From Small Data to MMM-Style Budget Decisions: Reproducible Conjugate Bayesian Log-Response Regression with ElasticNet Benchmarks and Explainable Budget Scenario Rollouts

Abstract

Marketing mix modeling (MMM) is often treated as a large-sample, time-series task, but many organizations must make near-term budget decisions with limited historical data. This study develops a reproducible small-sample MMM-style workflow using the public ISLR Advertising.csv dataset (n=200 markets; TV, radio, newspaper spend; sales). Reproducibility is ensured by using a public dataset, fixed random seeds, explicitly stated preprocessing, analysis scripts supplied with the manuscript, and specified Python libraries. The workflow combines predictive benchmarking and prescriptive decision support. First, OLS, Ridge, Lasso, and ElasticNet are evaluated under 5-fold cross-validation and a fixed 80/20 hold-out split. Second, a conjugate Bayesian regression on log(1+spend) features models diminishing returns and yields closed-form posterior and Student-t predictive distributions. Third, for each total budget B, the allocation that maximizes the posterior mean prediction is solved under non-negativity, budget-balance, and observed-maximum channel caps; posterior samples are then evaluated at each optimized allocation to form 90% credible bands for the budget-sales curve. On the fixed split, test RMSE ranges from 1.829 to 1.871, while 100 repeated splits indicate that raw-feature OLS is most accurate on average (mean RMSE 1.671). At B=200, the bounded optimum allocates $147.25k to TV, $49.60k to radio, and $3.15k to newspaper, predicting 18.149 sales. The results suggest that, in small samples, regularization and log-response modeling mainly support stable, interpretable budget recommendations rather than improving point prediction alone.