# Theoretical Addendum – Block 8.5.B:

### Comparison of the AMM-side T-learner against External Meta-learners: Canonization of the Pluggable Adapter Contract

#### José Mauricio Gómez Julián

#### 2026-07-06

# 1. Purpose and position in the package

This vignette canonizes the comparator `gdpar_compare_meta_learners`, the companion of the T-learner AMM-side bridge of Sub-phase 8.5.A. The bridge produces a per- observation posterior estimate of the conditional average treatment effect (CATE), \(\widehat\tau_{\text{bridge}}(x) = \mathbb{E}\big[\widehat\mu^{(\text{treat})}(x) - \widehat\mu^{(\text{ctrl})}(x) \mid \text{posterior}\big]\), documented in `vignette("v08b_cate_ite_bridge_implementation")` and positioned in the meta-learner literature in `vignette("v08_cate_ite_positioning")`. The comparator extends that machinery by accepting external meta-learner implementations (point estimators with their own native uncertainty quantification mechanisms) and reporting the discrepancy between \(\widehat\tau_{\text{bridge}}\) and each external \(\widehat\tau_{\text{ext}}\) on a common evaluation grid.

Two reference adapters are distributed with the package:

- `gdpar_adapter_grf()` wraps `grf::causal_forest` (Athey, Tibshirani, Wager, 2019) R-side, with native CIs from the built-in variance estimator under honesty.
- `gdpar_adapter_econml()` wraps `econml.dml.CausalForestDML` (Chernozhukov et al., 2018) Python-side via `reticulate`, with native CIs from `effect_interval()`.

Both are reference implementations of a pluggable contract. Users may add adapters for `DoubleML`, doubly-robust estimators, or any custom learner without touching the package; the operational recipe (grf, EconML, a custom adapter using DoubleML as a worked example, plus Python troubleshooting) lives in the companion operational vignette `vignette("vop06_meta_learner_comparison")`.

This addendum has the same canonical status as `v08b`: definitions, identification, estimation, the concordance criterion, and limits are stated here once and not reopened during implementation.

# 2. Notation inherited from `v08` and `v08b` (rapid reference)

We work in the same setup of `v08b`:

- A treatment indicator \(T \in \{0, 1\}\) and a continuous outcome \(Y \in \mathbb{R}\).
- A vector of covariates \(X \in \mathcal{X} \subseteq \mathbb{R}^p\).
- Per-observation potential outcomes \(Y^{(0)}, Y^{(1)}\) in the Neyman-Rubin sense (Rubin, 1974). Identification of \(\tau(x) = \mathbb{E}[Y^{(1)} - Y^{(0)} \mid X = x]\) rests on the conditional ignorability \(T \perp\!\!\!\perp (Y^{(0)}, Y^{(1)}) \mid X\), the overlap \(0 < \Pr(T = 1 \mid X = x) < 1\), and SUTVA, as detailed in `v02` and recalled in `v08b` §4.
- Two disjoint training samples (the treatment arm and the control arm) of sizes \(n_T\) and \(n_C\), with \(n_T + n_C = n\) observations total.
- An evaluation grid \(\mathcal{N} = \{x_i^{\text{new}}\}_{i = 1}^{n_{\text{new}}}\) on which the CATE is reported.

The bridge produces, for each \(x_i^{\text{new}}\), a posterior distribution over \(\widehat\tau(x_i^{\text{new}})\) of which the mean \(\widehat\tau_{\text{bridge}}(x_i^{\text{new}})\) and the central \(\alpha\)-credible interval are stored.

An external meta-learner is any procedure that, given a single stacked dataset \((X_i, T_i, Y_i)_{i = 1}^n\) assembled from the two training arms, produces a point estimate \(\widehat\tau_{\text{ext}}(x_i^{\text{new}})\) at each evaluation point and (optionally) a frequentist confidence interval. The inferential origin of that interval is heterogeneous across methods (the asymptotic Gaussian CI of `grf`, the bootstrap-or- asymptotic CI of EconML, the partially linear regression CI of DoubleML), and the comparator does not paper over the heterogeneity by pooling.

