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
• Need the simulation DGP, Table 1 parameters, the Eq. 17 signal-strength metric, or Table 3 priors? → Jin 2017 MMM - Simulation Design, Signal Strength and Priors
• Need simulated ROAS/mROAS recovery (Fig. 5) or the unimodal vs trimodal optimal mix (Fig. 6)? → Jin 2017 MMM - Simulated ROAS, mROAS and Optimization Results
• Need the sample size vs bias simulation / full Table 4 (two vs sixty years)? → Jin 2017 MMM - Sample Size vs Bias Study (Two vs Sixty Years)
• Need the prior-sensitivity experiments on beta and K (Tables 5-6, Figs. 9-12)? → Jin 2017 MMM - Prior Sensitivity Experiments on Beta and Half-Saturation
• Need the shampoo data, control orthogonalisation, full Tables 8-9, Figs. 13-17 diagnostics? → Jin 2017 MMM - Shampoo Case Study Data, Diagnostics and Posteriors
• Need the verbatim Stan program and the Gibbs/slice scheme? → Jin 2017 MMM - STAN Sampler Code and MCMC Implementation
• Need stated limitations, the adstock-then-shape ordering rule, extensions or an MMM audit checklist? → Jin 2017 MMM - Conclusions, Limitations and Extensions

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
Simulation design	Jin 2017 MMM - Simulation Design, Signal Strength and Priors	example	Overview, Carryover, Shape	DGP with Table 1 parameters; signal strength metric (Eq. 17); Table 3 priors
Simulated attribution and optimisation	Jin 2017 MMM - Simulated ROAS, mROAS and Optimization Results	example	ROAS, mROAS, and Optimal Media Mix	ROAS/mROAS recovered in large samples; optimal mix unimodal (Scenario I) vs trimodal (Scenario II)
Sample size vs bias	Jin 2017 MMM - Sample Size vs Bias Study (Two vs Sixty Years)	example	Bayesian Estimation and Priors for MMM	Two years: curves underestimated by about 18-40%; sixty years: bias near zero (500 replicates)
Prior sensitivity	Jin 2017 MMM - Prior Sensitivity Experiments on Beta and Half-Saturation	example	Bayesian Estimation and Priors for MMM	Prior on beta and K moves attribution by 15-20 points in short series
Shampoo case study detail	Jin 2017 MMM - Shampoo Case Study Data, Diagnostics and Posteriors	example	MMM Model Selection and Application	n = 106 weeks; orthogonalised controls cut Stan time 3040 s to 75 s; retention posterior equals prior
Stan implementation	Jin 2017 MMM - STAN Sampler Code and MCMC Implementation	method	Bayesian Estimation and Priors for MMM	Verbatim Appendix Stan program mapped to paper notation
Scope and extensions	Jin 2017 MMM - Conclusions, Limitations and Extensions	concept	whole cluster	Not causal by construction; adstock before shape; hierarchical/geo/simulator extensions

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.

• Jin 2017 MMM - Simulation Design, Signal Strength and Priors — CONTAINS: data-generating process; Table 1 generating parameters; Eq. 17 signal-strength metric; Table 2; Table 3 priors; Fig. 4 correlations; Fig. 18 single-dataset posteriors; sampler timing.

• Jin 2017 MMM - Simulated ROAS, mROAS and Optimization Results — CONTAINS: attribution windows and Fig. 3; Fig. 5 ROAS/mROAS recovery; optimisation design; two-approach optimal-mix algorithm; Fig. 6 Scenario I unimodal vs Scenario II trimodal.

• Jin 2017 MMM - Sample Size vs Bias Study (Two vs Sixty Years) — CONTAINS: ADEMP-style design of the 500-replicate study; full Table 4 (relative bias of betaHill at x=0.5 and x=1, two vs sixty years); Figs. 7-8; worked conversion of bias to curve units; prior-driven bias mechanism; sixty-years caveat.

• Jin 2017 MMM - Prior Sensitivity Experiments on Beta and Half-Saturation — CONTAINS: Sec. 7.1 design, full Table 5 and Figs. 9-10; Sec. 7.2 Scenarios I/II, full Table 6 and Figs. 11-12; footnote 3 definition of support.

• Jin 2017 MMM - Shampoo Case Study Data, Diagnostics and Posteriors — CONTAINS: data description and n = 106; Fig. 13 correlation tables; control-orthogonalisation algorithm; full Table 8; Table 9 with footnotes 5-7; Figs. 14-17 (per-model ROAS, bimodal optimum, posterior equal to prior, wider Hill bands, residual ACF to lag 15).

• Jin 2017 MMM - STAN Sampler Code and MCMC Implementation — CONTAINS: verbatim Appendix Stan program; code-to-notation map; notes on running it with current Stan; reconstructed Gibbs/slice scheme; timing table.

• Jin 2017 MMM - Conclusions, Limitations and Extensions — CONTAINS: stated scope limits (not causal, no synergy, independent channels, curvature forms, BIC not causal); adstock-then-shape ordering rule; Sec. 9 conclusions; hierarchical, geo and simulator extensions; MMM audit checklist.

Sources

• Jin-2017-Bayesian-MMM-Carryover-Shape — 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
• Chan Perry 2017 - Challenges in Media Mix Modeling - Overview — companion Google white paper on why MMMs are hard to validate
• Zhang Vaver 2017 - Aggregate Marketing System Simulator - Overview — simulator proposed in Jin et al. Sec. 9 for testing MMMs on known truth
• Model Validation and Interpretability - Overview — how to validate a model before it informs decisions