What is Bayesian statistics?

Bayesian statistics offers a flexible and easy-to-understand way to refresh our beliefs as we get new evidence. Instead of relying on strict significance thresholds, Bayesian methods help us make more informed, .

This approach is based on three key ideas: priors, likelihoods, and posteriors.

Priors

Priors are our initial expectations or assumptions about a parameter or hypothesis before we collect information.

For instance, if an is run to see if a new homepage design increases , the prior might be that the new design will perform at least 5% better than the old one.

These priors often come from past experience, expert knowledge, or a general sense of what we expect to see. Priors enable us to incorporate our existing understanding into the analysis rather than starting from scratch with each new set.

Likelihoods

Likelihoods tell us the probability of observing the results we collect, given a particular set of parameter values.

In other words, the likelihood shows how probable it is that the results we’re observing will be true if a certain hypothesis is true. How well does our model fit the observed data?

Posteriors

Posteriors is the end stage of the Bayesian process. The posterior probability is our updated belief about the parameter or outcome after considering both the prior and the likelihood. It gives us a revised, evidence-informed conclusion.

We use posteriors to refine our strategies and guide future actions. By combining our initial beliefs with the new data we’ve collected, posteriors help us better understand how likely a hypothesis or outcome is. This method allows us to make more and adjustments.

There are two main statistical approaches to analyzing evidence and making decisions: . Both aim to extract insights from data, but they have different underlying philosophies and methods.

Frequentist approach

The frequentist approach, used in conventional methods like null hypothesis significance testing (NHST), views probability as how often an event happens over many trials.

Parameters are seen as fixed but unknown values—if you flip a coin, the chance of landing head is set at 50%, even if you don’t know the outcome in advance. The focus is to figure out how likely the collected results are if a certain hypothesis is true.

Frequentist methods use and confidence intervals to measure uncertainty and decide if an experiment's results are . The goal is to make a clear yes-or-no verdict: either reject or fail to reject a null hypothesis based on pre-set thresholds.

Calculate the statistical significance using the

Bayesian approach

The Bayesian approach sees probability as a degree of belief. Parameters are treated as random variables with their own probability distributions.

Instead of only analyzing the data, Bayesian analysis uses your prior realizations and assumptions. As you gather new evidence, these expectations are updated—such as flipping a coin repeatedly and adjusting your expectations based on the outcomes of previous flips.

A Bayesian study results in a posterior probability distribution that reflects a revised understanding of the hypothesis or parameters. This method supports a more dynamic and nuanced way of interpreting information, acting on results, and handling ambiguity.

Choosing the right method

Each strategy has its strengths. Frequent methods often work well for large, standardized experiments. Bayesian methods excel when dealing with smaller samples, incorporating expert knowledge, and making more contextual decisions.

Many data analysts and product teams find combining elements of both approaches can be the most effective way to make informed, data-backed choices.

Bayesian statistics functions well in web and product , such as A/B testing.

Classic frequentist methods, such as null hypothesis significance testing (NHST), can be rigid and often fall short in real-world decision-making. Bayesian frameworks are more versatile and align better with how we naturally reach conclusions.

In an A/B test, a Bayesian assessment begins with your prior thoughts about how the new variation will likely perform—this is your parameter of interest, such as a or engagement . Then, you modify these opinions based on the results as you get more evidence.

The outcome is a posterior distribution, which gives you a complete picture. Instead of simply deciding if the results are significant or not, you get:

• A probability that the new variation is better
• An estimate of how much better it might be
• An understanding of the unknowns around these estimates.

This method enables more nuanced, data-driven resolutions. Rather than setting an arbitrary threshold, such as a 5% lift in , you can look at the entire distribution of possible outcomes and choose the option that best fits your business goals and risk tolerance.

Bayesian statistics also enables you to include external data or domain intelligence in your evaluation. If you have information from similar experiments, you can include that in your priors, leading to more reliable and actionable discoveries.

Bayesian statistics and sequential testing

One application of Bayesian methods in is .

Instead of running an experiment with a fixed sample size, you can continuously monitor the data and stop the experiment as soon as you have enough information to make a confident ruling.

This strategy can lead to significant time and cost savings compared to traditional fixed-sample-size approaches.

Compared to standard frequent strategies, Bayesian statistics offers several benefits that enhance . Bayesian methods provide insights that make your findings more flexible, contextual, and realistic.

Great for fast-changing environments

Bayesian enables you to update your beliefs as fresh figures become available. This element makes it well-suited for dynamic environments where conditions can change quickly, such as in digital product development.

Incorporates prior knowledge

Bayesian statistics enables you to easily combine your existing domain expertise and ideas—such as how a new or design might perform—with the latest data you’ve collected.

This technique is valuable when working with smaller sample sizes, where prior information can significantly influence your conclusions. By using all available knowledge, you can make more reliable rulings.

Easier to interpret

Bayesian posterior probabilities are much simpler to understand than the abstract concepts of p-values and confidence intervals. Posteriors show the direct probability that the hypothesis or parameter is true, which matches how we naturally make decisions. Insights from Bayesian analyses are easier to communicate and act on.

Supports ongoing learning

Traditional hypothesis testing views as a one-time task: you collect evidence, run a test, and get a final result. Bayesian methods treat testing as an ongoing process of learning and updating expectations.

As you gather more information over time, you continuously refine your understanding. You become more responsive to market changes and customer preferences and make better long-term choices.

