Module 6: Causal Forest

.title[
# Module 6: Causal Forest
]
.subtitle[
## Honest Splitting, Asymptotics, and HTE Diagnostics
]

---

# Course Map

<table>
<tr><th>#</th><th>Module</th><th>Status</th></tr>
<tr><td>1</td><td><a href="../module-01/slides.html">TWFE Diagnosed: Goodman-Bacon and the Zoo of 2x2s</a></td><td>done</td></tr>
<tr><td>2</td><td><a href="../module-02/slides.html">Heterogeneity-Robust DiD: CS, SA, BJS, dCDH</a></td><td>done</td></tr>
<tr><td>3</td><td><a href="../module-03/slides.html">Honest DiD: Sensitivity Bounds for Parallel Trends</a></td><td>done</td></tr>
<tr><td>4</td><td><a href="../module-04/slides.html">Synthetic Control: Estimator, Inference, Variants</a></td><td>done</td></tr>
<tr><td>5</td><td><a href="../module-05/slides.html">Synthetic DiD and the Bridge from SC to DiD</a></td><td>done</td></tr>
<tr><td><b>6</b></td><td><b>Causal Forest: Honest Splitting and Asymptotics</b> <i>(you are here)</i></td><td>current</td></tr>
<tr><td>7</td><td><a href="../module-07/slides.html">Policy Learning: From HTE to Deployment Rules</a></td><td>done</td></tr>
<tr><td>8</td><td><a href="../module-08/slides.html">Matrix Completion and the Modern Panel Toolbox</a></td><td>done</td></tr>
</table>

---

# This Module

The experimentation-refresher tour showed *that* a causal forest estimates
`$\tau(x)$` with honest trees. This module is the formal treatment.

**You will be able to:**

1. State the honesty split and explain why honesty plus *subsampling*
   (not bootstrap) yields a pointwise central limit theorem for
   `$\hat\tau(x)$`, and list the conditions.
2. Read a forest as an adaptive-kernel solver of a local moment condition,
   and say why that view generalizes to quantile, IV, and survival forests.
3. Explain the R-learner orthogonalization `grf` runs, and why it helps even
   in a clean RCT.
4. Run the full diagnostic stack live: calibration test, best linear
   projection, RATE, and (in simulation) coverage.

**Running application:** a randomized push-notification experiment on 6000
drivers. Outcome: weekly trips (read: engagement or revenue per customer in
a retail setting). Effect heterogeneity by city density, tenure, and
peak-hour share; `rating` is a deliberate nuisance covariate. The
experiment ran and gave you an ATE; the decision needs more than that
average.

---

# The Estimand and the Prize

The conditional average treatment effect:

`$$\tau(x) = \mathbb{E}\!\left[Y(1) - Y(0) \mid X = x\right]$$`

Never observed, not even noisily: no unit reveals both potential outcomes.
A forest estimates it through *local* treated-minus-control comparisons.

A T-learner, boosted trees, a neural net: a prediction `$\hat\tau(x)$`. No
honest uncertainty. You cannot say whether a high `$\hat\tau(x)$` is signal or
an artifact of where the model split.
]

A **pointwise confidence interval**: `$\hat\tau(x) \pm 1.96\,\widehat{\text{se}}(x)$`
with asymptotically valid coverage at each fixed `$x$`. That interval is the
entire reason to prefer a forest over an off-the-shelf regressor.
]

.highlight-box[
The rest of the module is about what makes that interval *valid*: honesty
kills the bias, subsampling delivers normality, and the infinitesimal
jackknife estimates the variance.
]

---

# Honest Trees: the Split-Sample Trick

An **adaptive** tree uses the same data to choose splits and to estimate
leaf effects. The split is chosen to make leaves look different, so the leaf
estimate reuses the noise that placed the split: a within-sample winner's
curse. Apparent heterogeneity appears even when `$\tau(x)$` is constant.