# 3. The pluggable adapter contract

An adapter is an R object of class `gdpar_meta_learner_adapter` constructed by

    gdpar_meta_learner_adapter(
      name,            # character scalar, unique within a comparison
      fit_predict_fun, # mandatory closure
      predict_fun,     # optional closure (default NULL)
      requires_r,      # character vector of R packages needed
      requires_py,     # character vector of Python modules needed
      native_ci,       # logical scalar
      description      # optional character scalar
    )

with closures of signature

\[\begin{aligned} \texttt{fit\_predict\_fun}: \quad & (X, Y, T, X_{\text{new}}, \alpha, \text{seed}_{\text{run}}) \;\longmapsto\; \big(\widehat\tau, \widehat{C}, s, \nu\big),\\ \texttt{predict\_fun}: \quad & (s, X_{\text{new}}, \alpha) \;\longmapsto\; \big(\widehat\tau, \widehat{C}\big), \end{aligned}\]

where \(X\) is the training covariate data frame, \(Y\) the training outcome, \(T\) the training treatment indicator, \(X_{\text{new}}\) the evaluation grid, \(\alpha\) the nominal credible level inherited from the bridge, \(\text{seed}_{\text{run}}\) the per-method seed, \(\widehat\tau \in \mathbb{R}^{n_{\text{new}}}\) the point estimate, \(\widehat{C} \in \mathbb{R}^{n_{\text{new}} \times 2}\) the native CI (or `NULL` when the adapter does not produce one), \(s\) the cached fitted state, and \(\nu\) a vector of free-form diagnostic notes emitted during the fit.

The two-layer structure is deliberate.

Theorem 3.1 (Soundness of the two-layer contract). Let \(\mathcal{A}\) be an adapter and assume `fit_predict_fun` and `predict_fun` are deterministic functions of their arguments. Then for any \((X, Y, T)\) training tuple and any evaluation grid \(X_{\text{new}}\), calling

\[\begin{aligned} (\widehat\tau, \widehat{C}, s, \nu) &= \mathcal{A}.\texttt{fit\_predict\_fun}(X, Y, T, X_{\text{new}}, \alpha, \text{seed}),\\ (\widehat\tau', \widehat{C}') &= \mathcal{A}.\texttt{predict\_fun}(s, X_{\text{new}}, \alpha), \end{aligned}\]

yields \(\widehat\tau' = \widehat\tau\) and \(\widehat{C}' = \widehat{C}\) modulo numerical noise of the same order as the floating-point arithmetic of the language wrappers (R for `grf`, Python via `reticulate` for EconML).

The “modulo numerical noise” qualifier is non-trivial in the EconML case because `reticulate::r_to_py(...)` may copy or re-allocate numpy arrays; the noise is bounded by IEEE 754 rounding in single-pass conversions and we do not claim bit-exact equality across language boundaries. The reference adapters satisfy Theorem 3.1 by construction: `predict_fun` re-uses the same fitted forest / EconML estimator that `fit_predict_fun` returned in \(s\).

Adapters that do not expose `predict_fun` are still valid; the comparator’s `predict()` method then performs a full refit on the original training data and emits a structured `gdpar_diagnostic_warning`. The fallback is honest about its cost: it does not pretend to reuse cached state.

# 4. Identification under cross-method comparison

The bridge identifies \(\tau(x)\) on the AMM side under the assumptions of `v08b` §4: conditional ignorability per arm, overlap, SUTVA, and the residual no-confounding-given-AMM-design.

Each external meta-learner identifies \(\tau(x)\) under its own identification assumptions. For the two reference adapters:

- `grf::causal_forest` identifies \(\tau(x)\) under conditional ignorability and the regularity conditions of Athey, Tibshirani, Wager (2019, §2). It does not require a parametric model for the outcome surfaces; the AMM-side model is implicit in the splitting rule of the forest.
- `econml.dml.CausalForestDML` identifies \(\tau(x)\) under the Neyman orthogonality of Chernozhukov et al. (2018, §1.3) and the cross-fitting protocol; it factors the bias-variance trade-off via two nuisance-function fits (for \(\mathbb{E}[Y \mid X]\) and \(\mathbb{E}[T \mid X]\)) before estimating the heterogeneous effect.

