussian, Negative Binomial, Beta, Gamma, Lognormal-loc-scale, Student-t, Tweedie, ZIP, ZINB, Hurdle-Poisson, Hurdle-NB.

# 6. Path C (K > 1, p > 1): full K x p extension

Path C, canonized in Sub-phase 8.6.D under decision D36 = (alpha) + D37 = (i) + D38’’ = (h), composes Path A coordinate-wise factorization with Path B multi-parametric K-slot semantics: a single outcome matrix-column `y[n, p]` is shared across the K distributional slots, each carrying its own per-coordinate linear predictor.

The initial 8.6.D iteration restricts Path C to `family$stan_id %in% c(1, 3)` (Gaussian K=2, NB K=2) under decision D40’ to avoid the numerical caveat of Section 6.1 of the opening handoff (HMC condicional bajo plug-in EB cerca del borde de soporte logit/log links + warmup corto). Beta / Gamma / Lognormal / Student-t / Tweedie / mixtures are deferred to a later iteration.

    y_p2_int <- matrix(
      rnbinom(n * 2L, size = 5, mu = exp(0.5 + 0.2 * df$x)),
      n, 2L
    )
    df$y_p2_int <- y_p2_int
    fit_eb_C <- gdpar_eb(
      formula        = y_p2_int ~ x,
      family         = gdpar_family("neg_binomial_2"),
      amm            = list(
        mu  = amm_spec(p = 2L, dims = dimwise(a = ~ x)),
        phi = amm_spec(p = 2L, dims = dimwise(a = ~ x))
      ),
      data           = df,
      iter_warmup    = 500L,
      iter_sampling  = 500L,
      chains         = 2L,
      refresh        = 0L,
      seed           = 4L,
      skip_id_check  = TRUE
    )
    print(fit_eb_C)

The fit object carries new Path C-specific slots:

- `theta_ref_kp_hat`: 3D numeric array of shape `[J_groups, K, p]`.
- `theta_ref_kp_se`: same shape; per-coordinate marginal standard errors derived from each slot’s Laplace covariance block.
- `theta_ref_kp_cov_per_slot`: named list of `K` matrices each `p × p`; the per-slot blocks of the joint Laplace covariance after the block-diagonal extraction of decision D43 = (a).
- `correction_tensor_constant`: 3D array of shape `[K, p, p]` with the Proposition 7B* tensor correction of decision D37 = (i).
- `correction_tensor_dispositions`: named character vector reporting whether each slot’s correction was applied (`"ok"`), `"non_finite"`, `"non_psd"`, `"missing"`, or `"disabled"`.

# 7. Numerical diagnostics: how to read them

Every `gdpar_eb_fit` carries a `diagnostics_numerical` slot with four entries derived from the multi-start Laplace strategy of Charter Section 2.8:

- `kappa` (base regime / Path A / Path B) or `kappa_per_slot` (Path C): the condition number of the marginal Hessian at the chosen MAP. The guard `laplace_control$kappa_threshold` (default `1e10`) aborts the fit with `gdpar_eb_numerical_error` when exceeded. A `kappa` between `1e6` and `1e9` is a warning sign; consider tightening the prior on `theta_ref` or moving the anchor closer to the data via `anchor = "empirical_y"`.

- `lm_perturbation`: the Levenberg-Marquardt ridge actually added to the marginal covariance when the bare Hessian was singular or non-PSD. A non-zero value means the Laplace approximation needed numerical stabilization; the result is still valid but the effective sample size for the EB anchor is smaller than the nominal Laplace draws would suggest.

- `multi_start_dispersion`: standard deviation of the log marginal across the `multi_start_M` independent inits, normalized by the absolute mean. A value above `0.05` triggers a diagnostic warning because it suggests multi-modality of the marginal likelihood (open question O5*-EBFB of v07b Section 9.5). Consider raising `multi_start_M` from the default 5 to 10–20 to stress-test the optimum.

- `marginal_log_lik_history`: the per-init log marginal achieved by each `optimize()` call. Useful for forensics when the dispersion is large.

# 8. EB vs FB comparison via `gdpar_compare_eb_fb()`

The companion function `gdpar_compare_eb_fb()` (canonized in Sub-phase 8.6.E) takes a `gdpar_eb_fit` and a `gdpar_fit` fitted on the same dataset and reports three operational diagnostics of the EB-vs-FB theory of v07:

    fit_fb <- gdpar(
      formula        = y_scalar ~ x,
      family         = gdpar_family("gaussian"),
      amm            = amm_spec(a = ~ x),
      data           = df,
      iter_warmup    = 500L,
      iter_sampling  = 500L,
      chains         = 2L,
      refresh        = 0L,
      seed           = 1L
    )
    cmp <- gdpar_compare_eb_fb(fit_eb, fit_fb, level = 0.95,
                                tv_bins = 30L)
    print(cmp)
    summary(cmp)

The output carries three tables:

- `theta_diff_table`: per-anchor cell comparison of \(\widehat\theta_{\text{ref}}^{\text{EB}}\) and the FB posterior mean \(E_{\text{FB}}[\theta_{\text{ref}}]\), with the difference and the difference normalized by the FB standard error. Under the standing hypotheses of v07 Section 4 (EB-MARG-ID + PRIOR-FB-WEAK + HIER-COMPLEX), Theorem 7A predicts `diff_rel` close to zero up to \(O(n^{-1/2})\).