Handles uncertainty

Uncertainty is normal in data analysis, but frequentist methods often overlook or minimize it. Bayesian statistics, however, quantifies and weaves ambiguity into the study, helping you reach more informed, risk-aware conclusions. This approach enables you to better account for the variability in your experiments and estimates.

Useful for personalization and targeting

Bayesian techniques can be used to develop models that adapt to individual and preferences. These actions can improve the effectiveness of targeted marketing, recommendations, and other personalized experiences.

Provides a unified framework

Most importantly, Bayesian inference offers a consistent framework for a wide range of data analysis and modeling problems. Whether you’re running A/B tests, building models, or forecasting business trends, Bayesian methods apply coherently and methodically. They are a powerful and versatile tool for any data science task.

While the advantages of Bayesian statistics are substantial, it’s important to be aware of potential challenges and limitations. Understanding these can help you mitigate risks and overcome initial hurdles.

Specifying priors can be challenging

One of the core aspects of Bayesian analysis is the use of prior views or assumptions. However, accurately setting these priors can be difficult, particularly with complex models or unfamiliar information.

If the priors are too strong or biased, they can skew the results. Ensuring that priors are well-justified and based on genuine knowledge is crucial.

Requires advanced computational techniques

Calculating Bayesian posterior distribution often involves advanced computational techniques, which can be resource-intensive, especially for large or intricate models.

Although some tools have made Bayesian modeling more accessible, the barrier to entry is generally higher than for simpler frequentist methods.

Tricky to interpret

While Bayesian outcomes are often more intuitive than p-values, the full posterior distributions can still be complicated and challenging to interpret—more so for non-technical stakeholders.

Effectively communicating Bayesian results requires careful visualization and clear explanations.

Difficult to debug

Diagnosing and resolving issues can be challenging when Bayesian models don’t behave as expected. The iterative, probabilistic nature of Bayesian inferences can make problems subtle and hard to isolate. Debugging requires a deep understanding of both the model and the underlying data.

Scalability limitations

For large-scale web and product experiments with millions of users, the technical requirements of Bayesisan analysis can become a limitation. Frequentist methods may still be more practical in high-volume cases.

Resistance to change

Many data analysis workflows and tools are built around traditional frequentist statistics. Shifting to a Bayesian mindset can demand significant changes to processes, tools, and organizational mindsets, slowing adoption and creating resistance within teams.

Follow these best practices to set up your Bayesian experiments and analyses for success. With thoughtful planning and execution, you can maximize the framework’s potential.

Start with informed priors

Carefully consider your prior beliefs to set up well-justified priors. Use relevant historical evidence, domain expertise, and industry benchmarks to establish these priors. This practice helps ensure your initial assumptions are grounded in solid evidence.

Use existing tools and platforms

Take advantage of the Bayesian capabilities built into popular experimentation and . These tools handle much of the computational complexity, making it easier to start with Bayesian methods without building everything from scratch.

Communicate results effectively

Bayesian statistics can be complicated, so put effort into presenting discoveries clearly and intuitively. Use visualizations such as probability distributions to help stakeholders grasp the nuanced, probabilistic nature of the results. Translate Bayesian outputs into actionable business implications to make them more accessible.

Iterate and refine

Bayesian analysis is an ongoing process. As you collect more data, revisit and refine your pre-existing assumptions and models to improve the reliability of your insights. Treat the evaluation as an iterative process rather than a one-time event.

Combine with other methods

Bayesian approaches can be used alongside other statistical techniques, such as conventional A/B testing or causal modeling. Combining methods can provide a richer, more complete view of your experimental results.

Monitor for potential biases

While Bayesian methods can address certain biases, they are not immune to issues such as selection bias or measurement error. Carefully monitor data sources and experimental design to find and alleviate potential biases.

Involve cross-functional experts

Effective Bayesian analysis often requires expertise from various disciplines—such as statistics, analytics, product management, and domain-specific proficiency. Assemble a collaborative team to provide diverse perspectives and ensure the assessment aligns with your business needs.

Putting Bayesian principles into practice often means using analytical tools and platforms. This software makes Bayesian analysis more accessible for web and product experimentation.

One such platform is , which concentrates on providing . Amplitude’s Bayesian inference engine lets you run Bayesian A/B tests and gain deeper findings.

Instead of telling you whether a product version performed well, Amplitude shows you the full posterior probability distribution. You see the range of possible performance improvements and the likelihood of each.

This probabilistic view empowers you to make decisions tailored to your end goals and capacity for risk. You can see not just whether a new feature is likely to perform but by how much. Then, you can weigh this information against factors such as development cost, user experience impact, and strategic fit.

Amplitude also makes incorporating historical data or domain knowledge into your priors easy. If you know how similar experiments have been performed in the past, you can include that information in your analysis for even more reliable insights.

Beyond Amplitude, there are also several open-source Bayesian tools and frameworks that data scientists and analysts can use, such as:

• : A Python library for building and fitting Bayesian models
• : A probabilistic programming language that enables adaptable Bayesian modeling
• : A program for Bayesian data analysis using simulation.

These tools provide flexibility and control but may require more statistical expertise and coding skills to implement effectively.

Reach confident conclusions with Amplitude

Bayesian methods aren’t just theoretical—they’re becoming increasingly accessible and integrated into the tools product teams use daily, such as Amplitude.