**Honesty** partitions each tree's subsample into two disjoint halves:

.small[
| Half | Symbol | Job |
|---|---|---|
| Splitting | `$\mathcal{J}$` | decide the tree structure: covariate + threshold at every node |
| Estimation | `$\mathcal{I}$` | dropped down the finished tree; leaf effect uses **only** these points |
]

Because `$\mathcal{I}$` never touched the split placement, the leaf estimate is
conditionally unbiased for the average effect over that leaf. The forest
averages honest trees over many subsamples.

.blue-box[
The cost is efficiency: each tree estimates from only half its subsample. You
trade *variance* (which widens the CI) for *bias* (which would invalidate it).
]

<a href="#honesty-proof" class="nav-btn">why this gives a CLT</a>

---

# Subsampling, Not Bootstrap

Wager and Athey build the forest from **subsamples** of size `$s$` drawn
without replacement. This is an inferential choice, not a computational one.

- A forest of subsampled trees is a **U-statistic**: an average of a
  symmetric kernel over subsets of the data.
- The Hajek projection of a U-statistic is asymptotically normal, and its
  variance is estimable by an **infinitesimal jackknife**.
- Bootstrap resampling breaks the clean U-statistic structure that both the
  CLT and the variance estimator rely on.

.small[
The subsample size must grow slowly: `$s = n^{\beta}$` with `$\beta \in (0,1)$`
and `$\beta$` bounded below, so `$s/n \to 0$`. Trees must be
`$\alpha$`-regular: every split leaves at least a fraction `$\alpha$` of the
parent on each side, so no leaf collapses onto a vanishing neighborhood.
]

.highlight-box[
`grf` returns the jackknife variance as `variance.estimates`. That single
vector is what turns a prediction machine into an inference machine.
]

---

# Pointwise Normality: What You Get, What You Do Not

.small[
**Theorem (Wager and Athey 2018, informal).** Under honesty,
`$\alpha$`-regularity, an admissible subsample rate `$s = n^\beta$`, overlap
`$0 < \eta \le e(x) \le 1-\eta$`, and Lipschitz smoothness of `$\tau$` and the
nuisances,
]

`$$\frac{\hat\tau(x) - \tau(x)}{\sqrt{\operatorname{Var}[\hat\tau(x)]}}
\;\xrightarrow{d}\; \mathcal{N}(0, 1).$$`

Honesty makes the bias asymptotically negligible *relative to* the standard
error, so the ratio centers at zero and `$\hat\tau(x) \pm 1.96\,\widehat{\text{se}}$`
covers.

.pull-right[
.highlight-box[
**You do NOT get:** uniform inference. No confidence *band* over all `$x$`; no
"the effect is positive everywhere". Do not union pointwise CIs into a
whole-function claim.
]
]

For questions about the *shape* of `$\tau(\cdot)$`, use the aggregate
diagnostics later in this deck (calibration, BLP, RATE), never a naive union
of pointwise intervals.

---

# Generalized Random Forests: Forest as Adaptive Kernel

Athey, Tibshirani and Wager (2019) reframe the forest as a producer of
**data-adaptive weights**. For a target `$x$`, tree `$b$` drops `$x$` to leaf
`$L_b(x)$`; training point `$i$` gets weight `$1/|L_b(x)|$` if it shares that leaf.
Average over trees:

`$$\alpha_i(x) = \frac{1}{B}\sum_{b=1}^{B}
\frac{\mathbb{1}\{X_i \in L_b(x)\}}{|L_b(x)|},
\qquad \sum_i \alpha_i(x) = 1.$$`

The forest is a **weighted nearest-neighbor** method whose metric is learned:
points that keep landing in `$x$`'s leaf are its neighbors, and the splitting
rule decides "near" in the directions that matter for the *effect*, not the
outcome level.

