Bayesian marketing mix models

Day 4 Module 8 115 minutes (50 briefing, 55 lab, 10 debrief) 09:15 briefing · 10:05 lab · 11:00 debrief

Build a Bayesian marketing mix model (MMM) with adstock, saturation and seasonality in PyMC-Marketing, set priors from spend shares, check priors and posterior against the data, and report ROAS per channel with intervals against the known truth.

Open the Python lab in Colab Python

The decision. Decide which channels pay back: list channels whose ROAS exceeds the break-even ROAS (1 ÷ gross margin) with posterior probability ≥ 0.9, channels below it with probability ≥ 0.9, and channels the data cannot yet decide.

Day: Day 4 · Where the next dollar goes (09:15–11:10)
Time: 115 minutes (50 briefing, 55 lab, 10 debrief)
Languages: Python. PyMC-Marketing 1.2.0 is a Bayesian MMM that runs on Colab’s CPU; Google’s Meridian conflicts with it on package versions and its docs recommend a GPU (see DECISIONS.md).
Labs: Python: labs/python/08-bayesian-mmm.ipynb
Data: Synthetic weekly MMM (mmm_weekly.csv, seed 2028), PyMC-Marketing mmm_example.csv (simulated demo data)

What you will be able to do

Write geometric adstock and logistic saturation as functions and explain what the decay α and the saturation λ do.
Set channel-coefficient priors from spend shares and run a prior predictive check.
Fit the model with nutpie and check divergences, R-hat and effective sample size before reading any result.
Compute ROAS per channel with a 94% HDI and check that the truth lies inside it for at least 3 of the 4 channels.
Run the same workflow on a second dataset whose true parameters you do not know.

You leave with:

Your own adstock and saturation functions, checked against PyMC-Marketing’s
A diagnostics table (divergences, R-hat, ESS) and prior and posterior predictive plots
A ROAS table with 94% HDIs next to the true ROAS

Before you start

Module 7 · Geo-experiments and quasi-experiments and its lab.
A Google account for Colab. The default Python runtime (CPU) is enough.
Or the workshop’s Docker image, which has every package installed.

Where we are. Day 3 measured incrementality directly: an email test with randomized customers, and a regional campaign with a synthetic control. Each answer covers one campaign, in one channel, in one window. A marketing team spends on TV, search, social and display every week and needs to know what each dollar returns, and what next month’s budget will. This module builds a model of how every channel drives sales over three years of weekly data, checks that it recovers known returns, and decides which channels pay back.

The lab

Python lab · `08-bayesian-mmm`

Open in Colab

Step	Time
Part A · The two transforms
Exercise 1 · Geometric adstock	7 min
Exercise 2 · Logistic saturation	5 min
Part B · Priors before data
Exercise 3 · Priors from spend shares	7 min
Part C · Can we trust the fit?
Exercise 4 · Diagnostics before results	7 min
Part D · What each dollar returned
Exercise 5 · ROAS against the truth	9 min
Part E · A dataset without truth
Exercise 6 · Contribution shares on `mmm_example`	5 min
Decision · Which channels pay back?	—
Python subtotal: 6 exercises	40 min

Stretch (optional) · Default priors, or another saturation curve. For anyone who finishes early.

6 exercises, 40 minutes between them, in a 55-minute lab slot. Each exercise has a stub to complete, a folded solution and a checkpoint.

1. What a marketing mix model is for

A marketing mix model (MMM) explains weekly sales with a baseline (trend, seasonality, price, holidays) plus one term per media channel. It uses aggregate data that every business has (sales and spend by week), not customer-level tracking. Its outputs are the inputs to budgeting:

ROAS (return on ad spend) per channel: the sales a channel’s spend caused, divided by that spend;
response curves: how a channel’s sales contribution grows as its spend grows, which Module 10 uses to divide a budget.

An MMM is an observational model. It sees spend and sales move together and attributes part of sales to spend. That is the strength and the weakness of the method: it covers every channel continuously, and it can be fooled by anything that moves spend and sales together for another reason (Chan and Perry 2017). This module builds the model and checks it on data where the truth is known; Module 9 confronts the weakness.

