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Model Form

This page describes the statistical model estimated by Epsilon's maintained MCMC path. The equations are written on the model scale: media channels and the target are scaled before the Turing model is built, and post-model summaries map fitted quantities back to original business units where appropriate.

Time-Series Mean Function

For observations indexed by time t = 1, ..., T, the maintained time-series path uses a Gaussian observation model:

\[y_t^{*} \sim \operatorname{Normal}(\mu_t, \sigma),\]

where y_t^{*} is the scaled target. The linear predictor is additive:

\[\mu_t = \alpha + m_t + c_t + e_t + h_t + s_t + r_t.\]

The terms are:

  • Intercept ($\alpha$).
  • m_t: total media contribution after adstock, saturation, optional media multiplier, and media coefficients.
  • c_t: optional control contribution.
  • e_t: optional event-window contribution.
  • h_t: optional holiday contribution.
  • s_t: optional Fourier seasonality contribution.
  • r_t: optional trend contribution.

The media term is a sum over configured channels:

\[m_t = \sum_{j=1}^{J} \beta_j f_j(x_{1:T,j})_t,\]

where x_{1:T,j} is the scaled spend or exposure series for channel j, f_j is the configured adstock-and-saturation transform, and $\beta_j$ is the media coefficient. In the default nonnegative media-coefficient path, $\beta_j$ has a positive prior.

Optional controls enter linearly:

\[c_t = \sum_{k=1}^{K} z_{t,k}\gamma_k.\]

Holiday, event, seasonality, and trend effects use the same additive structure: a design matrix multiplied by coefficient draws. If a component is absent from the config, its contribution is zero.

Panel Mean Function

For panel models, Epsilon uses a shared time axis and a deterministic flattened panel-cell axis. Let p = 1, ..., P index the flattened panel cells. The maintained panel path estimates:

\[y_{t,p}^{*} \sim \operatorname{Normal}(\mu_{t,p}, \sigma_p),\]

with

\[\mu_{t,p} = \alpha_p + m_{t,p} + h_{t,p} + s_{t,p}.\]

The panel media term is:

\[m_{t,p} = \sum_{j=1}^{J} \beta_{j,p} f_{j,p}(x_{1:T,j,p})_t.\]

Depending on the prior dimensions, panel parameters may be shared across panel cells, vary by channel, vary by panel cell, or vary by channel and panel cell. For multi-dimensional panels such as geo-by-brand, Epsilon stores the fitted model on one flattened panel-cell axis and keeps named coordinate metadata for interpretation and replay.

The current panel path is intentionally narrower than the time-series path: panel controls, panel trend, panel events, panel calibration, and panel holdout validation are not part of the maintained support surface.

Scaling

Epsilon scales media and target values before fitting:

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

For time-series models, a_j is the channel scale for channel j and b is the target scale. For panel models, channel scales are channel-by-panel-cell and target scales are panel-cell specific.

This means priors are interpreted on the model scale, not directly in pounds, dollars, impressions, or conversions. Response curves and decomposition artifacts are reported through Epsilon's post-model layer so analysts can inspect business-scale quantities without changing the fitted model.

Calibration Terms

Calibration does not replace the observation model. On the supported time-series path, Epsilon adds calibration information to the posterior as additional log-density terms:

\[\log p(\theta \mid y, q) \propto \log p(y \mid \theta) + \log p(\theta) + \log p(q_{\mathrm{lift}} \mid \theta) + \ell_{\mathrm{cost}}(\theta).\]

The lift-test term is available for the centered-logistic time-series MCMC path. The cost-per-target term is a soft penalty. Both are additive and independent of the Gaussian observation likelihood.

What The Fitted Mean Is Not

The fitted mean is not a causal identification guarantee. It is the posterior mean function implied by the supplied data, transform choices, priors, and calibration terms. Causal interpretation still depends on the data-generating setting, omitted-variable risk, media planning process, and whether calibration measurements are credible for the estimand being reported.