- `tv_table`: marginal empirical total variation distance per common \(\xi\) parameter, computed via histogram plug-in over the shared support. Under Theorem 7A, marginal TV \(\to 0\) in probability as \(n \to \infty\). A persistent large TV across many parameters suggests one of the discrepancy conditions of Proposition 7D (multi-modality of the marginal likelihood, near-singular Fisher information, informative prior, deep hierarchy).

- `coverage_table`: per anchor cell, the EB credible interval width (with inflation applied when `eb_correction = TRUE`), the FB credible interval width, and the ratio `width_eb / width_fb`. This operationally verifies the \(O(n^{-1})\) under-cover prediction of Proposition 7B (and its matricial / tensorial extensions). A ratio systematically below 1 confirms the EB under-cover; a ratio above 1 after correction suggests the inflation is over-correcting, which is acceptable in finite samples and consistent with the asymptotic guarantee.

# 9. Troubleshooting

## 9.1. `gdpar_eb_numerical_error: kappa = ...`

The marginal Hessian is too ill-conditioned for Laplace to be reliable. Options:

1. Tighten the prior on `theta_ref` via a stronger `gdpar_prior(theta_ref = "normal(0, 0.5)")`.
2. Move the anchor closer to the data via `anchor = "empirical_y"`.
3. Raise `laplace_control$multi_start_M` and rerun (the per-init seed offset will sample different unconstrained-space inits).
4. Fall back to FB via `gdpar()` (the most reliable option when the marginal likelihood is genuinely flat).

## 9.2. `gdpar_unsupported_feature_error` on Path C

The initial 8.6.D iteration restricts Path C to `family$stan_id %in% c(1, 3)` (Gaussian K=2, NB K=2) per decision D40’. For other distributional families under \(K > 1 \wedge p > 1\), fall back to FB via `gdpar()` or split the multivariate outcome into \(p\) separate Path B fits as a workaround.

## 9.3. Multi-start dispersion warning

A `multi_start_dispersion` above 0.05 suggests multi-modality of the marginal likelihood (open question O5*-EBFB of v07b Section 9.5). The result returned by `gdpar_eb()` is the mode with the highest log marginal across the `multi_start_M` inits, which may differ from the mode picked by a single run. Options:

1. Raise `multi_start_M` from the default 5 to 10–20.
2. Use a stronger prior on `theta_ref` to break the multi-modality.
3. Fall back to FB; HMC explores multi-modal posteriors better than single-mode Laplace.

## 9.4. Path B conditional HMC instability under `logit`-strict links

Path B with strict \((0, 1)\) inverse-link families (Beta, Bernoulli) can exhibit conditional HMC instability when the EB plug-in anchor falls near the support boundary and the HMC warmup is short. The caveat is documented in the closure of Sub-phase 8.6.C (`HANDOFF_SUBFASE_8_6_C_CIERRE.md` Section 3.3). Workarounds:

1. Raise `iter_warmup` to 1000 or more.
2. Use a tighter prior on `sigma_a_k` (the HMC has fewer chances to sample large `a_coef_k` values that saturate the logit).
3. Use `anchor = "prior_median"` or an explicit numeric anchor that keeps the EB plug-in in the interior of the support.

## 9.5. Path C smoke validation

A representative end-to-end Path C smoke (Gaussian K=2 + p=2, \(n = 80\), `iter_warmup = iter_sampling = 200L`, `chains = 2L`) takes approximately 50 seconds on contemporary hardware (`amm_eb_marginal_KxP.stan` and `amm_eb_conditional_KxP.stan` each compile once and are cached by cmdstanr). If the smoke takes significantly longer or aborts with `gdpar_eb_numerical_error`, the geometry of the K × p marginal is likely the culprit; check `kappa_per_slot` and the slot dispositions returned by the correction tensor.

# 10. Where to go next

- For the theoretical canonization of EB-vs-FB at the scalar level, read `vignette("v07_eb_vs_fb")`.
- For the multivariate / multi-slot extension of the theory (Theorem 7A*, Proposition 7B*, Theorem 7C* compound multi-slot, open question O5*-EBFB), read `vignette("v07b_eb_multivariate")`.
- For the AMM canonization on which both EB and FB operate, read the AMM family of canonical vignettes (v01–v06).
- For CATE / ITE applications and the EB-vs-FB shrinkage discussion in the context of heterogeneous treatment effects, read `vignette("v08_amm_for_cate_ite_positioning")` and the implementation vignettes `v08b_cate_ite_bridge_implementation` and `v08c_meta_learner_comparison`.
- If your data are dependent — temporally (a time series) or spatially — the EB posterior understates uncertainty because the likelihood assumes conditional independence. Measure the violation with `gdpar_dependence_diagnostic()` (lag-1 autocorrelation, Durbin-Watson, Ljung-Box) or its spatial sibling `gdpar_spatial_dependence_diagnostic()` (Moran’s I), and obtain dependence-robust standard errors and percentile intervals with `gdpar_dependence_robust()` (temporal block bootstrap) or `gdpar_spatial_dependence_robust()` (spatial block bootstrap), Block 9, Axis 2. This makes the uncertainty robust to dependence; gdpar does not model the dependence. The full recipe is in `vignette("vop09_dependence_robust")`.

# References

The canonical theoretical references for EB-vs-FB in the AMM are Petrone, Rousseau, and Scricciolo (2014) and Rousseau and Szabo (2017) for the asymptotic equivalence and merging results; Carlin and Gelfand (1990) for the higher-order coverage discrepancy and the inflation correction; and Robbins (1956) / Efron (2010) for the compound decision framework. Full bibliographic details are in `vignette("v07_eb_vs_fb")` Section “References Cited in This Block”.