.highlight-box[
This is the unifying idea. Every GRF estimate, causal, quantile, IV, is a
*local* estimate using these same `$\alpha_i(x)$` weights. `get_forest_weights`
returns the full matrix.
]

<a href="#adaptive-nn-detail" class="nav-btn">who gets weight, visualized</a>

---

# The Local Moment Condition

GRF estimates `$\theta(x)$` defined by a **local estimating equation**, solved
with the forest weights:

`$$\mathbb{E}\!\left[\psi_{\theta(x)}(O_i)\mid X_i = x\right] = 0
\;\;\Longrightarrow\;\;
\hat\theta(x) = \arg\min_\theta
\left\|\sum_i \alpha_i(x)\,\psi_\theta(O_i)\right\|.$$`

For the causal forest the score is the **residual-on-residual** moment

`$$\psi_\tau(O_i) = \big(Y_i - \hat m(X_i) - \tau\,(W_i - \hat e(X_i))\big)
\,(W_i - \hat e(X_i)),$$`

so the local solution is a weighted regression of residualized outcome on
residualized treatment: a **local Robinson (1988) partially linear**
estimator.

.blue-box[
Swap `$\psi$` and you get a different forest with the *same* weights: the IV
moment gives instrumental forests, the check function gives quantile forests,
a partial-likelihood score gives survival forests. That is why it is
"generalized".
]

<a href="#grf-derivation" class="nav-btn">gradient-based splitting</a>

---

# Local Centering and the R-Learner

Before growing the causal forest, `grf` **orthogonalizes**: fit
`$\hat m(x) = \mathbb{E}[Y\mid X=x]$` and `$\hat e(x) = \mathbb{E}[W\mid X=x]$`
with separate regression forests, then grow on the residuals. This minimizes
the **R-learner** objective (Nie and Wager 2021):

`$$\sum_i \big[(Y_i - \hat m(X_i)) - \tau(X_i)(W_i - \hat e(X_i))\big]^2.$$`

The estimate becomes **Neyman-orthogonal**: first-order insensitive to small
errors in `$\hat m$` and `$\hat e$`. Two payoffs.

- Confounding by a smooth prognostic signal is projected out through
  `$\hat m$`, so splits chase effect *heterogeneity*, not the main effect.
- Even in an RCT with `$e(x)\equiv 0.5$`, chance covariate imbalance means the
  finite-sample association between `$W$` and `$X$` is not exactly zero. Centering
  on `$\hat m(x)$` strips the prognostic variance out of `$Y$` and sharpens
  `$\hat\tau(x)$`.

.highlight-box[
In a randomized experiment, set `W.hat = 0.5`: the propensity is *known*, so
the forest should not spend data estimating it or risk injecting propensity
noise.
]

<a href="#rlearner-detail" class="nav-btn">why orthogonality protects</a>

---

# The Application: Fit the Forest

Shared HTE DGP: 6000 drivers, randomized push `$w$`, weekly trips `$y$`. True
`$\tau = 1.5\,\text{density} - 0.02\,\text{tenure} + 2.0\,\text{density}\cdot\text{peak share}$`;
`rating` is nuisance.

```r
X  <- as.matrix(d[, c("density", "tenure", "peak_shr", "rating")])
cf <- causal_forest(X, Y = d$y, W = d$w,
                    W.hat = 0.5,        # KNOWN propensity: it's an RCT
                    num.trees = 2000, seed = 42)
average_treatment_effect(cf)            # AIPW: doubly robust ATE
```

<table>
 <thead>
  <tr>
   <th style="text-align:right;"> estimate </th>
   <th style="text-align:right;"> std.err </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:right;"> 0.811 </td>
   <td style="text-align:right;"> 0.054 </td>
  </tr>
</tbody>
</table>

.pull-left[
.small[
True mean effect: **0.79** trips/wk.
AIPW ATE: **0.81** (se 0.05).
The forest recovers the average.
]
]

.pull-right[
.blue-box[
The AIPW score (augmented inverse-propensity weighting) combines the
outcome model and the propensity into a doubly-robust average: consistent if
*either* is right.
]
]

---

# The Same Problem at an Online Retailer

- The randomized signup discount is the experiment: `$W_i$` = offered discount, `$Y_i$` = 12-month spend, `$X_i$` includes account tenure, pre-period spend, primary purchase category, and device type.
- Causal forest estimates `$\hat\tau(x)$`: the conditional average treatment effect of the signup offer varies substantially across the covariate space.
- Honest splitting uses separate subsamples for tree building and leaf estimation, preventing the winner's-curse inflation that occurs when the same data selects the subgroup boundaries and evaluates the treatment effect within them.
- High pre-period spend and long tenure both moderate the offer's effect: already-heavy members may join regardless of the discount (never-takers) and respond little at the margin.
- The per-customer `$\hat\tau(x)$` estimate is the input to the policy-learning problem in Module 7.

