Coefficient Priors VS Contribution a broad technical conversation

[uomodellamansarda]

I’ve been reflecting on the discussion around prior choices in Meridian, studying again the official documentation in particular the section on choosing treatment prior types and reviewing the decision framework.

https://developers.google.com/meridian/docs/advanced-modeling/how-to-choose-treatment-prior-types#flowcharts_for_choosing_treatment_prior_types

On the Beta prior for coefficients, I understand that it is not a “coefficient” in the classical deterministic sense.
However, practically speaking, it does represent the expected value of the coefficient, with the key distinction being the variance of the prior distribution around that expectation.

Because of this, I’m aligned with the idea of not referring to it strictly as a “prior coefficient,” but rather more precisely as a prior on the expected coefficient value. The real degree of freedom—and modelling assumption—comes from how tight or loose that distribution is.

That said, I find the characterization of coefficient priors as “abstract” in the documentation a bit misleading.

Coefficient: The underlying statistical coefficient for a given treatment in Meridian's regression model. Unlike ROI, mROI, and Contribution—which are business metrics—the coefficient is a more abstract statistical parameter that is difficult to translate into business terms without context from other model parameters and data.

Even though they operate at a more statistical level, they still encode a clear belief about the impact of the variables, especially in cases where all inputs are comparable (e.g., impressions). In that context, they are not arbitrary—they are simply expressed in a different parameterization.
If I misunderstood how the beta coefficient prior works or something is wrong, please just let me know.

Where I think the distinction becomes critical is in how we initialize beliefs across channels:
• Equal contribution priors imply:

• "Regardless of impressions, channels will contribute similarly overall.”

• Equal coefficient priors imply:

• “The impact per impression is similar, but differences in cost (e.g., CPM) will drive different overall effectiveness.”

These are fundamentally different assumptions.

Would be great to conduct or have lift test from the client to calibrate the models. But these are not always available, and we often deal with collinearity and commercial constraints.
Additionally, clients often have “always on approach” on media phasing and this makes correlation with dependent KPI weak with some rare exception.
Given that, I’ve been thinking about how to better initialize contribution priors when experimental data is missing, and correlation is not significantly different among media vars and the KPI.

My current view is:

• When using coefficient priors, starting from similar expected values across channels is reasonable, as it encodes comparable per-impression effectiveness.

• However, for contribution priors, applying the same logic is less appropriate.

Instead, a more grounded approach could be to:

• Scale the expected total contribution proportionally to impression share (or reach) across channels
• While maintaining sufficient variance to allow the model flexibility to adjust.

This approach better reflects reality:

• Channels with higher impressions and reach inherently have more opportunity to generate impact and therefore should start with a higher expected contribution.

• At the same time, we acknowledge that effectiveness per impression is heterogeneous, and the model should be free to deviate from this proportional allocation.

This reasoning is also consistent with established marketing science. In the Google-sponsored study “Media in Focus: Marketing Effectiveness in the Digital Era” by Les Binet and Peter Field, the concept of ESOV (Excess Share of Voice) shows that a brand’s growth is strongly linked to how its share of exposure exceeds its market share.

From a modelling perspective:

• Share of Voice is fundamentally driven by impressions and reach, i.e., how much visibility a channel delivers relative to competitors.

• ESOV → growth implies that relative exposure levels are a key driver of incremental outcomes.

• Therefore, initializing contribution priors proportionally to impression share is effectively a proxy for embedding a baseline share-of-voice assumption into the model.

While we rarely have perfect ESOV or competitive SOV data, using impression-based scaling allows us to:

• Ground priors in observable exposure data
• Stay consistent with empirical evidence from marketing science
• And avoid imposing unrealistic symmetry across channels

Overall, I see this as a more consistent way to bridge domain knowledge and statistical modelling when experiments are not available.

What do you think?

[uomodellamansarda]

Just to add one point on the population scaling aspect because Jeff shared this part of the documentation from meridian discord channel:

To ensure I’m not missing anything: beyond population scaling and standardization (mean 0, std 1), does Meridian apply any additional transformations to the KPI that could affect how priors map to model parameters?

In particular, in scenarios where geo modeling is not feasible—due to limited data granularity, strong heterogeneity across regions, or simply real-world business constraints—are there any implicit transformations at the national level that could impact the interpretability or comparability of beta coefficients across channels?

[cpulavarthi]

Hello @uomodellamansarda,

Thank you for reaching out! You make some excellent and highly sophisticated points in your discussion. Here are some clarifications and thoughts.

Interpretation of the coefficients:

It's not quite true that "equal coefficient priors imply the impact per impression is similar". Coefficients cannot be interpreted as the "impact per media unit" or "impact per impression". The coefficient interpretation is complicated by the transformations applied to the data, so a common coefficient value could imply vastly different "impact per media unit" across channels.

• Adstock and diminishing returns transformations : Because Meridian applies adstock and diminishing returns (Hill) transformations before applying the coefficient, $\beta$ cannot be interpreted as "marginal effectiveness per unit exposure." The coefficient interpretation is highly dependent on the Hill and adstock parameter values, which vary with each draw from the posterior distribution. This is why the documentation refers to coefficient priors as "abstract" and why Meridian offers a reparameterization framework to allow users to set priors based on business metrics like ROI or Contribution.
• Media Variables: In addition to the parametric adstock and Hill transformations, the media variables undergo population and median Scaling (dividing by the non-zero median). This anchors the Hill Function’s ec parameter, making ec=1 the point of half-saturation relative to that channel's median volume per capita.
• KPI: The KPI is also scaled by population and standardized (to have Mean 0, Std 1). Standardizing the data makes the default priors reasonable across a variety of datasets, and it also changes the interpretation of the model parameters. Because the KPI is standardized, the resulting $\beta$ coefficients represent the change in "standard deviations of the KPI".

Our documentation on Input Data details the transformations performed on all the input variables in the input data. The logic behind the transformation choices are also detailed in this page.

Meridian utilizes a reparameterization framework, detailed in the paper titled Media Mix Model Calibration with Bayesian Priors, to allow users to provide priors in the form of ROI or Contribution. This allows priors to be assigned based on business knowledge of these metrics, which would be difficult to achieve by setting coefficient priors. When a prior is assigned on the ROI or Contribution, the prior on the coefficient is implicitly specified. Note that even if all channels are assigned the same ROI prior and all channels have the same cost per media unit, the resulting coefficient priors could differ significantly across channels.

Initializing contribution priors:

In the absence of experimental lift data, scaling expected contribution proportionally to impression volume is a reasonable approach. You could also consider setting a common ROI prior on all channels, which is equivalent to scaling the contribution proportionally to spend. A common ROI prior makes sense if you believe that the cost per impression correlates with the impact per impression.

Regarding national-level data:

There are no implicit national-level transformations. A national model is essentially a geo-level model with one geo. Without geographic variance to provide strong statistical information/signal, the choice of prior is often more critical for stable and logical results.

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Hope this adds information that can further your insights and understanding into prior selection and result analysis. Feel free to share your thoughts or any further questions you may have.

Google Meridian Support Team