Models

A model says how the outcome Y is generated from the linear index η = α(X) + β(X)·T. Picking the model is the same decision as picking a generalized linear model: you choose it by the type of your outcome. The neural network then learns the heterogeneous parameters θ(X) = (α(X), β(X)); the model supplies the loss, the link, and the derivatives that the inference machinery needs.

Every model page below follows the same layout, so once you have read one you can navigate any of them:

1. Source papers that introduce or use the model.

2. When to use it, with concrete examples.

3. Model: the data-generating process and link function.

4. Loss / likelihood: the negative log-likelihood that is minimized.

5. Score and Hessian: the gradient and curvature, and whether they depend on θ or Y.

6. Target: the estimand you get a confidence interval for.

7. Influence function: the residual, weight, and score that build the correction.

8. Algorithm and regime: which of the three regimes applies and why.

9. Usage: runnable Python.

10. Evidence: recovery and coverage numbers, with the source.

11. Diagnostics and pitfalls.

12. References and API links.

Pick your model by outcome type

Outcome	Model	Page
Continuous, unbounded	Linear	Linear
Continuous, with modelled variance	Gaussian	Gaussian
Binary (0/1), logistic link	Logit	Logit
Binary (0/1), normal link	Probit	Probit
Counts	Poisson	Poisson
Overdispersed counts	Negative Binomial	Negative Binomial
Counts with excess zeros	Zero-Inflated Poisson	Zero-Inflated Poisson
Positive, skewed	Gamma	Gamma
Time-to-event	Weibull	Weibull
Extreme values	Gumbel	Gumbel
Censored	Tobit	Tobit
Bounded in (0,1)	Beta	Beta
Discrete choice (3+ options)	Multinomial Logit	Multinomial Logit
Quantiles	Quantile	Quantile
Multiple treatments	Combinatorial	Combinatorial
Before/after, treated/control	Difference-in-Differences	Difference-in-Differences

Which regime does each model land in?

The regime (covered in Estimation) follows from one fact: does the Hessian depend on the parameters? Linear-style squared-error losses have a parameter-free Hessian and use the fast two-way path (Regime B); everything nonlinear estimates the curvature on a three-way split (Regime C), unless the treatment is randomized with a known law (Regime A).

Regime	Models
B (analytic, 2-way)	Linear, Difference-in-Differences
C (estimated, 3-way)	Logit, Probit, Poisson, Negative Binomial, ZIP, Gamma, Weibull, Gumbel, Tobit, Beta, Gaussian, Multinomial, Quantile, Combinatorial
A (computed, randomized)	Any model when the treatment law is known