Bayesian Media Mix Modeling: Axiomatic Budget Optimization with Non-Linear Saturation and Temporal Dynamics

Bayesian Media Mix Modeling: Axiomatic Budget Optimization with Non-Linear Saturation and Temporal Dynamics Media Mix Modeling (MMM) has traditionally relied on frequentist OLS regressions that struggle with the non-linearities and temporal dependencies inherent in modern marketing ecosystems. This paper presents a comprehensive Bayesian MMM Framework designed for enterprise-scale marketing optimization. We integrate: (1) Geometric and Weibull adstock transformations to model carryover effects and advertising memory; (2) Hill functions to capture diminishing returns and saturation plateaus with explicit parameter uncertainty; (3) Hierarchical Bayesian inference via No-U-Turn Sampling (NUTS) providing full posterior distributions for Return on Investment (ROI); (4) Convex budget optimization solving the constrained optimal spend problem under saturated market conditions. The framework provides explicit uncertainty quantification through Bayesian credible intervals, enabling marketers to distinguish between high-confidence and high-uncertainty channels. We validate the model through comprehensive diagnostic tests, sensitivity analyses, and hold-out forecasting, demonstrating superior predictive accuracy compared to traditional frequentist approaches while providing actionable budget reallocation insights. This work extends the First-Principles Hybrid Attribution Framework (Robinson, 2026) with time-series focus and Bayesian methodology.