These three identification routes coincide at the population level under conditional ignorability: each \(\widehat\tau_{\text{method}}\) converges (in its own asymptotic regime) to the same \(\tau(x)\). Differences in finite samples are not pathologies; they are the joint footprint of (i) the identification route’s bias-variance trade-off, (ii) the estimator’s finite-sample behaviour, (iii) the regularization imposed by each method.

What the comparator does not assume:

- The comparator does not require that the inferential origins of the CIs match. The bridge reports a posterior credible interval; `grf` reports a frequentist asymptotic Gaussian CI; EconML reports a bootstrap or asymptotic CI depending on the estimator. Pooling them is conceptually unsound and the comparator does not do it (see §5).
- The comparator does not claim algorithmic equivalence across methods. Two methods producing similar \(\widehat\tau_{\text{method}}\) over the evaluation grid is evidence of consistency in the chosen direction but not a certificate that the methods compute the same quantity in every regime.
- The comparator does not verify identification assumptions per method. Conditional ignorability is the responsibility of the study design; the comparator is descriptive.

# 5. Concordance criterion

Let \(m\) be the number of methods compared (always \(1 + \text{length(methods)}\), counting the bridge as method 0). Let \(\widehat\tau_k = (\widehat\tau_k(x_i^{\text{new}}))_{i = 1}^{n_{\text{new}}}\) denote the per-observation point estimate of method \(k\). For every ordered pair \((k, l)\) the comparator reports three scalar metrics:

\[\begin{aligned} \text{RMSE}_{k,l} &= \sqrt{\frac{1}{n_{\text{new}}}\sum_{i = 1}^{n_{\text{new}}} (\widehat\tau_k(x_i^{\text{new}}) - \widehat\tau_l(x_i^{\text{new}}))^2},\\ \text{Pearson}_{k,l} &= \frac{\sum_{i} (\widehat\tau_k(x_i^{\text{new}}) - \bar\tau_k)(\widehat\tau_l(x_i^{\text{new}}) - \bar\tau_l)}{\sqrt{\sum_{i} (\widehat\tau_k(x_i^{\text{new}}) - \bar\tau_k)^2 \sum_{i}(\widehat\tau_l(x_i^{\text{new}}) - \bar\tau_l)^2}},\\ \text{MAD}_{k,l} &= \frac{1}{n_{\text{new}}}\sum_{i = 1}^{n_{\text{new}}} \big|\widehat\tau_k(x_i^{\text{new}}) - \widehat\tau_l(x_i^{\text{new}})\big|. \end{aligned}\]

with \(\bar\tau_k = (1 / n_{\text{new}})\sum_i \widehat\tau_k(x_i^{\text{new}})\).

These metrics are computed exclusively on `cate_mean`. The native CIs \(\widehat{C}_k\) are reported per method but never aggregated across methods, for the inferential-origin reasons of §4.

Property 5.1 (Symmetry). The matrices `RMSE`, `MAD`, and `Pearson` are symmetric. The diagonal of `RMSE` and `MAD` is identically zero; the diagonal of `Pearson` is identically 1 (by convention; the standard Pearson formula is undefined on zero-variance inputs, and the diagonal entries reflect the trivial self-correlation).

Property 5.2 (Triangle inequality for `RMSE` and `MAD`). As both quantities are \(L^2\) and \(L^1\) norms over a discrete probability measure, they satisfy the triangle inequality: \(\text{RMSE}_{k,l} \le \text{RMSE}_{k,j} + \text{RMSE}_{j,l}\) and similarly for `MAD`. Pearson, by contrast, is not a metric.

Property 5.3 (Invariance). `Pearson` is invariant under affine transformations of each \(\widehat\tau_k\) (translation + positive rescaling). `RMSE` and `MAD` are not. This is the algebraic reason behind the operational guidance of `vop06` §5: high Pearson with high RMSE means agreement in shape but not in level.