The data are three years of weekly sales (156 weeks, 2023-01-02 to 2025-12-22) for the synthetic retailer, with spend on four channels, a price index and a holiday flag. Media cause about a third of sales in the truth. The channels are bought differently: TV in bursts with weeks off, search steady and rising with the season, social in a few four-week pushes, display small and slowly growing. How much a channel’s spend moves on its own decides how well its effect can be measured, and that sentence explains most of the results below.

Notation for this module

Symbol	Meaning
y_t	sales in week t
x_{c,t}	spend on channel c in week t
a_{c,t}	adstocked spend: x_{c} spread over the following weeks
\alpha_c	adstock decay of channel c, between 0 and 1
L	maximum lag of adstock, 8 weeks
\lambda_c	saturation parameter: how fast the channel’s effect approaches its ceiling
\beta_c	the channel’s effect size: the most it can add in a week
s(\cdot)	the logistic saturation function
\boldsymbol\gamma	coefficients of the controls (price, holiday, trend) and the seasonality terms
\text{share}_c	channel c’s share of total spend
\text{ROAS}_c	sales caused by channel c’s spend over a window, divided by that spend
m	gross margin rate (30%)

PyMC-Marketing scales each channel’s spend by its maximum and sales by their maximum before fitting, so \lambda_c is scale-free and \beta_c is in units of the best week’s sales.

2. Adstock and saturation

Objective for this section: write geometric adstock and logistic saturation as functions and explain what the decay \alpha and the saturation \lambda do.

The model for weekly sales is

\begin{gathered} y_t = \underbrace{\gamma_0 + \boldsymbol\gamma^{\top}\mathbf{z}_t}_{\text{baseline: trend, seasonality, price, holiday}} \;+\; \sum_{c} \beta_c\, s\big(a_{c,t};\, \lambda_c\big) \;+\; \varepsilon_t , \\[4pt] \varepsilon_t \sim \text{Normal}(0, \sigma^2), \end{gathered} \tag{1}

where \mathbf{z}_t holds the controls and two orders of yearly sine and cosine terms. Each channel’s spend passes through two transforms.

Adstock spreads one week’s spend over the following weeks, because people remember an ad. Geometric adstock with decay \alpha and maximum lag L is

a_t = \sum_{l=0}^{L-1} w_l\, x_{t-l}, \qquad w_l = \frac{\alpha^{l}}{\sum_{k=0}^{L-1} \alpha^{k}} , \tag{2}

with weeks before the data counted as zero spend. The weights are normalized to sum to 1, so a dollar’s effect is spread over later weeks but its total is unchanged: \alpha decides when the effect arrives, not how large it is. With \alpha = 0.5 the weights are 0.502, 0.251, 0.126, …: half the effect lands in the week of spend. In the truth, TV’s \alpha is 0.6 (a long memory) and search’s 0.2 (mostly immediate).

Saturation bends the curve, because each extra dollar reaches people who have already seen the ad or were never going to buy. Logistic saturation is

s(a; \lambda) = \frac{1 - e^{-\lambda a}}{1 + e^{-\lambda a}} = \tanh\!\left(\frac{\lambda a}{2}\right), \qquad s'(a) = \frac{\lambda}{2}\,\big(1 - s(a)^2\big). \tag{3}

It is zero at zero spend, rises steeply, and flattens toward 1; in Equation 1 the channel’s effect is \beta_c\, s(\cdot), so \beta_c is the ceiling. At \lambda a = 1, s = 0.462, just under half the ceiling. The slope s' (the return on the next dollar) falls as spend grows, so the average return over all dollars is higher than the marginal return on the last one. That is why “the ROAS of a channel” depends on how much you already spend on it, and why Module 10 allocates budgets by marginal, not average, returns.

The lab asks you to write both functions and checks them against PyMC-Marketing’s own on several spend series and parameter values.

Check yourself. A channel sits at \lambda a = 1. Using Equation 3, by how much does its contribution grow if spend doubles (to \lambda a = 2)? What happens to its average return per dollar?

Answer

s(1) = \tanh(0.5) = 0.462 and s(2) = \tanh(1) = 0.762: the contribution grows by 65% while spend grows by 100%. The average return per dollar falls to 0.762 / 2 \div 0.462 = 0.82 of what it was, and the marginal return on the last dollar falls further. A ROAS measured at one spend level does not carry over to another.

