Bayesian Time Varying Coefficient Model with Applications to Marketing Mix Modeling (2106.03322v4)

Abstract: Both Bayesian and varying coefficient models are very useful tools in practice as they can be used to model parameter heterogeneity in a generalizable way. Motivated by the need of enhancing Marketing Mix Modeling at Uber, we propose a Bayesian Time Varying Coefficient model, equipped with a hierarchical Bayesian structure. This model is different from other time varying coefficient models in the sense that the coefficients are weighted over a set of local latent variables following certain probabilistic distributions. Stochastic Variational Inference is used to approximate the posteriors of latent variables and dynamic coefficients. The proposed model also helps address many challenges faced by traditional MMM approaches. We used simulations as well as real world marketing datasets to demonstrate our model superior performance in terms of both accuracy and interpretability.

• The paper introduces a Bayesian Time Varying Coefficient model that mitigates challenges like endogeneity, multicollinearity, and sparse data in MMM.
• The model applies kernel regression and a hierarchical Bayesian structure with Stochastic Variational Inference to dynamically estimate marketing elasticities.
• Real-world deployment at Uber via Orbit and Michelangelo ML demonstrates improved forecasting, attribution accuracy, and scalable campaign insights.

Bayesian Time Varying Coefficient Model for Marketing Mix Modeling

This paper presents a Bayesian Time Varying Coefficient (BTVC) model designed specifically for Marketing Mix Modeling (MMM), motivated by the practical needs of optimizing marketing investments at Uber. The paper introduces a hierarchical Bayesian structure enhanced by kernel regression techniques to estimate dynamic coefficients, addressing challenges inherent in traditional MMM such as endogeneity, multicollinearity, and data granularity issues.

Introduction to Marketing Mix Modeling Challenges

MMM has long been utilized to quantify relationships between marketing efforts and demand, with a major focus on understanding the marginal effects of different marketing tactics rather than merely predicting sales. Traditional MMM approaches, however, face several challenges:

1. Model Complexity: Modern marketing demands consideration of numerous evolving variables, leading to the "small $n$, large $p$" problem, making it difficult to build scalable models.
2. Sparse Data and Granularity: High granular datasets lead to sparse observations, requiring a balance between data granularity and historical data reliability.
3. Sequential and Correlated Errors: The sequential nature of data often results in correlated errors that violate OLS assumptions.
4. Endogeneity and Multicollinearity: Common practices like setting marketing budgets based on expected revenue introduce complexities in model performance due to endogeneity and multicollinearity.
5. Interpretability and Stakeholder Alignment: Models need high interpretability to align investments and strategies across organizational stakeholders.

The proposed BTVC model effectively addresses these issues by providing a robust framework adaptable to such complexities.

Proposed Model Framework

Time Varying Coefficient Regression

The BTVC model expresses regression coefficients as a weighted sum of local latent variables, providing a dynamic view of time-varying elasticity with respect to marketing spend. The model allows incorporation of seasonality and trend components through kernel functions.

• Kernel Functions: A mixture of Gaussian and custom kernels are proposed for regression, trend, and seasonality components to optimize the weighting scheme across time points.
• Hierarchical Bayesian Structure: A two-layer hierarchy is applied for robustness in coefficient estimation during sparse data periods, utilizing shrinkage properties inherent to Bayesian models.

Bayesian Framework and Inference

The Bayesian framework leverages Stochastic Variational Inference (SVI) to approximate posterior distributions of the latent variables and dynamic coefficients. Customizable priors, including experimentation results, enhance model calibration, offering robust solutions for regression component estimation.

Implementation

The BTVC model is implemented as a feature branch in Uber’s open-source package, Orbit. Orbit is structured to simplify the deployment of structural time series models in real-world applications. Deployment at scale is facilitated through integration with Uber's Michelangelo ML platform, enabling automation of the entire modeling process.

Evaluation and Results

Simulation Studies

Simulation studies demonstrate the BTVC model’s superior coefficient curve fitting against established models like BSTS and tvReg, highlighting its accuracy and adaptability.

Real Case Studies

• Forecasting Benchmark: BTVC exhibited improved forecasting accuracy over SARIMA and Facebook Prophet in real-world datasets, validating its efficacy across varying market dynamics and seasonal patterns.
• Attribution Accuracy: The model effectively integrates multiple lift test insights, demonstrating enhanced attribution accuracy and improved extrapolation capabilities compared to baseline models relying on single experimentation insights.

Conclusion

The Bayesian Time Varying Coefficient model offers a significant advancement in Marketing Mix Modeling by integrating Bayesian principles and kernel regression to produce dynamic coefficients. This framework addresses traditional MMM challenges and provides marketers with a potent tool for deriving actionable insights in campaign management. Through robust model design and deployment systems, BTVC powers improved prediction accuracy and enables scientific calibration of marketing strategies with experimentation data, marking an effective solution for current MMM complexities. Future developments could involve extending the model's adaptability to emerging marketing channels and further integrating with causal inference models.