We do not define a Mahalanobis-style metric across methods. Such a metric would require pooling per-method posterior or asymptotic covariances, which we explicitly avoid.

# 6. Identifiability per arm under the bridge

The bridge requires the structural compatibility checks listed in `v08b` §3 (matching family, link, AMM level, modulating basis type, anchor, covariate column structure of every AMM component, and absence of the hierarchical regime). Once the bridge is built, those checks have already passed and the comparator does not re-run them.

The external adapters, on the other hand, do not see the AMM spec or the anchor. They consume the stacked dataset and produce \(\widehat\tau_{\text{ext}}\) without inspecting any AMM-specific identifiability machinery. In particular:

- (C7) anti-aliasing of Block 6.5 is not invoked because the bridge has already excluded hierarchical fits (see `v08b` §7).
- The basis-restricted identifiability check of `v01` does not apply to the external adapters; the external methods use their own regularization to ensure well-posedness (the trees of `grf`’s causal forest, the cross-fitting of EconML).

This is a structural feature of the comparison, not a defect. The exercise is to measure how the AMM-side T-learner agrees with methods of different inferential origin, with each method identified under its own canonical assumptions. Forcing the external adapters to verify AMM-specific identifiability would constitute the imposition of an AMM bias on the comparator.

# 7. Limits of the comparison

The comparator is descriptive, not inferential. Three concrete limits worth stating:

(i) No claim of algorithmic equivalence. A small RMSE between two methods is evidence that they agree on the shape and the level of \(\widehat\tau\) on the evaluation grid; it is not a certificate that they compute the same quantity, asymptotically or even in another finite sample. The comparator is silent on the mechanism of agreement.

(ii) No pooling of CIs. The native CIs of each method live on their own inferential plane: posterior credible (bridge), asymptotic Gaussian under honesty (`grf`), Neyman-orthogonal CI (EconML). They are reported per method and never aggregated across methods. A user wanting an overall uncertainty envelope should pick one method as the canonical inferential source.

(iii) Sensitivity to the evaluation grid. Both `RMSE` and `MAD` depend on the choice of \(\mathcal{N}\). A grid that oversamples a region where two methods disagree will inflate the discrepancy; a grid that focuses on the “boring” mid-region will deflate it. The grid is the user’s responsibility; we recommend the same grid used for the bridge’s posterior summaries (the default `newdata` of `gdpar_compare_meta_learners` reuses `bridge$newdata`).

A fourth limit, conceptually distinct, is inherited from `v08b` §9: the T-learner itself is known to suffer from regularization-induced bias under unbalanced samples (Kuenzel et al. 2019, §3.4). That bias travels from the bridge into the comparator unchanged; the comparator can detect it (as disagreement with an X-learner or with DR-style methods) but does not correct it. S-learner, X-learner, and DR-style adapters AMM-side are queued for Block 9 (`v08b` §10 (O4-CATE), (O5-CATE)).

# 8. Minimum reproducible example (CRAN-valid, R-only)

The example below runs only when `grf` is installed and the package’s Bayesian path (Path 1, `cmdstanr`) is operative. It uses synthetic data to keep the chunk fast (~5–10 seconds with `num_trees = 500L`).

    library(gdpar)
    if (requireNamespace("grf", quietly = TRUE) &&
        requireNamespace("cmdstanr", quietly = TRUE)) {

      set.seed(2026L)
      n <- 300L
      df <- data.frame(x1 = rnorm(2L * n))
      df$arm <- rep(c("treat", "ctrl"), each = n)
      df$y <- with(df, ifelse(arm == "treat", 0.5, 0) +
                       0.8 * x1 +
                       rnorm(2L * n, sd = 0.5))
      df_t <- subset(df, arm == "treat"); df_t$arm <- NULL
      df_c <- subset(df, arm == "ctrl");  df_c$arm <- NULL