3. Priors before data

Objective for this section: set channel-coefficient priors from spend shares and run a prior predictive check.

A prior is what the model believes about a parameter before it sees sales (Module 2). PyMC-Marketing’s default prior for every channel’s \beta_c is HalfNormal with scale 2, the same for every channel: a small channel gets as much room as the largest. The lab encodes a weak, defensible belief instead: without other information, expect a channel to matter in proportion to what the business spends on it,

\beta_c \sim \text{HalfNormal}(\sigma_c), \qquad \sigma_c = C \cdot \text{share}_c , \tag{4}

where C is the number of channels, so the scales average 1. A HalfNormal puts about 95% of its mass below 2\sigma, so the scale caps what the model expects a channel can do before seeing data. TV, the largest spender, gets the widest prior and display the narrowest. Spend shares are not evidence of effectiveness (a lift test is, Module 9); they keep a small channel from being handed an implausibly large effect by noise.

The prior predictive check

Before fitting, simulate sales from the priors alone and compare with the data. A prior that allows absurd data is not an innocent “uninformative” choice: it is a model that believes absurd things. FULL run, 500 simulated series:

the prior’s 94% band for weekly sales, in its widest week, runs from −$344 million to +$239 million, against observed sales of at most $0.51 million;
the prior’s 94% interval for media’s share of all sales runs from 50% to 381%.

The band covers the data, and it is absurd. The main cause is not the channel priors but the trend control: t runs from 0 to 155 and its coefficient has a default Normal(0, 2) prior on sales divided by their maximum, so the trend alone can move sales by hundreds of times by the last week. Here 156 weeks of data pin it down, so the lab keeps it. On your own data, rescale the trend (to years, say) or narrow its prior, and run this check again. This is the first step of a Bayesian workflow: check the prior, fit, check the computation, check the fit, then read the results (Gelman et al. 2020).

4. Can we trust the fit?

Objective for this section: fit the model with nutpie and check divergences, R-hat and effective sample size before reading any result.

The lab fits Equation 1 by MCMC with the nutpie sampler, 2 chains of 2,000 draws, with target_accept=0.9 (smaller sampler steps than the default 0.8, which left an occasional divergence in the workshop’s test fits). On Colab this fit is the longest cell of the lab; its measured time is on the Readiness page.

MCMC draws are a sample from the posterior only if the sampler worked. Three checks come first, and the lab reads no result until all three pass:

Check	What it detects	Threshold
Divergences	steps where the sampler’s numerical path broke down, usually in a region of the posterior that is hard to explore; draws near them may be biased	0
R-hat (\hat R)	chains (and the halves of each chain) that disagree with each other	at most 1.01
Bulk ESS	how many independent draws the correlated draws are worth, for estimating the posterior’s center	at least 400

The R-hat and ESS thresholds come from Vehtari et al. (2021), who define the rank-normalized versions ArviZ computes. They also recommend at least four chains; the workshop runs two because Colab’s default runtime has two CPUs, and applies the total ESS threshold of 400 to them.

FULL run: 0 divergences, max \hat R = 1.004, min bulk ESS = 493. The gate passes. The lowest ESS belongs to saturation_lam[search], then the intercept and saturation_beta[search]. Over the range of spend a channel actually had, a steeper curve with a lower ceiling (larger \lambda, smaller \beta) fits about as well as a flatter one with a higher ceiling, so the sampler moves slowly along that ridge. The channel’s ROAS, which depends on both together, can still be estimated well.

The fourth check is the posterior predictive check: data simulated from the fitted model should look like the data. In the lab, 8 of 156 weeks fall outside the 94% posterior predictive band, against about 9 expected by chance.

Check yourself. A colleague’s MMM reports 12 divergences, max \hat R = 1.03 and a bulk ESS of 150, and their ROAS table looks sensible. What do you tell them, and what would you try first?

Answer

That the ROAS table cannot be read yet: all three checks fail, so the draws may not represent the posterior, and a sensible-looking table is no evidence that they do. Try a smaller step size first (target_accept closer to 1) for the divergences, then more draws (and chains) for \hat R and ESS. If divergences persist, look at the priors and the parameterization: a ridge between \lambda and \beta, or a prior that allows absurd values (section 3), is a common cause.