---

# Does `$\hat\tau(x)$` Track the Truth?

Points hug the 45-degree line; the spread is honest estimation noise, not
bias. Density (color) orders cleanly, exactly the DGP's dominant driver.

---

# Pointwise Intervals and Their Coverage

`predict(cf, estimate.variance = TRUE)` returns the jackknife variance. On a
**held-out grid** (fresh 2000 drivers) we can check coverage against the
known `$\tau$`.

.small[
Held-out coverage: **0.91** at nominal 0.95. Close, and the exercise verifies it over many refits.
]

---

# Where the Effect Lives

The forest reproduces the DGP's structure without being told the functional
form: strong positive gradient in density, mild negative slope in tenure.

---

# Diagnostic 1: the Calibration Test

`test_calibration` regresses the effect on two constructed regressors: the
**mean** forest prediction and the **differential** (demeaned) prediction.

```r
test_calibration(cf)
```

.small[
- `mean.forest.prediction` `$\approx$` **1**: the forest's average effect is calibrated (coefficient near 1).
- `differential.forest.prediction` `$\approx$` **1.05**, highly significant: the forest's *ranking* of who responds carries real signal.
]

.highlight-box[
This is the omnibus "is there any heterogeneity?" test. A differential
coefficient near zero or insignificant means stop: report the ATE, do not
model heterogeneity that is not there.
]

---

# Diagnostic 2: Best Linear Projection

`best_linear_projection` reports the best linear approximation to `$\tau(x)$`
with robust SEs, the interpretable summary you show a stakeholder.

```r
best_linear_projection(cf, X)   # X = density, tenure, peak_shr, rating
```

<table>
 <thead>
  <tr>
   <th style="text-align:left;"> term </th>
   <th style="text-align:right;"> Estimate </th>
   <th style="text-align:right;"> Std. Error </th>
   <th style="text-align:right;"> t value </th>
   <th style="text-align:left;"> p value </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> (Intercept) </td>
   <td style="text-align:right;"> 1.103 </td>
   <td style="text-align:right;"> 1.318 </td>
   <td style="text-align:right;"> 0.84 </td>
   <td style="text-align:left;"> 0.40 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> density </td>
   <td style="text-align:right;"> 2.224 </td>
   <td style="text-align:right;"> 0.182 </td>
   <td style="text-align:right;"> 12.23 </td>
   <td style="text-align:left;"> &lt;0.001 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> tenure </td>
   <td style="text-align:right;"> -0.016 </td>
   <td style="text-align:right;"> 0.003 </td>
   <td style="text-align:right;"> -5.03 </td>
   <td style="text-align:left;"> &lt;0.001 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> peak_shr </td>
   <td style="text-align:right;"> 0.970 </td>
   <td style="text-align:right;"> 0.267 </td>
   <td style="text-align:right;"> 3.63 </td>
   <td style="text-align:left;"> &lt;0.001 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> rating </td>
   <td style="text-align:right;"> -0.322 </td>
   <td style="text-align:right;"> 0.279 </td>
   <td style="text-align:right;"> -1.15 </td>
   <td style="text-align:left;"> 0.25 </td>
  </tr>
</tbody>
</table>

.small[
- density **2.22**, tenure **-0.016**, peak_shr **0.97**: signs and magnitudes track the DGP.
- `rating` coefficient is indistinguishable from zero, `$p = 0.25$`, so the nuisance covariate outs itself.
]

.blue-box[
BLP is a projection, not the truth: it is the best *linear* read on a
nonlinear `$\tau(x)$`. Use it to communicate direction and rough magnitude,
not as the model.
]

---

# Diagnostic 3: RATE and the TOC Curve

"Is the heterogeneity *targetable*?" Treat the units with the highest
`$\hat\tau$` first; plot the benefit among the top fraction `$q$` minus the ATE.
That is the
**Targeting Operator Characteristic** curve; its area is the **RATE**.

