5. Three Regimes for Lambda

The expected Hessian \(\Lambda(X) = \mathbb{E}[\nabla_\theta^2 \ell(Y, T, \theta) \mid X]\) from the IF correction depends on how the model and data are structured. Farrell, Liang, and Misra (2025) identify three cases that determine how \(\Lambda\) is obtained and how many sample splits are needed.

Regime	When	Lambda method	Cross-fitting
A	RCT, known \(F_T\)	Compute (MC integration)	2-way
B	Linear model	Analytic (closed-form)	2-way
C	Observational + nonlinear	Estimate (ridge)	3-way

Regime A — Randomized experiment with known treatment distribution

If \(T\) is randomly assigned with known distribution \(F_T\), and the Hessian does not depend on \(Y\), then \(\Lambda(X)\) can be computed via Monte Carlo integration:

\[ \Lambda(X) = \int \nabla_\theta^2 \ell(y, t, \theta) \, dF_T(t). \]

This is the simplest case and requires only two-way cross-fitting. Typical examples include A/B tests where treatment assignment probabilities are known by design, and randomized pricing experiments where the price distribution is controlled by the researcher. The inference() API accepts is_randomized=True and a treatment_dist argument to trigger this regime.

Regime B — Linear model

For linear models the Hessian is constant (\(\nabla_\theta^2 \ell = 2\)), so \(\Lambda(X) = 2\,\mathbb{E}[T T' \mid X]\), which can be estimated analytically from the data without a separate estimation step. This avoids three-way splitting entirely, making the method fast and reliable even at moderate sample sizes. The package detects Regime B automatically when model='linear' is specified.

Regime C — Observational data with nonlinear model

In most applied settings — including the H&M application — the Hessian depends on \(\theta\) (e.g. through \(p(1-p)\) in logit or the softmax probabilities in multinomial logit). Since \(\theta\) is estimated, we must estimate \(\Lambda\) via ridge regression, which requires a three-way sample split. The package handles the splitting automatically.

Warning

For Regime C, use ridge (the default), aggregate, or lgbm for \(\Lambda\) estimation. Never use mlp: it attains the highest correlation with the oracle Hessian but produces only ~67% coverage, because high correlation does not guarantee a low-variance estimate. Valid inference needs both.

Regime selection in code

The regime is selected automatically based on the model and data:

from deep_inference import inference
from deep_inference.lambda_.compute import Normal

# Regime A: randomized experiment with known F_T
result = inference(Y, T, X, model='logit', target='beta',
                   is_randomized=True, treatment_dist=Normal(0, 1))

# Regime B: linear model (auto-detected)
result = inference(Y, T, X, model='linear', target='beta')

# Regime C: observational + nonlinear (auto-detected)
result = inference(Y, T, X, model='logit', target='beta')