Navigating Uncertainty: How Bayesian Linear Regression Enhances Marketing Decisions

Marketing Mix Modeling combines AI with advanced statistical analysis for campaign measurement, with clarity. It predicts holistic revenue by weighing ad spend variables across media channels, both offline and online. It is compatible with evolving ad platforms, and the latest testing methods.

We’re publishing a series of articles that explore the future of Marketing Mix Modeling, in an effort to empower marketers to get more insight and technical knowledge about predictive measurement. This one will focus specifically on Bayesian modeling, a foundational concept to MMM, that can produce highly accurate predictive performance outcomes by arriving at causal relationships through dynamic probability. Our goal is to help you build up a better understanding of Bayesian modeling by starting with the basics and gradually explaining each concept. And we’ll do it without using any math or statistics.

Bayesian Statistical Analysis

In short, Bayesian modeling is a method of statistical analysis that relies upon the Bayes Theorem to update the probability of a hypothesis as more evidence or information becomes available, over time. The Bayes Theorem was important because it provided a method for linking an independent variable with a dependent variable for the first time in human history over 300 years ago. This provided a method for people to go beyond assumptions and target causal relationships in data specifically (for example, the relationship between illness and age).

In marketing, Bayesian modeling can be used to assess relationships between pertinent variables, such as the efficacy of spend on media channels that leads to a purchase or, for example, when calculating Customer Acquisition Costs (CAC) where we want to know how many marketing dollars we need to spend to acquire one new customer over time.

Linking Bayesian Modeling with Linear Regression Analysis

Linear regression is a subset of regression analysis. In its simplest form, regression is a type of model that uses one or more independent variables to estimate the actual values of another dependent variable. A regression analysis is a way for us to measure the changing relationship between these variables.

It’s a given that ad spend on a platform will change from day to day, over time. We can use linear regression to reveal how the ad spend (independent variable) relates to other marketing factors, and to our KPI (dependent variable). Linear regression analysis establishes a correlation between these variables, and it helps us understand whether the correlation is strong or weak.

The great benefit of linear regression analysis is its simplicity because it produces an equation that clearly explains how an independent variable affects a dependent variable.

The Coefficient

Let’s assume that we use linear regression to assess the confidence we have that paid social is increasing online sales. This is where the coefficient comes into play.

The coefficient is a number that represents how a one-unit increase in paid social impacts the value of sales revenue. If increasing ad spend on social media by a certain amount leads to a corresponding increase in revenue, we can calculate that coefficient.

In linear regression, the coefficient can be said to help us understand the magnitude and direction of the relationship between our two variables. The coefficient unlocks the equation so that, going forward, we can quantitatively estimate the influence of our independent variable (changes in ad spend on social) on our dependent variable, (our KPI in this case, online sales). The coefficient reveals how each factor influences the outcome we want to predict.

Bayesian Linear Regression

Bayesian statistics enhance the capabilities of linear regression. Its major benefit is that it considers uncertainty which helps us get to more informed, predictive decisions.

Traditional linear regression is simple and easy to interpret. The coefficient is a single value that directly shows the change in the dependent variable for each unit change in the independent variable. Bayesian linear regression expands the number of coefficients. These become represented as distributions that indicate a range of possible values and their probabilities. And that provides the additional information we want about uncertainty in order to gain deeper insights into the significance and reliability of the relationships between our variables.

Using Bayesian linear regression, we can consider the uncertainty in the relationship between our marketing channels and their impact on our KPI across a range of different coefficients. Here’s how it plays out. A marketing channel with a large coefficient that also has a broad distribution (aka, a wider range of coefficient values) suggests that the model is unsure about that channel’s effectiveness. If there’s a lot of uncertainty about its precise impact, we should consider shifting resources to a different channel with a narrower distribution that has a better-understood impact. Generally, a narrow distribution indicates more certainty (a more confident estimation of the relationship) between the marketing channel and revenue.

In short, Bayesian linear regression helps us capture uncertainty, enabling us to make better-informed decisions in the real world. In Bayesian linear regression, we model the complete data set to form a distribution of linear functions, that each represent the relationship between our independent and dependent variables. This enables us to sample from this distribution and test linear regression lines. This approach makes a lot of sense because it considers the uncertainty in our linear regression estimate.

Although Bayesian linear regression feels more complicated because it involves computations with probability density functions, the ultimate outcome is undoubtedly more gratifying.

By delving into the connection between various marketing perspectives and actions, we establish a foundation for eventually testing causality,

Bayesian Beauty

Unlike traditional methods of statistical analysis that estimate fixed values, Bayesian Linear Regression gives us a range of possible values, providing a deeper understanding of uncertainty.

The beauty of Bayesian modeling lies in its treatment of probability as a measure of belief, allowing us to assign a probability distribution directly to the parameter or parameters that quantify our belief. Bayesian statistical methods leverage Bayes Theorem to calculate and adjust probabilities once we acquire fresh data. It plays a crucial role by describing the conditional probability of an event, by considering both the data we have on hand with any prior information or beliefs pertaining to the event.

Bayesian linear regression is a crucial tool for dialing-in your ad spend using the predictive powers of Marketing Mix Modeling. We hope we have helped to shed light on Bayesian modeling as an intuitive thinking process and demystified some of its complexity by breaking it down to its fundamentals.