A Bayesian MMM combines a prior distribution, data information and a likelihood to produce a posterior distribution.
A Bayesian MMM combines a prior distribution, data information and a likelihood to produce a posterior distribution. Its value is not making a model subjective: it exposes assumptions, stabilises weakly identified parameters and directly expresses decision probability. A prior must never force an expected result; it represents documented knowledge, a plausible constraint or regularisation, and its influence is tested.
Decision and method.
Use posterior distributions to select allocations whose probability of exceeding the economic threshold remains sufficient under several priors and calibrations. Specify outcome, controls, trend, season, adstock, saturation, interactions and hierarchical level, linking each to a testable business assumption. Convert internal experiments, studies, comparable benchmarks and constraints into distributions; publish plausible quantiles rather than arbitrary precision and retain a weak reference prior. Check convergence, effective sample size, autocorrelation, divergences and posterior prediction; test recovery on simulated data and held-out periods. Compare posteriors with experiments, vary priors and simulate decisions; allow an allocation only if it stays acceptable in credible variants.
Worked example.
Weekly data give video a highly unstable ROAS of 4.1 ± 3.0 because spend follows seasonality. A comparable prior experiment estimates 1.8 ± 0.5. The retained prior is 1.9 ± 0.8, wider than the experiment. Under a normal approximation, prior precision is 1/0.8² = 1.5625, data precision 1/3² = 0.1111. Posterior mean: (1.9 × 1.5625 + 4.1 × 0.1111) ÷ 1.6736 = 2.05. Posterior standard deviation: √(1/1.6736) = 0.77. Probability ROAS exceeds 1.6 is about 72%. Do not extend at maximum: 72% is below the 80% gate. Fund a geo experiment to improve information before the next allocation.
Checks and limits.
Document prior sources, scales and transport; verify convergence, prediction and parameter recovery; use threshold probabilities and loss risk; compare several credible priors and experimental calibration. A precise prior can dominate weak data; numerical convergence guarantees neither causality nor sound specification; posteriors remain conditional on selected variables, transformations and data.
Resources and sources.
Download the illustrative Bayesian MMM model. Related: validate MMM, calibrate with experiments, decide under uncertainty. Sources: Gelman et al. (2013); Jin et al. (2017); Vehtari, Gelman & Gabry (2017).