A consulting client needed a mathematical model to optimally allocate advertising budgets across eighteen-plus channels. I built the complete theoretical framework from scratch: Poisson process models for individual media consumption, compound Poisson expressions for cumulative reach, and budget allocation formulated as a constrained nonlinear optimization problem.
Highlights
- Modeled each respondent's channel engagement as a Poisson process, deriving analytical expressions for individual ad-exposure probabilities and expected cumulative reach over a campaign period.
- Built a Bayesian framework for targeted channels, modeling imperfect demographic classification via sensitivity/specificity parameters and deriving conditional probabilities for ad selection given audience targeting accuracy.
- Derived closed-form exponential expressions for expected campaign reach from compound Poisson distributions, transforming a stochastic multi-channel problem into a computationally tractable optimization objective.
- Formulated budget allocation as a constrained nonlinear optimization problem and implemented the solution in Python/SciPy, integrating directly into the client's production web application.
- Incorporated RIM weighting for population scaling and uncertainty handling from survey data on media consumption.
- Authored a 14-page LaTeX mathematical specification deriving all results from fundamental probability principles — complete with model assumptions, all probability derivations, and the optimization formulation.
Technology
- Python
- NumPy / SciPy (constrained optimization)
- Bayesian probability theory
- Stochastic process modeling (Poisson, compound Poisson)
- LaTeX (mathematical specification)