How Marketing Mix Modeling Actually Works: Adstock, Saturation, Priors

Start with the regression, then fix it four times

Most explanations of marketing mix modeling stop at the concept. MMM is a top-down statistical model that estimates each channel's contribution to sales without touching user-level data. True, and useless if you want to know whether the number is any good. That is the definition, not the mechanism.

Here is the mechanism. MMM is a regression, and everything hard about it is the set of corrections you bolt onto that regression so it resembles how advertising actually behaves.

The naive version is what you would write on a whiteboard in thirty seconds: take weekly revenue, regress it on weekly spend per channel, read off a coefficient for each channel, call that coefficient the channel's effectiveness. It is wrong in four specific ways, and MMM is the accumulated fix for all four.

Google's Jin, Wang, Sun, Chan, and Koehler put the diagnosis plainly in their 2017 paper on Bayesian media mix modeling: advertising usually has lag effects and diminishing returns that are hard to capture with linear regression. So they propose flexible functional forms for the carryover and the shape of the response, and estimate the whole thing with a Bayesian approach that folds in prior knowledge. That paper is the spine of this article. The four corrections below are the same four moves, named.

Fix 1: adstock, because last week's ad still sells this week

A campaign you run this week keeps moving sales next week. Someone sees a connected-TV spot on Monday, thinks about it, buys the following Thursday. A naive regression, which lines up this week's spend against this week's sales, credits that later purchase to whatever happened to be running when the order landed, not to the spot that actually planted it.

Adstock is the operation that fixes this. It takes raw spend and leaks it forward in time on a decay curve, producing an effective exposure the model can regress on. The simplest form is geometric decay with one parameter, the decay rate. Robyn also offers two Weibull forms, the CDF and the PDF, each with two parameters, which let the decay have a flexible shape and even a delayed peak, so the response can build before it fades rather than starting at its maximum.

Fix 2: saturation, because the tenth dollar does less than the first

A linear coefficient says every dollar in a channel returns the same amount. Advertising does not work that way. Early spend reaches the people most ready to buy. Later spend reaches people who need more convincing, then people who were going to buy anyway, then nobody new. The return per dollar bends down.

You model that bend by passing adstocked spend through a concave or S-shaped curve before it enters the regression. Robyn uses the Hill function, which has two parameters: alpha controls the shape of the curve, anywhere between a plain exponential decline and a full S-shape, and gamma controls where the curve inflects, meaning where the channel starts to bend over. Fit those two on the history and the model has learned each channel's diminishing-returns profile instead of pretending there isn't one.

The single most useful thing this produces is the gap between average and marginal return.

Fix 3: the baseline, because most sales owe marketing nothing

Turn off every campaign and you still sell. Brand equity, distribution, price, repeat buyers, weather, holidays, the macro trend of your category. That floor is the baseline, and it is usually the largest single term in the model. If you do not estimate it, and estimate seasonality on top of it, marketing steals credit for sales it never caused. A model with no baseline will hand your entire Christmas peak to whatever channel happened to be spending in December.

So MMM carries control variables that have nothing to do with your media: seasonal dummies or Fourier terms, price and promotion, distribution, competitor activity where you can get it, and a trend. The media coefficients only get to explain what is left after the baseline and controls have taken their share. This is also where daily data hurts you. Daily sales are noisy, and the noise does not carry marketing signal, which is why the practitioner consensus lands on weekly.

Fix 4: priors and calibration, because the model is under-identified

Here is the uncomfortable part. Feed two or three years of weekly data and a dozen correlated channels into this model and there is no single best answer. Many combinations of adstock rates, saturation curves, and coefficients fit the history about equally well. Practitioners call this equifinality, and it is why Robyn deliberately surfaces a set of candidate models rather than one. The model is really a family, and picking a member of that family is a judgment call.

Bayesian priors are how you pin it down. Instead of letting the data alone choose from infinitely many equally-good fits, you tell the model what you already believe about each channel before it looks at the history, and let the data update that belief. The strongest version of this comes from experiments.

Google's Zhang, Wurm, Li, Wakim, Kelly, Price, and Liu described the formal method in their 2024 paper on calibrating media mix models with Bayesian priors. The trick is to reparameterize the model so that a channel's ROAS is itself a parameter. Once ROAS is a parameter, you can set its prior distribution directly from an incrementality experiment. That is the exact mechanism by which a lift test corrects an MMM.

The one honest weakness: multicollinearity versus omitted-variable bias

There is a structural trap under all of this, and a senior reader will trust the rest of the piece more if it is named. MMM's inputs are channels that rise and fall together. Brand campaigns get flighted alongside search. Promotions run across paid social and email at once. That correlation between predictors is multicollinearity, and it has a nasty property: it barely hurts prediction, but it wrecks attribution. The model can still forecast total sales, yet it cannot cleanly say which correlated channel earned the credit.

