6. The Multinomial Choice Model

The H&M application uses a multinomial logit (conditional logit) model (McFadden, 1974; Berry, Levinsohn & Pakes, 1995; Train, 2009). Consumer \(i\) chooses from \(J\) alternatives, each described by \(K\) attributes \(x_{ij} \in \mathbb{R}^K\). Utilities are

\[ V_{ij} = x_{ij}' \theta(X_i), \qquad P(Y_i = j \mid X_i) = \frac{e^{V_{ij}}}{\sum_{m=1}^J e^{V_{im}}}. \]

Heterogeneity without a distributional assumption

The FLM framework estimates \(\theta(X_i)\) for each individual using individual-level data and neural networks, rather than assuming a parametric distribution for heterogeneity. This is the key departure from mixed/random- coefficient logit, which integrates over an assumed taste distribution. See also Hetzenecker & Osterhaus (2024) for a related approach to heterogeneous discrete choice.

Connection to two-tower embeddings

The connection between two-tower embeddings and the FLM framework is natural:

The consumer embedding \(X_i\) captures the latent heterogeneity dimensions.
The neural network maps \(X_i \to \theta(X_i)\), the consumer’s taste parameters.
The item attributes \(T_i\) include prices and (PCA-reduced) item embeddings.

This is what lets the package recover a full taste vector \(\theta(X_i) = (\beta_{\text{price}}(X_i), \beta_{\text{style},1}(X_i), \ldots)\) for every consumer, and then average any target \(H\) over the population to obtain \(\mu^*\) with a valid confidence interval.