.small[
RATE with the forest's own `$\hat\tau$` priority: **0.59** (se 0.05). With the useless `rating` priority it collapses to **-0.05** (se 0.05): a null.
]

<a href="#rate-detail" class="nav-btn">AUTOC vs Qini</a>

---

# The Same Objects Under Industry Names

.small[
| Econometrics term | Industry name | What it is here |
|---|---|---|
| CATE `$\tau(x)$` | uplift | the causal forest's estimate (this module) |
| RATE / AUTOC with Qini weighting | Qini curve / uplift curve | ranking quality of a targeting score |
| treat the top fraction `$q$` by predicted effect | incremental targeting | the TOC curve's x-axis |
| ATE | incrementality / lift | what an A/B test reports |
]

.blue-box[
In an uplift-modeling interview these are the same objects under different
names. Say so explicitly: "uplift modeling" is CATE estimation, a "Qini
curve" is a RATE/TOC curve, and "incremental targeting" is thresholding
`$\hat\tau(x)$` against a cost, exactly Module 7.
]

---

# Diagnostic 4: the Honesty Ablation

Refit with `honesty = FALSE` and evaluate both forests on the held-out grid:

<table>
 <thead>
  <tr>
   <th style="text-align:left;"> forest </th>
   <th style="text-align:right;"> CATE RMSE </th>
   <th style="text-align:right;"> mean SE </th>
   <th style="text-align:right;"> 95% coverage </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> honest </td>
   <td style="text-align:right;"> 0.277 </td>
   <td style="text-align:right;"> 0.245 </td>
   <td style="text-align:right;"> 0.911 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> adaptive </td>
   <td style="text-align:right;"> 0.595 </td>
   <td style="text-align:right;"> 0.987 </td>
   <td style="text-align:right;"> 0.997 </td>
  </tr>
</tbody>
</table>

.small[
- Adaptive **doubles** the CATE RMSE (0.6 vs 0.28): the winner's curse re-enters.
- Its pointwise SEs **balloon** (0.99 vs 0.25): unstable trees inflate the jackknife variance.
]

.highlight-box[
Adaptive intervals are not "too narrow", they are *uninformative*: badly
centered and four times too wide. Honesty buys a tight, well-centered
interval, which is the entire point. The exercise shows the winner's curse in
isolation on a constant-effect DGP.
]

---

# Tuning `grf`: What Moves Results

.small[
| Parameter | Default | What it controls | Moves results? |
|---|---|---|---|
| `num.trees` | 2000 | Monte Carlo error, variance stability | yes for the *SE*; use 2000+ |
| `honesty.fraction` | 0.5 | share of subsample placing splits | changes *inference*, not just fit |
| `min.node.size` | 5 | leaf-size floor, smoothing | yes, main bias-variance knob |
| `sample.fraction` | 0.5 | subsample size `$s/n$`, CLT rate | yes |
| `mtry` | `$\sqrt{p}+20$` | covariates tried per split | mainly with many junk covariates |
| `honesty.prune.leaves` | TRUE | drop leaves empty of treated or control | rarely, but keep on |
]

`tune.parameters = "all"` cross-validates against the R-learner debiased
error. In practice `min.node.size` and `sample.fraction` do the real work;
`mtry` matters when noise covariates dominate; turning honesty off is not
tuning, it is abandoning valid inference.

<a href="#tuning-detail" class="nav-btn">the debiased-error objective</a>

---

# Practitioner Checklist

.small[
| Step | Call | What you learn |
|---|---|---|
| 1. Set the propensity | `W.hat = 0.5` in an RCT | do not estimate a known quantity |
| 2. Fit with enough trees | `causal_forest(num.trees = 2000)` | stable variance estimates |
| 3. Sanity-check the average | `average_treatment_effect()` | AIPW ATE matches the experiment |
| 4. Test for heterogeneity | `test_calibration()` | omnibus: is there any signal? |
| 5. Summarize it | `best_linear_projection()` | direction + rough magnitude |
| 6. Quantify targetability | `rank_average_treatment_effect()` | RATE: worth personalizing? |
| 7. Report uncertainty | `estimate.variance` CIs | pointwise only, never a band |
]

