Pricing

DML

Two models: One regresses Y on X, another one regresses T on X

Obtain the residuals from the T regression and the Y regression, then regress the latter on the former.

It does not require a randomized experiment and theoretically removes the simultaneous influence of X on both T and Y.

The underlying principle is Neyman orthogonality.


PSM

PSM removes bias by mimicking a randomized experiment.

But PSM itself introduces some bias, so in theory it performs the worst.


Meta-learner

A randomized experiment ensures that T is independent of X.

After that, it becomes a standard machine-learning regression problem.

Train a model to estimate, for any given X, the difference Y(T=1) – Y(T=0), i.e., the treatment effect.

  1. S-learner

    Put both T and X into a single model for training.

    Prediction is direct: simply feed X and T into the model.

  2. T-learner

    Train two separate models: one for the treated group and one for the control group.

    For a given X, predict Y under T=1 and Y under T=0, then take the difference.

  3. X-learner

    Train two models that regress X → Y separately for the control and treatment groups.

    Then cross-predict: use the control model to predict the treatment-group data, etc.

    The predicted value minus the true Y gives the ITE; then fit two more models to regress these pseudo-ITEs to smooth noise.

    For inference, weight the predictions using the probability of receiving treatment P(T=1 X) (the propensity).

    You need another model to estimate the propensity.

    The weighting addresses treatment imbalance.

    In total, five models are trained; only three are used in prediction.

    X-learner = T-learner + pseudo-ITE estimation + propensity weighting (The latter two components can be used independently.)

  4. R-learner

    Train one model to regress Y on X → outcome model. Train another model to regress T on X → propensity model.

    Then a third model regresses the residual of Y on the residual of T, using a loss function based on the MSE of: (residual of T * treatment effect) vs. (residual of Y).

    This is essentially equivalent to DML.