The obvious fix, dropping one of the two collinear variables, walks you straight into the opposite failure. Now a real driver is missing from the model, so its effect leaks into whatever correlated variable remains. That is omitted-variable bias, it produces endogeneity, and it breaks any causal claim you wanted to make. Aryma Labs calls multicollinearity one of the enemies of MMM and frames the choice as a predict-or-explain dilemma. You are trapped between two failure modes, and the honest response is to report ranges and posteriors rather than single point estimates, and to lean on calibration to break the tie. This is exactly why fix 4 is not optional.

The tooling: what changed in 2020 and again in 2025

The econometrics here is decades old. What changed is that the full build is now open source, so a competent team can run MMM without a six-figure vendor engagement.

Meta open-sourced Robyn in 2020 under an MIT licence. It uses ridge regression, which regularizes to handle the correlated media variables, the three adstock options and the Hill saturation described above, a multi-objective evolutionary optimizer called Nevergrad to tune the hyperparameters, and a downstream gradient optimization for budget allocation. Runge, Skokan, Zhou, and Pauwels documented the approach in their 2024 introduction to Robyn's open-source method.

Google open-sourced Meridian to everyone on 29 January 2025, announced by Harikesh Nair, Senior Director of Data Science and Engineering at Google. Meridian is fully Bayesian, built on TensorFlow, and works on geo-level hierarchical data because, in Google's words, geo data contains more statistical information and improves the media-effect estimates. It builds in adstock and saturation, and it can use reach and frequency rather than only impressions, on the reasoning that ten impressions could mean ten people reached once or one person reached ten times. Meridian supersedes Google's earlier LightweightMMM. Google reports testing it with hundreds of brands, with Finder, an Australian financial-services company, saying the model gave it more confidence in YouTube's contribution. Treat that as illustrative, not a hard adoption statistic.

The word "open source" carries a trap of its own. Free to run is not free to operate. The real cost of MMM is not the licence. It is the data engineering, the two to three years of clean weekly history, and the data-scientist hours to calibrate, validate, and re-run every quarter. Both Robyn and Meridian assume a competent modeler on the other side of the keyboard.

What MMM can and cannot honestly claim

Keep the hierarchy of measurement methods explicit, because this is where teach separates from pitch. MTA is user-path correlational. MMM is aggregate correlational. Only experiments measure incrementality. MMM's claim to causality is borrowed from the experiments used to calibrate it, never manufactured by the regression. Never write that MTA measures incrementality, and never treat MMM and MTA as interchangeable.

With that fixed, here is the balanced view.

The counterargument deserves a straight answer. If MMM is correlational and uncertain, why run it at all? Because it is the only privacy-durable, channel-complete, cookie-independent way to see every dollar of spend in a single frame, and when calibrated with experiments it is the strongest available approximation of incrementality at the portfolio level. That is a real claim with real bounds. It is also why measurement-mature teams run MMM alongside experiments and MTA, each doing the job it is honest about, rather than asking any one of them to be the whole answer.

Sources

• Jin, Wang, Sun, Chan, Koehler (2017), Bayesian Methods for Media Mix Modeling with Carryover and Shape Effects, Google. research.google/pubs/bayesian-methods-for-media-mix-modeling-with-carryover-and-shape-effects
• Zhang, Wurm, Li, Wakim, Kelly, Price, Liu (2024), Media Mix Model Calibration With Bayesian Priors, Google. research.google/pubs/media-mix-model-calibration-with-bayesian-priors
• Runge, Skokan, Zhou, Pauwels (2024), Packaging Up Media Mix Modeling: An Introduction to Robyn's Open-Source Approach, arXiv:2403.14674 / MSI Working Paper 24-147.
• Meta Robyn, MIT-licensed. github.com/facebookexperimental/Robyn
• Nair, H. (29 January 2025), Meridian is now available to everyone, Google Ads & Commerce Blog. blog.google/products/ads-commerce/meridian-marketing-mix-model-open-to-everyone; developers.google.com/meridian
• Data-requirement consensus figures are convergent practitioner guidance (Improvado 2026, Measured 2026, Hopmann Marketing Analytics, Analytical Alley), not a single authoritative study.
• Multicollinearity and omitted-variable-bias framing: Aryma Labs, corroborated by standard econometrics treatments of endogeneity.

Further reading

• MTA vs MMM: when each method is the right tool
• Triangulating MTA, MMM and incrementality: how the three methods check each other
• Diminishing returns on ad spend: the saturation curve, from the budget side
• The measurement maturity map: where MMM fits as your spend grows

Holly Mehakovic

Co-Founder, mbuzz

Holly Mehakovic is Co-Founder of mbuzz. With 10+ years in marketing including roles at Westpac, Avon, and Forebrite, she's obsessed with making measurement actually useful.

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