.blue-box[
**The interview arc:** ATE from AIPW, heterogeneity confirmed by the
calibration test, summarized by BLP, its business value quantified by RATE.
Then Module 7 turns `$\hat\tau(x)$` into a deployment rule.
]

---

# Interview Questions

.small[
| Question | Core of a strong answer |
|---|---|
| "Why do we need *honest* trees, and what does it cost?" | Adaptive trees reuse the noise that placed a split to estimate the leaf: a winner's curse that invents heterogeneity. Honesty sample-splits (structure on one half, leaf effects on the other), removing the bias so the CLT holds. Cost is efficiency: half the subsample estimates, so wider CIs. |
| "The forest gives a CI at each x. Can I say the effect is positive everywhere?" | No. Wager-Athey give *pointwise* normality, not uniform. A union of pointwise CIs is not a confidence band. For whole-function claims use the calibration test, BLP, or RATE. |
| "It's a randomized experiment. Why bother orthogonalizing on m(x) and e(x)?" | Local centering (the R-learner) makes the estimate Neyman-orthogonal. It strips the prognostic-covariate variance out of Y, sharpening tau-hat, and protects against chance covariate imbalance even when e(x) is known. Set W.hat to the design propensity. |
| "How do you know the heterogeneity is real and not overfitting?" | `test_calibration`: a significant differential-forest coefficient near 1. Then `rank_average_treatment_effect`: a positive RATE means the ranking is targetable; feed it a nuisance covariate and it returns a null, a built-in placebo. |
| "Forest vs a two-model T-learner for CATE?" | The T-learner differences two noisy regressions and has no honest inference. The causal forest targets tau directly via a residual-on-residual moment, honest splitting gives valid pointwise CIs, and orthogonalization guards against confounding. Use the T-learner only as a quick baseline. |
| "So can I tell leadership the feature works for everyone in segment X?" | No: pointwise CIs are not a uniform band, so a segment-level claim needs more than reading off one `$\hat\tau(x)$`. Use the best linear projection or RATE to make a defensible segment-level statement. |
]

---

# Going Deeper

.small[
| Paper | What it adds |
|---|---|
| Wager and Athey (2018), *JASA* | Honest forests: the pointwise CLT, subsampling as a U-statistic, the infinitesimal-jackknife variance. |
| Athey, Tibshirani and Wager (2019), *Annals* | GRF: forests as adaptive kernels, the local moment condition, gradient-based splitting. Unifies causal, quantile, IV, survival. |
| Nie and Wager (2021), *Biometrika* | The R-learner objective and quasi-oracle error bounds behind `grf`'s local centering. |
| Yadlowsky et al. (2021) | RATE / TOC: rank-weighted ATE for evaluating (and comparing) targeting rules with a valid SE. |
| Kunzel et al. (2019), *PNAS* | Meta-learners (S/T/X-learner): the baselines a causal forest improves on. |
]

**Next module:** turn `$\hat\tau(x)$` into an interpretable, cost-aware
deployment rule with `policy_tree`, and evaluate it off-policy.

**Drill:** `exercise.R` hand-codes honest vs adaptive partition estimation to
expose the winner's curse, then verifies `grf`'s pointwise CI coverage over
many refits.

---

# Backup Slides

---

# Backup: Why Honesty Gives a CLT

.small[
Write the forest estimate as an average of a tree kernel `$T$` over subsamples
`$s$` of the `$n$` points:

`$$\hat\tau(x) = \binom{n}{s}^{-1} \sum_{|S| = s} T(x;\, S),$$`

a **U-statistic** of order `$s$`. Its Hajek projection is the sum of leave-one-in
first-order terms, and Hoeffding's decomposition gives asymptotic normality
provided the higher-order terms vanish, which holds when `$s/n \to 0$` at the
admissible rate.

