What is a posterior?

A posterior is the result of training a Bayesian model. The model starts from a prior for each parameter, meets the data, and ends with a posterior: a distribution that combines the two. Its centre is the most likely value; its width is how much the data left undecided. Every number a Bayesian MMM reports, from channel contribution to marginal ROAS, is read off a posterior.

Put the prior and the posterior side by side and the picture tells you what the data taught. A posterior that moved far from the prior and narrowed means the history had a lot to say about that channel. A posterior that barely moved and stayed wide means the data was thin or the channel always ran at the same level, so the model still leans on what it was told. Both are useful: the second one is the signal to run a test.

The width is the confidence range. "Most likely 3.2, probably between 2.4 and 4.1" is a posterior; "3.2" on its own is a posterior with its honesty removed.

Posteriors are what make a budget decision plannable at a chosen level of certainty. Because the model holds a full distribution for every channel, a scenario can be forecast at the cautious end, the middle or the optimistic end, and the budget that puts a quarterly target at the cautious end is a different number from the one that puts it in the middle. The gap between them is the price of certainty.

At Odins every scenario and recommendation carries its range, and the posteriors are retrained monthly on data that arrives daily. With about two years of history behind each new month, they move slowly, which is what makes the model stable enough to plan on.

• Prior: What the model believes about a channel before it sees the data. The part of MMM that decides how much the data can teach.
• Confidence range: The range a Bayesian model reports around every number. A recommendation without one is hiding something.
• Bayesian marketing mix modeling: A model that starts from what you already know and lets the data correct it, with a range on every answer.
• Scenario planning: Forecasting what a budget would return before committing it, and comparing several futures side by side.

Go one level deeper: Why Bayesian, and what it changes. Or go back to all terms in the glossary.