Index: Bayesian Media Mix Modeling

Deep ingestion of Jin, Wang, Sun, Chan & Koehler (Google, 2017), “Bayesian Methods for Media Mix Modeling with Carryover and Shape Effects” — a flexible-functional-form media mix model (MMM) estimated by MCMC, with attribution metrics (ROAS/mROAS), optimal-mix derivation, prior-sensitivity analysis, BIC model selection, and a shampoo-advertiser application.

Routing Summary

• Need what MMM is / why Bayesian / key findings? → Bayesian Media Mix Modeling - Overview
• Need the carryover / adstock formulas (geometric decay, delayed/peak adstock, finite-window normalization)? → Carryover (Adstock) Functional Forms
• Need the saturation / shape forms (Hill function, $\beta$Hill, reach transform, identifiability)? → Shape (Saturation) Effects
• Need the likelihood, MCMC samplers, prior choices, and small-sample prior dominance? → Bayesian Estimation and Priors for MMM
• Need ROAS / mROAS definitions, optimal media mix, and the high-variance caveat? → ROAS, mROAS, and Optimal Media Mix
• Need BIC model selection, simulation recovery results, or the shampoo case study? → MMM Model Selection and Application

Concept Map

Concept	Note	Type	Depends On	Key Result
MMM overview & combined model	Bayesian Media Mix Modeling - Overview	overview	Carryover, Shape	Additive model $y_t=\tau +{\sum }_m{\beta }_m\text{Hill}(\text{adstock}(x))+{\sum }_c{\gamma }_c{z}_{t,c}+{ϵ}_t$; Bayesian to offset weak single-dataset signal
Adstock / carryover	Carryover (Adstock) Functional Forms	definition	Carryover Effects and Distributed Lags	Geometric decay ${\alpha }_m^l$ (= Koyck analog) and delayed/peak ${\alpha }_m^{(l-{\theta }_m{)}^2}$ (Gaussian kernel), finite-window normalized
Shape / saturation	Shape (Saturation) Effects	definition	Shape of the Marketing Response Function	Hill $\frac{1}{1+(x/\mathcal{K}{)}^{-\mathcal{S}}}$; $\beta$Hill poorly identifiable → prefer reach ($\mathcal{S}=1$)
Bayesian estimation & priors	Bayesian Estimation and Priors for MMM	concept	Overview, Carryover, Shape, MCMC Basics	Gibbs/HMC MCMC; small samples → posterior dominated by prior → biased estimates
ROAS, mROAS, optimal mix	ROAS, mROAS, and Optimal Media Mix	definition	Bayesian Estimation	Counterfactual zero/perturb spend incl. post-change period; optimal mix has large (multimodal) variance
Model selection & application	MMM Model Selection and Application	example	all of the above	BIC picks parsimonious Model IV (geometric + reach), BIC 51.49; large samples recover, two-year samples biased

Notes

• Bayesian Media Mix Modeling - Overview — CONTAINS: definition of MMM and the 4Ps lineage; the combined additive regression equation (Eq. 7); motivation for the Bayesian approach; carryover + shape framing; headline findings (large-sample recovery, small-sample prior bias, high optimal-mix variance).
• Carryover (Adstock) Functional Forms — CONTAINS: the normalized finite-window adstock (Eq. 1); geometric decay weights and retention rate ${\alpha }_m$ (Eq. 2); delayed/peak adstock with delay ${\theta }_m$ as a Gaussian radial kernel (Eq. 3); choice of carryover length $L$; explicit Koyck-lag connection.
• Shape (Saturation) Effects — CONTAINS: the Hill function (Eq. 4); the $\beta$Hill scaled transform (Eq. 5); the one-parameter reach transformation (Eq. 6); slope $\mathcal{S}$ vs half-saturation $\mathcal{K}$ behavior (concave vs S-shape); the near-unidentifiability problem.
• Bayesian Estimation and Priors for MMM — CONTAINS: MLE vs posterior (Eqs. 8-9); Gibbs (slice/BOOM) vs STAN/HMC samplers; per-parameter prior specifications and rationale; prior-dominance-in-small-samples finding; $\beta$- and $\mathcal{K}$-prior sensitivity studies.
• ROAS, mROAS, and Optimal Media Mix — CONTAINS: ROAS (Eq. 10) and mROAS (Eq. 11) definitions with post-change-period accounting; plug-posterior-samples-not-means rule; constrained optimal-mix optimization (Eqs. 12-16); the bimodal/high-variance optimal-mix caveat.
• MMM Model Selection and Application — CONTAINS: BIC formula (Eq. 18); the four candidate specifications (Table 7); sample-size recovery simulations (Tables 4); sampler timing (Table 8); the shampoo dataset, BIC table (Table 9, Model IV wins), real-data ROAS/mROAS, optimization, and residual-autocorrelation misspecification diagnostic.

Sources

• Jin-2017-Bayesian-MMM-Carryover-Shape.pdf — Jin, Y., Wang, Y., Sun, Y., Chan, D., & Koehler, J. (2017). Bayesian Methods for Media Mix Modeling with Carryover and Shape Effects. Google Inc., 14 April 2017. 34 pp.

See Also

• Market Response Models
• Carryover Effects and Distributed Lags — Koyck / distributed-lag foundations
• Shape of the Marketing Response Function — concave vs S-shaped response
• Advertising and Promotion Effects — advertising elasticity (~0.10) benchmarks
• Activity Bias in Advertising — observational-causality confound for MMM
• MCMC Basics · Bayesian Linear Regression — Bayesian computation