Introduction to Bayesian A/B Testing
Bayesian A/B testing is a statistical approach to comparing two or more versions of a product, webpage, or application to determine which one performs better. This method has gained popularity due to its simplicity, flexibility, and ability to provide intuitive results, especially when dealing with small sample sizes.
Key Differences from Frequentist Approaches

Interpretation of Probability:
 Frequentist Approach: Probability is viewed as the likelihood of an event occurring based on repeated trials. This approach relies on hypothesis testing with fixed sample sizes and pvalues to determine statistical significance[2].
 Bayesian Approach: Probability is seen as a measure of belief that is updated with new data and prior knowledge. This allows for continuous updating of beliefs as more data becomes available[2].

Sample Size:
 Frequentist Approach: Requires a predefined sample size to ensure statistical significance.
 Bayesian Approach: Does not require a fixed sample size, allowing for more flexible and adaptive testing[2].

Peeking at Data:
 Frequentist Approach: Peeking at the data while the test is running is generally not allowed to avoid bias.
 Bayesian Approach: Allows for peeking at the data, although with caution, to make more informed decisions[2].
Steps of Bayesian A/B Testing

Select Your Distribution:
 Choose a distribution based on the metric of interest. Common distributions include binomial (for true/false outcomes), multinomial (for categorical outcomes), and exponential (for timetoevent outcomes)[1].

Calculate Your Prior:
 Select a conjugate prior that matches the chosen distribution. This prior reflects any preexperiment data or beliefs. Parameters can be chosen manually or using statistical libraries[1].

Run the Experiment:
 Collect data from the experiment, ensuring random assignment of participants to different variants.

Calculate Key Metrics:
 Use Monte Carlo simulations to calculate key metrics such as percent lift, probability of one variation being better than another, and expected loss[1].
Advantages of Bayesian A/B Testing

Smaller Sample Sizes:
 Bayesian methods can often achieve reliable results with smaller sample sizes compared to frequentist methods, reducing the time and resources needed for testing[1].

Intuitive Results:
 Bayesian A/B testing provides results that are easier to interpret, such as the probability that one variation is better than another, rather than relying on pvalues[2].

Flexibility:
 Bayesian methods allow for continuous updating of beliefs as new data arrives, making them more adaptable to changing conditions[2].

Incorporating Prior Knowledge:
 Bayesian statistics naturally incorporate prior knowledge into the analysis, which can be particularly useful when leveraging information from previous experiments[5].
Practical Considerations

Computational Intensity:
 While Bayesian methods offer several advantages, they are computationally more intensive than frequentist methods. However, these computations are typically performed offline, reducing performance requirements[1].

Choosing Effective Priors:
 The choice of prior distribution is crucial. It is advisable to err on the side of a weak prior (smaller hyperparameter values) to avoid overly influencing the results with preconceived beliefs[1].

Early Stopping and Power Analysis:
 Practical considerations include balancing the need to detect true differences quickly while minimizing false discoveries (the early stopping problem) and planning the length and size of A/B tests using power analysis[3].
Conclusion
Bayesian A/B testing offers a powerful and flexible approach to experimentation, allowing for more intuitive and datadriven decisions. Its ability to handle small sample sizes, incorporate prior knowledge, and provide continuous updates makes it an attractive choice for many industries, particularly in tech where rapid experimentation is common. However, it requires careful consideration of prior distributions and computational resources. As the industry continues to move towards Bayesian methods, understanding these principles is essential for making informed decisions based on empirical data.
Citations:
 [1] https://towardsdatascience.com/bayesianabtestinganditsbenefitsa7bbe5cb5103
 [2] https://www.dynamicyield.com/lesson/bayesiantesting/
 [3] https://www.pymc.io/projects/examples/en/latest/causal_inference/bayesian_ab_testing_introduction.html
 [4] https://www.youtube.com/watch?v=nRLI_KbvZTQ
 [5] https://matteocourthoud.github.io/post/bayes_reg/