Have you ever scrolled through your spam folder and wondered how your email provider decided to filter emails there? How is it so accurate in determining which emails you *don’t *want to see?

The answer lies in **Bayesian modeling**, a statistical approach based on Bayes’ theorem that was first discovered in the 18th century.

So, how is this 300-year-old mathematical technique influencing AI and everyday technology today? Among the many advantages of this type of modeling is that it combines prior information with new data, all within a solid theoretical framework. In this case, you can consider initial beliefs alongside new discoveries to influence the probabilities of a certain outcome.

In this guide, we dive deeper into Bayesian modeling, outline how Bayes’ theorem applies to AI, and give advice for how to leverage it at your organization.

## What Is Bayesian Modeling & How Does It Work?

Bayesian modeling is a statistical model where probability is influenced by the belief of the likelihood of a certain outcome. A Bayesian approach means that probabilities can be assigned to events that are neither repeatable nor random, such as the likelihood of a new novel becoming a *New York Times *bestseller.

In traditional statistics, it wouldn’t make much sense to assign probabilities to this situation, but with a Bayesian approach, you can use prior probability to inform the outcome and then continually update that probability when new evidence is received. For example, was the author’s previous book on the bestseller list? Has there been a lot of buzz about the book on social media?

One common example of Bayesian modeling is Naïve Bayes classifiers—this model is used in classification tasks only, and it falls under the umbrella of supervised ML algorithms, meaning that the model is trained with both categorical outputs and input features.

**But, why is the algorithm considered “naïve”? **

This particular model assumes that the input features used are independent of each other. Therefore, changing one input feature will not affect the others.

Given two events “A” and “B”, where P(x) represents the probability of “x,” the Naïve Bayes theorem states that:

**P (A/B):**How often A occurs given that B happens, also known as posterior probability.**P (B/A):**How often B occurs given that A happens, also known as likelihood probability.**P (A):**How likely it is for A to occur on its own.

**P (B):**How likely it is for B to occur on its own.

Both P(A/B) and P(B/A) are also known as conditional probabilities.

The formula for Bayes Theorem looks like:

P (A/B) = [P(B/A) P(A)] / P(B)

Simply put, Bayes’ theorem is a model for determining probability when you know certain other probabilities. It’s often used in machine learning to predict classes with high accuracy using previously collected data.

## How Can Businesses Use Bayesian Modeling to Their Advantage?

Bayesian modeling is an efficient way to make real-time predictions—that’s why email servers like Google use this algorithm to determine whether emails belong in the spam folder or not. Let’s check out some other ways organizations can use Bayes’ theorem to their advantage.

### Testing & Comparing Models

Bayes’ theorem offers an effective way to combine existing datasets with new observations within a theoretical framework. You can include past data about a certain variable and create a subsequent distribution for future analysis. Once you have new observations, the previous prediction can be used as old data.

With these predictive distributions, businesses can perform in-depth testing of any aspect of their operations, including the effectiveness of a marketing campaign that’s currently running. As new data is collected (like click-through rate), marketers can update the probability of the campaign driving traffic, revenue, etc. and make updates as needed.

### Making the Most Accurate Predictions

One of the best aspects of Bayesian modeling is its effectiveness at generating real-time predictions due to the fact that the probability of a hypothesis being true or not is constantly updated as new evidence is received.

For instance, doctors can use a Naïve Bayes’ classifier to determine if a patient is at high risk for cancer, heart disease, diabetes, etc. based on previous medical history and conditions. They can then use new data, like how a patient is feeling on a certain day, or how they respond to treatment, to update their hypothesis and treatment plan with a high degree of accuracy.

### Interpreting Results Quickly & Effectively

Unlike other approaches which might just show a confidence interval, Bayesian modeling provides easily interpretable answers on the modular level. It’s very easy to understand how much each feature contributed to a certain prediction, as we can measure which probabilities were used to determine the outcome.

## How Does a Naïve Bayes Model Work?

Suppose we have a training dataset of weather conditions and a corresponding target variable, “Play,” which reflects the probability of playing a tennis match depending on the weather.

We’ll follow these steps to determine if the conditions indicate good playing weather:

- First, put the information into a frequency table.
- Then, construct a probability table by calculating the probabilities of different conditions.
- Finally, use the Bayesian Model to compute the posterior probability (the probability that an outcome will occur given new data). The class with the highest posterior probability is the most likely outcome.

If the weather is sunny and you want to calculate if the tennis match will go on, you can set the problem up like this and calculate the probability by inputting values appropriately:

**P(Yes/Sunny) = P(Sunny/Yes) * P(Yes)/P (Sunny)**

## Explore More Machine Learning Algorithms

Bayesian modeling is widely deployed for accurate, easy-to-interpret results in spam filtering, sentiment analysis, and campaign success rate. It’s a go-to algorithm to update predictions in real-time when new discoveries come to light.

Recently, naïve Bayesian learning systems have been applied to predict the likelihood of a positive COVID-19 test given the current infection rate and test accuracy—this is just one example of how statistical modeling has an impact on our everyday lives.

*Want to learn more about how statistical approaches like Bayesian modeling can combine with machine learning to make a big impact? Check out our detailed guide on **machine learning algorithms** to get started!*