5. What each dollar returned

Objective for this section: compute ROAS per channel with a 94% HDI and check that the truth lies inside it for at least 3 of the 4 channels.

The return on ad spend of channel c over a window W is the sales its spend caused, divided by that spend:

\text{ROAS}_c = \frac{\sum_{t \in W} \big(\hat y_t - \hat y_t^{(x_c = 0)}\big)}{\sum_{t \in W} x_{c,t}} , \tag{5}

where \hat y_t^{(x_c = 0)} is the model’s sales had channel c spent nothing in the window. “Caused” is a counterfactual, as in Module 7, but here the counterfactual comes from the model rather than from control regions. PyMC-Marketing computes Equation 5 for every posterior draw (incrementality.contribution_over_spend), so ROAS has a full posterior. The truth uses the same definition with the noiseless true contributions over all 156 weeks.

FULL run:

Channel	Posterior mean ROAS	94% HDI	True ROAS
TV	1.61	1.52 to 1.69	1.6
search	2.57	1.17 to 4.49	3.0
social	1.19	0.61 to 1.90	1.4
display	0.29	0.00 to 0.84	0.6

All four true values lie inside their 94% HDIs. (Four 94% intervals all cover only about 0.94^4 \approx 78\% of the time, so the checkpoint asks for 3 of 4.) The widths tell the identification story:

TV switches on and off, so its effect stands out against everything else: its interval is narrow.
Search has the highest true ROAS and the widest interval. Its spend varies little and rises with the season, which also raises sales, so the model cannot separate the two well and many ROAS values fit.
Display is small and slowly growing: little variation, little information.

The adstock decays tell a related story: TV’s is recovered precisely (0.598, 94% HDI 0.573 to 0.620; truth 0.6), while social’s true 0.4 falls outside its interval (0.199, 0.000 to 0.387) even though social’s ROAS interval covers the truth. The data constrain a channel’s total effect better than when the effect arrives, and the two trade off.

Search is the warning for Module 9. Here search spend only correlates with the season, and the model still gets its ROAS roughly right with a wide interval. If spend followed demand (more search ads bought when people were already searching), the model would credit search with sales the demand caused, and could do so with a narrow, confident, wrong interval.

6. A dataset without truth

Objective for this section: run the same workflow on a second dataset whose true parameters you do not know.

On real data nobody knows the true ROAS. The lab repeats the workflow on mmm_example.csv, a demo file from the PyMC-Marketing repository: 179 weeks of sales, two media variables x1 and x2, two event flags and a trend. Its documentation calls it a simulated example and gives no true parameters, so the fit can be checked but not scored. x1 and x2 run between 0 and 1, so they are not dollars; the lab reports each channel’s share of the media contribution rather than ROAS, computed per draw so that every draw’s shares sum to 1.

FULL run: 0 divergences, max \hat R = 1.005, min bulk ESS = 818; x1 contributes 80.2% of the media effect (94% HDI 77.9% to 82.5%) and x2 19.8%.

Without truth, three things stand between you and a wrong answer: the sampler diagnostics, the predictive checks, and outside evidence. Only the third can catch a model that fits the data well and gets the causal effect wrong. The strongest outside evidence is an experiment, such as the geo test of Module 7, which Module 9 feeds into the MMM.

7. The decision

A dollar of sales earns the retailer its gross margin, 30% here, so a channel pays back on short-run sales if m \times \text{ROAS} > 1:

\begin{aligned} \text{break-even ROAS} &= \frac{1}{m} = \frac{1}{0.30} \approx 3.33 , \\[4pt] P(\text{ROAS}_c > 3.33) &= \frac{1}{J}\sum_{j=1}^{J} \mathbb{1}\big[\text{ROAS}_c^{(j)} > 3.33\big]. \end{aligned} \tag{6}

Decision. Classify a channel as pays back if P(\text{ROAS} > 3.33) \ge 0.9, does not pay back if P(\text{ROAS} < 3.33) \ge 0.9, and undecided otherwise. The 0.9 is a business choice: how sure we want to be before moving money. The diagnostics gate passed, so the results can be read. FULL run:

Channel	ROAS (94% HDI)	P(\text{ROAS} > 3.33)	Verdict
TV	1.61 (1.52 to 1.69)	0.000	does not pay back
search	2.57 (1.17 to 4.49)	0.175	undecided
social	1.19 (0.61 to 1.90)	0.002	does not pay back
display	0.29 (0.00 to 0.84)	0.000	does not pay back

On short-run sales, no channel is shown to pay back at a 30% margin; TV, social and display do not, with high probability, and search cannot be decided from these data. The truth agrees (no true ROAS reaches 3.33; search’s is 3.0). Two things would change the answer: a lift test on search, which would narrow its interval (Module 9), and the value of the customers a channel brings beyond this quarter’s sales, which this short-run ROAS ignores (Module 10 adds it, using Module 5’s customer values).

Summary

An MMM explains weekly sales as a baseline plus one saturating, carried-over term per channel (Equation 1).
Normalized geometric adstock (Equation 2) moves an effect in time without changing its total; logistic saturation (Equation 3) makes the return on the next dollar fall as spend grows.
Priors scaled by spend shares (Equation 4) are a weak, stated belief. The prior predictive check exposed an absurd default prior on the trend.
Read nothing before the gate: 0 divergences, \hat R \le 1.01, bulk ESS \ge 400; then a posterior predictive check. The FULL fit passed (1.004, 493).
ROAS (Equation 5) has a posterior; all four true ROAS values were inside their 94% HDIs. Intervals are narrow for channels whose spend moves on its own (TV) and wide for channels whose spend moves with demand (search).
At a 30% margin the break-even ROAS is 3.33 (Equation 6): three channels do not pay back on short-run sales and search is undecided.

Equations and where you implement them

Equation	What it is	In Lab 8
Equation 2	normalized geometric adstock	Exercise 1: `geometric_adstock(x, alpha, l_max, normalize)`; Checkpoint 1 compares it with PyMC-Marketing’s
Equation 3	logistic saturation	Exercise 2: `logistic_saturation(x, lam)`; Checkpoint 2 compares it with PyMC-Marketing’s
Equation 4	spend-share priors	Exercise 3: `beta_prior_sigma(spend)`; Checkpoint 3 checks the scales; then the provided prior predictive check
Equation 1	the model	Provided: `MMM(...)` and the nutpie fit
Divergences, \hat R, ESS	the diagnostics gate	Exercise 4: `diagnose(idata)`; Checkpoint 4 uses a fake posterior with known problems, then the real fit
Equation 5	ROAS per channel with a 94% HDI	Exercise 5: `roas_table(roas_draws, true_roas)`; Checkpoint 5 asks for 3 of 4 true values inside
Shares of the media contribution	the workflow without truth	Exercise 6: `contribution_share(idata)`; Checkpoint 6 checks that the mean shares sum to 1 and the intervals lie in [0, 1]
Equation 6	which channels pay back	Provided: the Decision cell

Gotchas

Normalized adstock keeps a dollar’s total effect fixed; without normalization, a larger \alpha also makes the effect larger and \beta absorbs the difference.
Scaled units. PyMC-Marketing’s \beta_c is in units of maximum weekly sales and \lambda_c applies to spend divided by its maximum. Convert before comparing with anything in dollars.
Default priors are choices. The default trend prior allowed sales of hundreds of millions; run the prior predictive check on your own data.
ArviZ 1.x defaults to an 89% equal-tailed interval. Name the interval: here, 94% HDI.
Two chains are below the four that Vehtari et al. recommend. Colab’s two CPUs are why; on a bigger machine, run four.
A good fit is not a correct ROAS. Predictive checks test the fit to sales, not the causal attribution between channels.
Average ROAS is not marginal ROAS. Budget decisions need the return on the next dollar (Module 10).
Spend that follows demand inflates a channel’s ROAS; the model cannot tell from these data alone (Module 9).

Next steps

This MMM recovered the truth because the synthetic data contain nothing to fool it: no channel’s spend is driven by the same demand that drives sales. Real search budgets often are. When spend follows demand, an MMM credits the channel with sales it did not cause, fits the data just as well, and can report a narrow interval around the wrong ROAS. Module 9 builds that failure on purpose, then calibrates the model with lift tests (Module 7’s geo-test results as extra likelihood terms) and validates it out of sample with time-slice cross-validation.