1. Setup

Note

This is the first of seven theory pages. They build on each other — read them top to bottom. Use the Next button at the bottom of each page to follow the phases in order.

This section lays out the Farrell–Liang–Misra (FLM) framework as a practitioner would encounter it. We describe the structural model, define target functionals, explain why naive neural network inference fails, present the influence function correction, and classify problems into three “regimes” that determine how a key quantity — the expected Hessian \(\Lambda(x)\) — is handled.

The structural model

Following Farrell, Liang, and Misra (2021, 2025), we observe \(n\) i.i.d. observations \((Y_i, T_i, X_i)\) where

  • \(Y_i\) is the outcome (chosen product, sales quantity, duration),

  • \(T_i\) is the treatment or “action” variable (price, product attributes, dosage),

  • \(X_i \in \mathbb{R}^{d_x}\) are individual covariates that drive heterogeneity.

A structural model specifies a loss function \(\ell(y, t, \theta)\) parameterized by structural parameters \(\theta \in \mathbb{R}^{d_\theta}\). The true parameters for individual \(i\) solve

\[ \theta^*(X_i) = \arg\min_\theta \; \mathbb{E}[\ell(Y, T, \theta) \mid X = X_i]. \]

The key move of the framework is to let the structural parameters be functions of the covariates \(X\), rather than fixed constants. A deep neural network learns the map \(X \mapsto \theta^*(X)\), capturing rich heterogeneity while the structural loss \(\ell\) preserves economic interpretability.

Concrete example (H&M application)

In the H&M application, \(Y_i \in \{0, 1, \ldots, J-1\}\) is the chosen product from a set of \(J\) alternatives, \(T_i \in \mathbb{R}^{J \times K}\) contains the attributes of each alternative (log-price, style embeddings), and \(X_i \in \mathbb{R}^{64}\) is the consumer’s learned embedding. The structural parameters

\[ \theta^*(X_i) = \big(\beta_{\text{price}}(X_i),\, \beta_{\text{style},1}(X_i),\, \ldots,\, \beta_{\text{style},5}(X_i)\big) \]

are the consumer’s taste parameters, and the loss is the multinomial logit negative log-likelihood:

\[ \ell(y, t, \theta) = -V_{y} + \log \sum_{j=0}^{J-1} e^{V_j}, \qquad V_j = x_j' \theta. \]

Available structural models

The framework is not limited to discrete choice. The table below lists the structural models currently available in the deep-inference package, along with their loss functions and parameter dimensions.

Model

Loss \(\ell(y,t,\theta)\)

Link

\(d_\theta\)

Linear

\((y - \alpha - \beta t)^2\)

Identity

2

Logit

\(\log(1+e^\eta) - y\eta\)

Logistic

2

Poisson

\(e^\eta - y\eta\)

Log

2

Gamma

\(y/\mu + \log \mu\)

Log

2

Multinomial

\(-V_y + \log\sum_j e^{V_j}\)

Softmax

\((J\!-\!1)+K\)

Custom

Any differentiable \(\ell(y,t,\theta)\)

Any

User-specified

Here \(\eta = \alpha + \beta t\) and \(\mu = g^{-1}(\eta)\). The package includes over a dozen families (Weibull, Gumbel, Tobit, NegBin, Probit, Beta, ZIP, and others). Custom losses can be supplied via the loss= argument — see Available Models and Targets.