Conjugate Variational Inference for Large Mixed Multinomial Logit Models and Consumer Choice

2026-02-13

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• User Modeling, Personalization and Recommendation: Practical large-scale studies of user experience
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

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.

Problem

• mixed multinomial logit
• consumer choice
• random coefficients
• large datasets
• heterogeneity

Innovation

• Variational Inference
• Mixed Logit Models
• Random Coefficients
• Scalable Bayesian Inference
• Consumer Choice Heterogeneity