**Where each condition earns its keep:**

| Condition | What it prevents |
|---|---|
| Honesty | leaf estimate correlated with split placement, i.e. first-order bias |
| `$\alpha$`-regularity | leaves collapsing onto a vanishing neighborhood (inconsistency) |
| `$s = n^\beta$`, `$\beta$` in range | U-statistic remainder dominating the linear term |
| Overlap `$\eta \le e(x) \le 1-\eta$` | leaves with no treated or no control units |
| Lipschitz `$\tau,\, m,\, e$` | leaf-averaging bias shrinking faster than the SE |

The bias is `$o(\text{se})$`, so the standardized statistic centers at zero. The
variance is estimated by the infinitesimal jackknife (`variance.estimates`).
Consistency of *that* estimator is the second theorem, and needs `$B$` large,
hence "use enough trees".
]

---

# Backup: Who Gets Weight, Visualized

.small[
Pick one dense high-peak target driver. `get_forest_weights` returns
`$\alpha_i(x)$` for every training point; the forest's "neighbors" of `$x$` are
the points with large weight, clustered near `$x$` in the directions the splits
found relevant (density, peak share), and indifferent to `rating`.
]

---

# Backup: Gradient-Based Splitting

.small[
Solving the local moment `$\sum_i \alpha_i(x)\psi_\theta(O_i) = 0$` exactly at
*every* candidate split would be too slow. GRF uses a **one-step (Newton)
approximation**.

At a parent node with estimate `$\hat\theta_P$`, compute the score gradient
`$A_P = \frac{1}{n_P}\sum_{i\in P}\nabla_\theta \psi_{\hat\theta_P}(O_i)$` and
form a **pseudo-outcome** (influence function) for each observation:

`$$\rho_i = -A_P^{-1}\,\psi_{\hat\theta_P}(O_i).$$`

For the causal forest this is the familiar
`$\rho_i \propto (W_i - \bar W_P)\big[(Y_i - \bar Y_P) - (W_i - \bar W_P)\hat\tau_P\big]$`.
Then split to **maximize the between-child variance of `$\rho_i$`**, exactly a
CART regression split on the pseudo-outcomes:

`$$\Delta(j, c) \;\approx\; \frac{n_L n_R}{n_P^2}\,(\bar\rho_L - \bar\rho_R)^2.$$`

This is the fast label-based approximation to "split where `$\theta(x)$` changes
most". Leaf estimates are then re-solved exactly (and honestly) on the
estimation half. Same `$\rho$` machinery with a different `$\psi$` gives the
splitting rule for quantile, IV, and survival forests.
]

---

# Backup: Why Orthogonality Protects

.small[
Partition the outcome as `$Y = m(X) + (W - e(X))\tau(X) + \varepsilon$`, where
`$m(x) = \mathbb{E}[Y\mid X=x]$` is the **prognostic** part. Residualizing both
sides on `$\hat m$` and `$\hat e$` leaves the Robinson moment

`$$\tilde Y_i = \tilde W_i\,\tau(X_i) + \varepsilon_i,
\qquad \tilde Y_i = Y_i - \hat m(X_i),\;\; \tilde W_i = W_i - \hat e(X_i).$$`

The R-learner loss `$\sum_i(\tilde Y_i - \tau(X_i)\tilde W_i)^2$` has a score
whose derivative with respect to the nuisances `$(m, e)$` is **zero in
expectation at the truth**. So an `$O(h)$` error in `$\hat m$` or `$\hat e$`
contributes only `$O(h^2)$` to the `$\tau$` estimate: **Neyman orthogonality**.

