Scaling And Priors

Epsilon fits the maintained MCMC models on a transformed numerical scale. This keeps the sampler away from raw business-unit magnitudes and makes default priors more reusable across small demo datasets and larger user datasets.

The important practical rule is simple: priors live on the model scale unless a page or artifact explicitly says otherwise.

Target Scaling

For a time-series target, Epsilon uses max-absolute scaling:

\[b = \max_t |y_t|, \qquad y_t^{*} = \begin{cases} y_t / b, & b > 0, \\ y_t, & b = 0. \end{cases}\]

The fitted likelihood is attached to y_t^{*}. Post-model quantities such as contributions and decomposition totals are mapped back by multiplying scaled contributions by b.

For panel models, the target scale is panel-cell specific:

\[b_p = \max_t |y_{t,p}|, \qquad y_{t,p}^{*} = y_{t,p} / b_p.\]

If a target scale would be zero, Epsilon stores scale 1.0 so replay remains well defined.

Media Scaling

For a time-series media channel j, Epsilon stores:

\[a_j = \max_t |x_{t,j}|, \qquad x_{t,j}^{*} = x_{t,j} / a_j.\]

For panel models, channel scales are channel-by-panel-cell:

\[a_{j,p} = \max_t |x_{t,j,p}|, \qquad x_{t,j,p}^{*} = x_{t,j,p} / a_{j,p}.\]

The model applies adstock and saturation to the scaled media series. Response and contribution artifacts then replay the fitted transform path and report quantities in original target units where appropriate.

Controls

Controls can be standardised in the time-series path:

\[z_{t,k}^{*} = \frac{z_{t,k} - \bar{z}_k}{s_k}.\]

The scale uses the population standard deviation over the fitted data. A constant control column receives scale 1.0, which avoids division by zero and leaves the centred column at zero.

Panel controls are not part of the current maintained panel surface.

Default Prior Locations

The default priors are deliberately weak and model-scale oriented. The exact config surface can override them, but the defaults in the maintained path are:

Adstock and saturation parameters have family-specific priors. For example, geometric and binomial adstock use an alpha prior on [0, 1], while the centered-logistic saturation path uses a positive lam prior.

Prior Dimensions

Priors can be scalar, channel-specific, panel-specific, or channel-by-panel depending on their configured dimensions and the model type.

For time-series models, channel-level media parameters have one value per configured media channel:

\[\beta = (\beta_1, \ldots, \beta_J).\]

For panel models, the flattened panel-cell axis allows:

\[\beta_j, \qquad \beta_p, \qquad \beta_{j,p}.\]

Epsilon rejects partial panel-dimensional priors in the maintained panel path: a prior must either include all declared panel dimensions or none. This avoids ambiguous indexing when a model has more than one panel dimension, such as geo-by-brand.

Reading Priors In Business Terms

Because priors are applied on scaled variables, a coefficient prior is not a direct prior over pounds, dollars, impressions, sales, or conversions. A rough business-scale interpretation requires the fitted scales:

\[\Delta y \approx b \, \beta_j \, \Delta f_j(x^{*}).\]

For panel cells:

\[\Delta y_{p} \approx b_p \, \beta_{j,p} \, \Delta f_{j,p}(x^{*}_{j,p}).\]

This is why the stored model spec includes channel and target scales. Prediction and post-model replay use those scales to keep fitted artifacts consistent with the original data.

What To Check When Priors Dominate

Signs that priors are doing too much work include:

• posterior response curves that stay close to the prior shape despite clear media variation,
• very wide posterior predictive intervals,
• weak movement in media coefficients from prior to posterior,
• implausible response curves outside the observed spend range,
• sensitivity of conclusions to small prior changes.

Those are modelling diagnostics, not package failures. The remedy is usually a clearer design, better calibration evidence, narrower prior assumptions, or a more honest support boundary for the decision being made.