Budget Optimisation

Budget optimisation is a post-model decision aid. It takes one fitted InferenceResults artifact, replays its response curves, and searches for a channel allocation that maximises total model-implied response under a fixed budget and user-supplied constraints. It never refits the model, and it never introduces information that was not already in the fitted posterior.

This page describes exactly what is solved, what the constraints mean, how the solver works, and, just as importantly, what an allocation result does not tell you.

What `optimize_budget` Solves

For time-series results, the maintained objective is total response. Let $a_j$ denote the total spend assigned to channel $j$ over the same aggregation window as the historical data, and let $R_j(a_j)$ be the channel response surface described below. The problem is:

\[\max_{a_1, \ldots, a_J} \;\; R_0 + \sum_{j \in \mathcal{O}} R_j(a_j) \quad \text{subject to} \quad \sum_{j \in \mathcal{O}} a_j + \sum_{j \in \mathcal{F}} \bar{x}_j = B, \qquad \ell_j \le a_j \le u_j .\]

Here $\mathcal{O}$ is the optimised channel set, $\mathcal{F}$ is the set of channels held fixed at their observed spend $\bar{x}_j$, $B$ is the supplied total_budget, $R_0$ collects the baseline and fixed-channel response, and $\ell_j, u_j$ are the effective per-channel bounds. The budget constraint is an equality: the optimiser reallocates the budget, it does not decide whether the budget should be spent at all.

Only objective = :total_response is currently supported. The utility functions described in Post-Model Quantities score evaluated allocations; they do not change the solved objective.

How Response Surfaces Feed The Solve

Each optimised channel contributes a one-dimensional response surface built from the fitted model's response curve machinery:

A spend grid is constructed for the channel, covering zero, the observed spend, the effective bounds, and enough intermediate points (64 by default) to make interpolation stable. Grid points are in original spend units.
response_curve_results is evaluated on that grid, giving a matrix of responses indexed by posterior draw and grid point.
The posterior mean across draws is taken, yielding a single deterministic response value per grid point.
A smooth interpolant through those points, together with its first and second derivatives, is registered with the solver as a nonlinear operator.

Two consequences follow. First, the surfaces inherit the response-curve semantics exactly: a grid point asks "what total contribution would the fitted model assign to this channel if its historical spend path were rescaled to this total?", with the historical spend shape preserved. Second, because the objective uses posterior-mean curves, the solve itself does not propagate posterior uncertainty into the allocation. Uncertainty enters afterwards, through evaluation and decision summaries, not through the optimiser.

Constraints And Bounds

The supported constraint surface is deliberately small:

Total-budget equality. The sum of optimised and fixed-channel spend must equal total_budget. All quantities, including the budget, observed spend, explicit bounds, and response grids, must be in the same original input units as the channel columns supplied to the model. Epsilon does not convert currencies or rescale thousands/millions at the optimiser boundary.
Channel subset. Passing channels restricts optimisation to those channels; every other fitted channel is held at its observed spend. This is how you answer "reallocate within TV and search, hold everything else".
Absolute bounds (budget_bounds). A channel-keyed mapping with optional lower and upper spend limits in original units.
Relative bounds (relative_bounds). Guardrails expressed relative to each channel's observed spend; for example, "no more than 50 per cent above or below what was historically spent". Relative bounds exist because response curves are only credible near the observed spend range; they stop the solver from extrapolating deep into untested regions of the curve.
Effective bounds. Where absolute and relative bounds are both present, the effective bound is the intersection. Infeasible combinations (for example, a budget that cannot be reached within the bounds) fail with an explicit error rather than a silently relaxed constraint.

After the nonlinear solve, Epsilon performs a small bound-projection pass to snap near-bound solver drift back onto the exact bounds and to restore the budget equality. This is post-solve hygiene, not a second optimisation; if the residual cannot be absorbed within valid bound slack, the call fails closed.

The Solver

The problem is solved with Ipopt through JuMP. The channel response surfaces are registered as nonlinear operators with analytic first and second derivatives taken from the interpolants, and the solve starts from a feasible allocation constructed from the effective bounds and observed spend.

Ipopt is a local nonlinear solver. Epsilon accepts locally feasible optima; the response surfaces are smooth interpolations of posterior-mean grids, not proven globally concave functions. For well-behaved saturation families the surfaces are typically concave over the credible spend range, but analysts should treat the returned allocation as a strong local candidate, not a certified global optimum. Solver metadata, including termination status, primal and dual status, and solve time, is retained in the result's convergence metadata. A solve that does not reach a feasible solution raises an error rather than returning a dubious allocation.

