[Paper Review] Conjugating Variational Inference for Large Mixed Multinomial Logit Models and Consumer Choice

Weiben Zhang, Rubén Loaiza‐Maya+2 more|arXiv (Cornell University)|Feb 13, 2026

Consumer Market Behavior and Pricing0 citations

TL;DR

The paper develops a new variational inference method that scales Bayesian estimation of large mixed multinomial logit models by efficiently updating a Gaussian approximation to the conditional posterior of random coefficients, and applies it to standard, nested, and bundle choice models.

ABSTRACT

Heterogeneity in multinomial choice data is often accounted for using logit models with random coefficients. Such models are called "mixed", but they can be difficult to estimate for large datasets. We review current Bayesian variational inference (VI) methods that can do so, and propose a new VI method that scales more effectively. The key innovation is a step that updates efficiently a Gaussian approximation to the conditional posterior of the random coefficients, addressing a bottleneck within the variational optimization. The approach is used to estimate three types of mixed logit models: standard, nested and bundle variants. We first demonstrate the improvement of our new approach over existing VI methods using simulations. Our method is then applied to a large scanner panel dataset of pasta choice. We find consumer response to price and promotion variables exhibits substantial heterogeneity at the grocery store and product levels. Store size, premium and geography are found to be drivers of store level estimates of price elasticities. Extension to bundle choice with pasta sauce improves model accuracy further. Predictions from the mixed models are more accurate than those from fixed coefficients equivalents, and our VI method provides insights in circumstances which other methods find challenging.

Motivation & Objective

• Address heterogeneity in multinomial choice data by estimating large mixed logit models with random coefficients.
• Develop a scalable variational inference (VI) method that overcomes computational bottlenecks in updating the conditional posterior of random coefficients.
• Demonstrate the method on simulations and a large scanner panel dataset to study pasta choice and bundle effects.
• Investigate how store characteristics, prices, promotions, and geography drive heterogeneity in price elasticities and model accuracy.

Proposed method

• Review existing Bayesian variational inference methods for mixed logit models and identify bottlenecks.
• Introduce a new VI step that efficiently updates a Gaussian approximation to the conditional posterior of random coefficients.
• Apply re-parameterization tricks and stochastic gradient techniques to scale to large datasets.
• Estimate standard, nested, and bundle mixed logit models using the proposed VI method.
• Evaluate performance via simulations and a large scanner panel pasta choice dataset.
• Compare predictive accuracy of mixed models against fixed coefficients models.

Experimental results

Research questions

• RQ1 Can a variational inference procedure be designed to efficiently update the Gaussian approximation for random coefficients in large mixed logit models?
• RQ2 How does the proposed VI method perform in standard, nested, and bundle mixed logit structures relative to existing VI methods?
• RQ3 What is the impact of model features such as store size, premium status, and geography on estimated price elasticities and heterogeneity?
• RQ4 Does extending to bundle choices improve model accuracy and predictive performance?

Key findings

• The new VI method improves efficiency by addressing a key bottleneck in updating the Gaussian approximation to the conditional posterior of random coefficients.
• In simulations, the method outperforms existing VI approaches in estimating large mixed logit models.
• Applied to a large pasta choice scanner panel, consumer response to price and promotions shows substantial heterogeneity at store and product levels.
• Store size, premium status, and geography are drivers of store-level price elasticity estimates.
• Extension to bundle choice with pasta sauce further improves model accuracy.
• Predictions from the mixed models are more accurate than those from fixed coefficients equivalents.