Bayesian MMM & Media Saturation Curves

Modern Bayesian Marketing Mix Modeling (MMM, powering Meta's LightweightMMM, Google Meridian, and Robyn) establishes aggregated top-down marketing attribution and budget optimization in privacy-first, cookie-less environments; Classical OLS fails by ignoring non-linear media physics; Modern Bayesian MMM introduces 2 foundational non-linear transformations: 1) Adstock Carryover Decay (Geometric or Weibull decay): modeling sustained brand memory persistence where past advertising impacts future weeks: $x_t^* = x_t + \theta x_{t-1}^*$; 2) Diminishing Returns Saturation (Hill Saturation Function): capturing saturation plateaus where marginal returns diminish at high spend: $\text{Saturation}(x) = \frac{x^S}{K^S + x^S}$; Calibrated using Markov Chain Monte Carlo (MCMC via NUTS in PyMC/Stan) incorporating informative Bayesian priors from randomized Geo-Lift experiments.

Multi-million dollar annual media budget allocation, top-down omni-channel ROI measurement, and privacy-safe marketing analytics.

Eliminates dependence on vulnerable user-level tracking cookies while mathematically modeling advertising carryover memory and saturation ceilings.

Derive Geometric and Weibull Adstock transformations modeling advertising carryover memory and lagged peak responses?

Derive the mathematical properties of the Hill Saturation Function and explain how parameters $K$ (half-saturation) and $S$ (slope) govern diminishing returns?

How to calibrate Bayesian MMM models by injecting empirical Geo-Lift causal experiment results as informative priors to resolve collinearity?

How does MMM decompose top-line sales into unprompted organic baseline, macroeconomic seasonality, and marketing-driven incremental volume?

How does the MMM Budget Optimizer utilize non-linear programming (SLSQP) to allocate media budgets along optimal marginal return frontiers?