What Panel Optimisation Does And Does Not Do

For panel results, optimize_budget allocates channel totals and then distributes each channel's total across panel cells in proportion to the historical within-channel spend shares:

\[a_{j,p} = a_j \cdot \frac{\bar{x}_{j,p}}{\sum_{p'} \bar{x}_{j,p'}} .\]

This is a bounded allocation rule, chosen because panel response curves are defined by a shared historical spend delta within each channel. They do not identify the effect of moving spend between panel cells independently. Free channel-by-panel allocation, panel-total bounds, and channel-panel bounds are intentionally unsupported and are rejected rather than approximated.

In practical terms: panel optimisation answers "how should the channel split of the budget change, holding each channel's geographic or brand mix at its historical shape?" It cannot answer "which geo should get more TV spend?"

Marginal Response Diagnostics

The result reports marginal response, the derivative of the channel response surface, at the observed spend and at the solved spend for each optimised channel. At an unconstrained interior solution, marginal responses should be approximately equal across channels: that is the economic content of the optimality condition, since a budget unit should flow to wherever it buys the most response.

Where a channel sits at an active bound, its marginal response is a diagnostic, not a first-order condition. A channel pinned at its upper bound with a high marginal response is telling you the constraint, not the model, is doing the deciding; treat that as a prompt to question the bound, not as a finding about the channel.

When Not To Optimise

Budget optimisation is optional. Do not treat it as the natural final step for every MMM fit.

It is usually better to skip optimisation, or label it explicitly as exploratory, when:

response curves are unstable across posterior draws, prior choices, or reasonable model specifications;
convergence diagnostics are poor, effective sample sizes are low, or the fitted model has obvious posterior predictive misfit;
channels are strongly collinear and the decomposition between them is not decision-grade;
the proposed allocation relies on spend ranges far outside the observed data, even if the solver can technically evaluate the curve there;
the KPI is a fragile proxy, such as web visits, clicks, sessions, or another upper-funnel metric with weak commercial interpretation;
channel response is dominated by active bounds, so the allocation mostly reflects constraints rather than model evidence;
external business constraints, such as availability, margin, fulfilment, price, or sales capacity, are missing from the decision problem.

For fragile KPIs, prefer descriptive scenario evaluation over prescriptive allocation. A model may explain movement in web visits well while still saying little about profit, acquisition quality, or retained revenue. Optimising that proxy can produce a precise answer to the wrong question.

Uncertainty Summaries And Utility Scoring

Because the solve uses posterior-mean surfaces, Epsilon provides a separate evaluation layer for uncertainty-aware comparison:

evaluate_budget_allocation replays full posterior response draws at a supplied allocation (current, solved, or manual), giving a posterior distribution of total response for each candidate.
budget_allocation_decision_summary compares a candidate against a reference allocation with paired draws, reporting the uplift distribution $\Delta^{(s)} = y^{(s)}_a - y^{(s)}_0$, the draw-wise percentage uplift, and the posterior probability of improvement $\Pr(y_a > y_0)$.
budget_utility_value scores allocations under documented utilities: mean response, a lower-interval quantile for downside-averse decisions, the probability of improvement, and a risk-adjusted mean-minus-penalty form $\bar{y}_a - \lambda\,\mathrm{sd}(y_a)$.

These summaries are the honest way to compare allocations. Two allocations can have nearly identical posterior-mean response but very different downside risk; the mean-surface solve cannot see that distinction, the evaluation layer can.

What Not To Infer From An Allocation Result

An allocation produced by optimize_budget is conditional on the fitted model. It is a rearrangement of the model's own response estimates under your constraints. Specifically:

It is not causal proof. If the fitted response curves are confounded, the optimal allocation inherits that confounding. Optimisation sharpens whatever the model believes; it does not test whether the model is right.
It is not a forecast. Response surfaces replay the historical spend shape under rescaling; they do not simulate a future baseline, future seasonality, competitor reactions, or saturation drift.
It is not robust to extrapolation. Spend points far outside the observed range rest on the weakest part of the response curve; relative bounds exist to keep the solver honest here, and loosening them should be a deliberate modelling decision.
It is not uncertainty-aware at the solve step. The probability-of-improvement and utility summaries can be close to ambiguous even when the point allocation looks decisive. Report both.
It is not a global guarantee. Local solver, interpolated surfaces, and an equality budget constraint mean small changes in inputs can move the allocation, especially where response curves are flat.

Used within these limits, budget optimisation is a disciplined way to explore the implications of a fitted model. Used outside them, it manufactures precision the data never supplied.