      fit_t <- gdpar(y ~ x1, amm = amm_spec(a = ~ x1), data = df_t,
                     iter_warmup = 300, iter_sampling = 300, chains = 2)
      fit_c <- gdpar(y ~ x1, amm = amm_spec(a = ~ x1), data = df_c,
                     iter_warmup = 300, iter_sampling = 300, chains = 2)
      newdata <- data.frame(x1 = seq(-2, 2, length.out = 21L))
      bridge <- gdpar_causal_bridge(fit_t, fit_c, newdata = newdata)

      cmp <- gdpar_compare_meta_learners(
        bridge,
        methods = list(grf = gdpar_adapter_grf(num_trees = 500L,
                                                seed = 2026L))
      )
      print(cmp)
      summary(cmp)
    }

The Python-side equivalent is in `vignette("vop06_meta_learner_comparison")` §4 with `eval = FALSE` chunks (the example cannot be CRAN-valid because it requires a working Python environment with `econml`).

# 9. Open questions (O*-CMP)

Each is anchored to a specific Block 9 sub-phase or to an ecosystem decision that exceeds the scope of 8.5.B.

> (O1-CMP) S-learner and X-learner AMM-side as comparators. Both S- and X-learner are queued AMM-side as `gdpar_causal_s_learner` and `gdpar_causal_x_learner` in Block 9 (`v08b` §10 (O4-CATE)). Once implemented, both will become adapters in this comparator (a separate sub-phase, not 8.5.B).

> (O2-CMP) Doubly-robust and DML AMM-side. DR and DML add a propensity-score model that the T-learner avoids. The AMM-side versions are queued as (O5-CATE) of `v08b`. Block 9 will deliver them along with their adapter wrappers.

> (O3-CMP) Overlap diagnostics integrated with the comparator. A canonical overlap plot per arm (propensity-score density, Hosmer-Lemeshow per stratum, ECDF comparison) would belong here. Queued for Block 9 together with the bridge diagnostic battery (`v08b` §10 (O6-CATE)).

> (O4-CMP) Heterogeneous CI calibration. Quantifying the coverage of each method’s native CI on synthetic ground-truth CATEs is the most demanding way to compare methods. The infrastructure is out of scope of 8.5.B (it requires a simulator and a coverage-counting orchestrator); it would be a dedicated sub-phase.

> (O5-CMP) Cross-validation across overlap and effect-size regimes. An adversarial battery (poor overlap, weak effect, heteroskedasticity, non-linear interactions) would stress-test the four canonical regimes of the comparator. Queued for the Block 9 release battery alongside the eBird re-validation of Block 9.

# Appendix A. Notational correspondence with the meta-learner literature

Concept | This vignette | Kuenzel et al. (2019) | Athey-Wager (2019) | Chernozhukov et al. (2018)
--- | --- | --- | --- | ---
Treatment indicator | \(T \in \{0,1\}\) | \(W \in \{0,1\}\) | \(W \in \{0,1\}\) | \(D \in \{0,1\}\)
Covariates | \(X \in \mathcal{X}\) | \(X\) | \(X\) | \(X\)
Outcome | \(Y\) | \(Y\) | \(Y\) | \(Y\)
Potential outcome | \(Y^{(t)}\) | \(Y(w)\) | \(Y(w)\) | \(Y(d)\)
CATE | \(\tau(x)\) | \(\tau(x)\) | \(\tau(x)\) | \(\theta_0(x)\) (heterogeneous case)
Bridge T-learner estimate | \(\widehat\tau_{\text{bridge}}(x)\) | \(\widehat\tau_T(x)\) | (n/a; uses causal forest) | (n/a; uses DML)
External T-learner estimate | \(\widehat\tau_{\text{ext}}(x)\) | \(\widehat\tau_T\) with their own \(M_w\) | \(\widehat\tau_{CF}(x)\) | \(\widehat\theta(x)\) via DML
Identification assumption | conditional ignorability + overlap + SUTVA | same | same + honesty | same + Neyman orthogonality
Nuisance functions | implicit in per-arm AMM | implicit in \(M_w\) | implicit in CF splits | explicit (\(e(X)\), \(g(X)\)) with cross-fitting
CI source | posterior credible | bootstrap or asymptotic on \(M_w\) | asymptotic Gaussian under honesty | Neyman-orthogonal CI on \(\theta\)

The correspondence is structural rather than numerical: when the per-arm base learners coincide and the assumptions match, the estimates agree at the population level; the finite-sample discrepancy is what the comparator measures.