Two consequences:

1. **Confounding by a smooth signal** (a covariate that shifts `$Y$` and
   correlates with `$W$`) enters through `$m$` and is differenced out, so the
   forest does not mistake a large main effect for heterogeneity.
2. **In an RCT**, `$e(x) = 0.5$` exactly, but a finite sample has chance
   imbalance, so `$\widehat{\text{Cov}}(W, X) \ne 0$`. Centering on `$\hat m(x)$`
   removes the prognostic-covariate variance from `$\tilde Y$`, so `$\hat\tau(x)$`
   is lower-variance and robust to that imbalance. Set `W.hat = 0.5` so the
   forest uses the true propensity and adds no estimation noise from `$\hat e$`.
]

---

# Backup: The HTE DGP

```r
make_hte_data <- function(n = 6000, seed = 42) {
  set.seed(seed)
  tibble(
    density  = runif(n),                        # city density percentile
    tenure   = pmin(rexp(n, 1 / 18), 90),       # months on platform
    peak_shr = rbeta(n, 2, 3),                  # share of hours in peak
    rating   = pmin(pmax(rnorm(n, 4.7, 0.2), 3.5), 5),   # NUISANCE
    w        = rbinom(n, 1, 0.5),               # randomized push (e = 0.5)
    tau      = 1.5 * density - 0.02 * tenure + 2.0 * density * peak_shr,
    y        = 10 + 5 * density + 0.05 * tenure + 3 * peak_shr +
               tau * w + rnorm(n, 0, 2)
  )
}
```
]

.small[
- `tau` is the **true CATE**: rises with density, falls slowly with tenure,
  and the `density * peak_shr` term makes peak-hour drivers in dense cities
  respond most. Mean effect `$\approx 0.79$` trips/wk.
- `rating` enters neither `tau` nor `y`: a pure nuisance, there to be caught
  by BLP and RATE.
- The big prognostic term `5 * density` in `y` (with none in the effect
  slope for its own sake) is exactly what local centering removes.
]

---

# Backup: AUTOC vs Qini, and Reading the TOC

.small[
The TOC at fraction `$q$` is
`$\text{TOC}(q) = \mathbb{E}[\tau(X)\mid \hat\pi(X) \ge \hat\pi_{(q)}] - \mathbb{E}[\tau(X)]$`,
the average effect among the top fraction `$q$` of prioritized units minus the
ATE. A curve that starts high and decays to zero at `$q = 1$` means the priority
successfully ranks responders.

**Two weightings**, both from `rank_average_treatment_effect`:

| Target | Weight on `$\text{TOC}(q)$` | Emphasis |
|---|---|---|
| `AUTOC` | `$1/q$` (log) | rewards concentrating effect in a *small* top slice |
| `QINI` | `$q$` (linear) | rewards total gain across the whole ranking |

AUTOC is the sharper test when only a minority carry large effects; Qini
tracks aggregate value. Both are estimated on held-out data (honest split
under the hood) with a standard error, so "is the ranking better than random?"
is a `$t$`-test: `estimate / std.err`.

**The built-in placebo:** feed a covariate you believe is irrelevant (here
`rating`) as the priority. A near-zero, insignificant RATE confirms the
diagnostic is not manufacturing targetability out of noise, and that the
*real* priority's positive RATE is signal.
]

---

# Backup: The Debiased-Error Objective

.small[
`tune.parameters` cannot cross-validate on CATE error (the truth is
unobserved). Instead it minimizes an estimate of the **R-learner objective**
on held-out folds, the same loss the forest is built to minimize:

`$$\widehat{\text{err}}(\text{params}) =
\frac{1}{n}\sum_i \Big[(Y_i - \hat m(X_i)) -
\hat\tau_{-i}(X_i)\,(W_i - \hat e(X_i))\Big]^2,$$`

using out-of-bag `$\hat\tau_{-i}$`. Because `$\tilde Y = \tilde W\tau + \varepsilon$`,
a `$\hat\tau$` closer to the truth lowers this residual in expectation, so it is
a valid proxy for CATE accuracy without ever seeing `$\tau$`.

**What tuning actually changes here:** `min.node.size` and `sample.fraction`
trade the leaf-averaging bias against variance; over-large leaves flatten real
heterogeneity (the calibration differential coefficient drops below 1),
over-small leaves inflate the SE. `mtry` matters once junk covariates (like
`rating`) compete for splits. `num.trees` barely moves the point estimate but
is essential for a stable `variance.estimates`, so never economize on trees
when you need the CIs.

**Rule of thumb:** tune, but if `test_calibration` already shows a
mean coefficient near 1 and a strong differential coefficient, the defaults
are usually fine and tuning buys little.
]