Post-Model Quantities

Epsilon separates fitting from post-model replay. The sampler estimates posterior draws for model parameters; post-model functions replay those draws through the stored model specification, scales, and coordinate metadata to produce interpretable artifacts.

This page defines the main quantities used in result folders and plotting outputs.

Fitted Mean

The fitted mean is the posterior draw of the model expectation:

\[\mu_t^{(d)}\]

for draw d. For panel models, the indexed form is:

\[\mu_{t,p}^{(d)}.\]

The fitted mean is a conditional expectation under the fitted model. It is not the same object as an observed outcome draw.

Posterior Predictive

Posterior predictive draws include observation noise:

\[\tilde{y}_t^{(d)} \sim \operatorname{Normal}(\mu_t^{(d)}, \sigma^{(d)}).\]

This is why an observed-versus-fitted plot and a posterior-predictive plot can look similar but are not conceptually identical. One summarises the fitted mean; the other summarises replicated outcomes from the full observation model. For low-noise fits or visually compressed y-axes, the two plots can be very close.

Residuals

Residual diagnostics use:

\[e_t = y_t - \mathbb{E}[\tilde{y}_t \mid \mathrm{posterior}],\]

or the corresponding panel-cell version where supported. Residual plots are diagnostics for fit quality and remaining structure. They are not causal validation.

Contributions

Contribution replay decomposes the additive mean into named components. For a time-series model, Epsilon stores draw-level contribution arrays with axes:

draw, observation, component

For panel models, the axes are:

draw, time, panel_cell, component

Media contribution for channel j is:

\[\mathrm{media}_{t,j}^{(d)} = b \, \beta_j^{(d)} f_j^{(d)}(x_j^{*})_t,\]

where b is the target scale. Panel replay uses the panel-cell target scale b_p.

Intercept, control, event, holiday, seasonality, and trend components are also reported in original target units when they are present in the fitted model.

Decomposition

Decomposition aggregates contribution draws over time:

\[T_k^{(d)} = \sum_t \mathrm{component}_{t,k}^{(d)}.\]

Shares are computed from component totals:

\[S_k^{(d)} = \frac{T_k^{(d)}}{\sum_{r} T_r^{(d)}}.\]

Panel decomposition aggregates over both time and flattened panel cells before component totals and shares are computed.

Response Curves

A response curve for channel j asks: what total contribution would the fitted model assign to this channel if the historical spend path were rescaled to a given spend point?

For time-series models, the grid is in original total-spend units. Let g be a grid point and $G_j = \sum_t x_{t,j}$ be observed total spend. Epsilon rescales the historical path by:

\[\delta(g) = g / G_j, \qquad x_{t,j}^{\mathrm{scenario}} = \delta(g) x_{t,j}.\]

It then scales the scenario path to model space, replays adstock and saturation, applies the media coefficient, sums over time, and maps to original target units.

For panel models, curves use a delta_grid: each delta rescales the observed historical spend path within every panel cell. This preserves panel-cell spend shape and avoids pretending that a single aggregate spend grid identifies a free channel-by-panel allocation.

Saturation Curves

Saturation curves bypass adstock and show the saturation-only contribution surface for the selected channel. They help answer whether fitted diminishing returns are visible over the plotted spend range.

If a saturation curve is almost linear, the fitted posterior may be placing the observed range in the near-linear part of the saturation function. That is a substantive diagnostic: check the saturation parameter, observed spend range, and prior strength before assuming the plot is wrong.

Adstock Curves

Adstock curves bypass saturation and downstream target coefficienting. Returned values are in original channel-spend-equivalent units. They describe carryover, not diminishing returns.

This distinction matters: adstock is a weighted-sum memory mechanism, so an adstock-only curve can look linear. Curvature normally appears in the saturation or full response curve.

Efficiency Metrics

Metrics are derived from response curves, not from a separate model:

\[\operatorname{ROAS}(g) = \frac{\operatorname{response}(g)}{g},\]

\[\operatorname{CPA}(g) = \frac{g}{\operatorname{response}(g)}.\]

Marginal metrics use finite differences over the response grid:

\[\operatorname{mROAS}(g_i) \approx \frac{\operatorname{response}(g_{i+1}) - \operatorname{response}(g_{i-1})} {g_{i+1} - g_{i-1}},\]

with one-sided differences at the edges. Marginal CPA is the reciprocal of the marginal response where the denominator is numerically nonzero.

The default headline metric follows the target type: revenue targets default to ROAS; conversion targets default to CPA.

Optimisation Inputs

Budget optimisation uses response-curve surfaces, observed spend, and explicit constraints. It does not refit the model. The current maintained optimisation path optimises over supported channel-level surfaces and holds unsupported or unselected channels fixed according to the optimisation contract.

For panel results, optimisation uses historical within-channel panel shares. That is a bounded allocation rule, not free channel-by-panel optimisation.