# Appendix B. Implementation notes for additional adapters

The contract of an adapter is documented in detail in `?gdpar_meta_learner_adapter`; the operational walkthrough lives in `vignette("vop06_meta_learner_comparison")` §6. This appendix records the theoretical obligations of a well-formed adapter.

B.1. Signature obligation. `fit_predict_fun` must accept the arguments in the exact order `(X, Y, T, X_newdata, level, seed_run)`. The comparator passes data frames `X` and `X_newdata`; the adapter is responsible for any conversion to numeric matrices or to language-native arrays (e.g. numpy via `reticulate::r_to_py`).

B.2. Output obligation. `fit_predict_fun` must return a list with components `cate_mean` (numeric vector of length \(n_{\text{new}}\)), `cate_ci` (numeric matrix of dimensions \(n_{\text{new}} \times 2\) with columns named `lower` and `upper`, or `NULL`), `state` (opaque object cached for the optional `predict_fun`, or `NULL`), and `notes` (character vector of free-form diagnostics).

B.3. CI obligation. If `native_ci = TRUE` was declared at adapter-construction time, `cate_ci` must be non-`NULL` on every successful run. The comparator does not synthesize a CI when the adapter declares `native_ci = TRUE` and returns `NULL`; this is detected at validation time and the comparator aborts with `gdpar_internal_error`.

B.4. Determinism obligation (when `predict_fun` is provided). If `predict_fun` is non-`NULL`, the adapter must guarantee that `predict_fun(state, X_newdata, level)` reuses the cached fit deterministically (Theorem 3.1). Refits inside `predict_fun` are disallowed; if the method does not expose cheap re-prediction, pass `predict_fun = NULL` and accept the fallback refit.

B.5. Serialization obligation. The `state` slot should ideally survive `saveRDS` round-trip. If it cannot (e.g. because it references a Python object via `reticulate`), the adapter’s `predict_fun` should detect the broken reference and abort with `gdpar_unsupported_feature_error` (the EconML reference adapter follows this pattern).

B.6. Side-effect prohibition. Adapters must not install packages, write files, modify the global environment, or perform network I/O. Resource handling (worker pools, Python sub- processes) is the adapter’s responsibility but must be transparent to the comparator.

B.7. Categorical covariates. Adapters are responsible for handling factor / character covariates. The reference adapters apply `model.matrix(~ . - 1, data = X)` after coercing characters to factors, and they remember the factor levels in `state$template` so re-prediction on a fresh grid aligns with the training design.

# References cited in this addendum

- Athey, S., Tibshirani, J., and Wager, S. (2019). Generalized random forests. The Annals of Statistics, 47(2), 1148-1178.
- Athey, S., and Wager, S. (2019). Estimating treatment effects with causal forests: An application. Observational Studies, 5, 37-51.
- Chernozhukov, V., Chetverikov, D., Demirer, M., Duflo, E., Hansen, C., Newey, W., and Robins, J. (2018). Double/debiased machine learning for treatment and structural parameters. The Econometrics Journal, 21(1), C1-C68.
- Kuenzel, S. R., Sekhon, J. S., Bickel, P. J., and Yu, B. (2019). Metalearners for estimating heterogeneous treatment effects using machine learning. Proceedings of the National Academy of Sciences, 116(10), 4156-4165.
- Rubin, D. B. (1974). Estimating causal effects of treatments in randomized and non-randomized studies. Journal of Educational Psychology, 66(5), 688-701.
- Wager, S., and Athey, S. (2018). Estimation and inference of heterogeneous treatment effects using random forests. Journal of the American Statistical Association, 113(523), 1228-1242.

End of Theoretical Addendum – Block 8.5.B.