Bayesian inference is a statistical model used to find the probability of an outcome, taking into account prior evidence. It's based on Bayes' theorem, a mathematical equation used to calculate the probability of future actions. Bayesian Inference consists of three components: Likelihood, prior, and posterior.

- Likelihood is the statistical probability of an outcome.
- Prior is a distribution based on previous knowledge (e.g. past results).
- Posterior combines likelihood and prior to make a new, potentially more accurate prediction.

The advantage of a Bayesian model is that it compares its predicted observations to actual observations and adjusts accordingly. It continuously uses this comparison to build increased knowledge over time.

In a Frequentist model, probability is the limit of the relative frequency of an event after many trials. In other words, this method calculates the probability that the experiment would have the same outcomes if you were to replicate the same conditions again. This model only uses data from the current experiment when evaluating outcomes.

With Bayesian statistics, probability simply expresses a degree of belief in an event. This method is different from the frequentist methodology in a number of ways. One of the big differences is that probability actually expresses the chance of an event happening. The Bayesian concept of probability is also more conditional. It uses prior and posterior knowledge as well as current experiment data to predict outcomes.

Since life doesn’t happen in a vacuum, we often have to make assumptions when running experiments. But the Bayesian approach attempts to account for previous learnings and data that could influence the end results. As the experiment ensues and data is collected, the results start to take shape and adjustments can be made to affect the outcome. On a graph, for example, a Frequentist approach would display a bell-shaped curve while the Bayesian approach would show changes based on data revelations, resulting in a steeper curve. This is attributable to the increase in certainty of the data’s behavior.

You're at a friend's house and can't find your phone. You ask them to call it for you. A faint noise is heard.

- A frequentist would use their ears to identify the most likely area the sound is coming from.
- A Bayseian would also use their ears, but they would additionally take into account areas of the home they've previously lost their phone in, when inferring where to look. This prior knowledge can help provide a better prediction of where the phone may be (couch cushions?).

Since this model takes into account both probability and observed data, it is very useful when the goal is pattern recognition and predicting an outcome. Some examples include:

- Detecting SPAM messages before they’re delivered to email inboxes.
- Analyze website activity in terms of visitor access, actions taken on the site, time spent on a web page, and other indicators that help with conversion rate optimization.

An important component of Bayesian Inference is the detection of prior actions and how those actions might predict future activity. Ongoing results can be applied to further prove or disprove the effectiveness of a hypothesis. Bayesian Inference is beneficial for website conversion testing because of its flexibility.

Bayesian testing can be applied to analyze website conversions by measuring the ratio of the number of times a conversion occurred out of the number of trials that were conducted (speaking to the Frequentist approach), and then testers can capture data and leverage learnings to improve the chance for conversions as they go. As more data becomes available, the Bayesian Inference Model can be amended to reflect updated probable behaviors.

Since the introduction of artificial intelligence (AI) and machine learning, the Bayesian Inference Model has been used to recognize patterns and predict the succeeding behavior of a given pattern. As such, it can be continually applied to datasets to analyze the influence of prior actions on future behavior.

In website optimization, for instance, data are gathered following a series of tests to determine which elements on the page lead to a desired result (i.e. conversions). The results can be leveraged to design and construct a web experience that leads visitors to a desired result.

Benefits of Bayesian inference include producing clear answers to direct questions in terms of what treatment will produce the best, most desired effect. Additionally, the model provides what is known as the Highest Posterior Density Region (HPDR), which allows researchers to set parameters (or boundaries) that would produce a target conversion rate.

You can use the Bayesian Inference Model as a means of data analysis to reveal patterns and structure of web activity. These patterns can inform future behavior, both positive and negative, leading to strategic decisions about how and where to make adjustments to a website to enhance performance.

As new data emerges, you can leverage the benefits of the Bayesian Inference Model to apply new probabilities and strategize for optimal web functionality. This simple, more flexible, and more reliable approach to A/B testing is fast becoming the norm in the field of statistical analysis.

If the idea of optimizing for conversions based on prior visitor engagement all while leveraging predictive models sounds appealing, consider leveraging Intellimize's Continuous Conversion™ experiences. All you have to do is launch a few variations and the system's machine learning optimizes for maximum conversions based on engagement and predictive models. The system continually adapts as new data comes in (e.g. visitor behavior changes or you launch a new idea) to ensure that the best combination of ideas are being shown to drive the most conversions possible.