Have you ever run a Bayesian test and wondered how your prior choices affect the outcome?

In the world of data analysis, balancing what we already know with new data can be tricky. But understanding this balance is key to making accurate predictions and smarter decisions.

In this blog, we'll explore how sample size and priors interplay in Bayesian tests. We'll talk about leveraging pre-experimental data, even when your dataset is small, and discuss practical considerations for Bayesian A/B testing. Let's dive in and see how we can make the most out of Bayesian methods!

The importance of sample size and priors in Bayesian tests

Sample size is super important when it comes to reliable Bayesian estimates. If you're working with small datasets without informative priors, you might end up with shaky results. That's because Bayesian methods hinge on how data and prior knowledge come together.

By bringing in what we already know, priors help us get better inferences, especially when we're short on data. This comes in handy in situations with rare events or tiny datasets, where standard frequentist methods often falter. Using prior information, Bayesian approaches can tackle the hurdles of insufficient data.

But here's the catch: priors need to be chosen carefully. If they're uninformative or just plain wrong, they can skew your results—even if you have lots of data. So, picking the right priors is key for accurate Bayesian inference.

Priors can come from historical data, expert opinions, or theory. For instance, commensurate priors let you blend pre-experimental data in a solid way. And doing a sensitivity analysis can show how different priors affect your results.

So, while Bayesian methods are flexible and let us include prior knowledge, it's crucial to think carefully about sample size and how we pick our priors. Balancing data and prior info can yield valuable insights, especially when the data is tricky.

Related reading: A beginner's guide to Bayesian experimentation.

Leveraging commensurate priors to incorporate pre-experimental data

With commensurate priors, we get to borrow strength from historical data, making our Bayesian estimates better. By mixing in pre-experimental info—from expert opinions to previous studies—we can crank up accuracy and make figuring out sample sizes more efficient.

This is super handy in fields like clinical trials. In rare disease trials, for example, where sample sizes are tight, bringing in pre-experimental data through Bayesian methods leads to more precise estimates and smarter decisions.

But we have to think about the relevance and quality of that pre-experimental data. The fit between historical and current data—how "commensurate" they are—should be checked to weight the prior info appropriately. Techniques like Gamma mixture priors help account for differences in data sources.

By carefully picking and blending in pre-experimental data, we can tap into the power of Bayesian stats to improve sample size determination and overall efficiency. This gives us a more complete, data-driven decision-making process, leading to results that are both reliable and actionable.

Understanding the impact of priors in large sample sizes

Even when we've got large sample sizes, priors still matter in Bayesian analysis. Sure, loads of data can overpower prior info, but priors still sway hypothesis testing and model selection. So, thinking carefully about prior knowledge is important, especially when data doesn't tell us everything about specific parameters.

Bayesian inference uses posterior distributions to figure out probabilities for competing hypotheses. We should still incorporate priors thoughtfully in model evaluation and prediction, even as sample sizes grow. The subtle dance between data and priors highlights the need to balance objective data and subjective beliefs in Bayesian stats.

Informative priors might still play a big role in model selection and hypothesis testing. They let us bring valuable prior knowledge into the mix. In this way, Bayesian methods bridge the gap between subjective beliefs and objective data.

At Statsig, we understand how crucial it is to consider priors, no matter the sample size. Our Bayesian approach helps you find the right balance between prior beliefs and data-driven insights.

Practical considerations in Bayesian A/B testing

While Bayesian methods have their perks, they're not bulletproof against issues like peeking and jumping to conclusions too soon. As David Robinson points out, even with Bayesian approaches, stopping tests early can mess up your results. So, it's crucial to understand the limitations and pitfalls to make them work effectively.

Picking the right priors is essential to avoid bias in Bayesian tests. Priors should be justified based on what you know and the context of your experiment. For example, Statsig uses naive priors by default to minimize bias while keeping up statistical power.

To get reliable results, follow best practices in your Bayesian experiments. First, clearly define your hypotheses and pick relevant metrics before kicking off the test. Second, avoid peeking at results too often—it can lead you to premature conclusions. Third, use appropriate priors based on what you know, and make sure to justify your choices. Lastly, interpret results carefully, keeping in mind the posterior distribution and any uncertainties.

By following these guidelines, you can tap into the benefits of Bayesian A/B testing while sidestepping potential issues. Bayesian methods are powerful decision-making tools, but they work best when you understand and implement them properly.

Closing thoughts

Understanding how sample size and priors interplay in Bayesian tests is key to making accurate inferences and better decisions. By carefully selecting priors and considering the size of your data, you can leverage Bayesian methods to their full potential—even in challenging situations. Remember, balancing prior knowledge and new data can make all the difference.

If you're keen to learn more, check out Statsig's beginner's guide to Bayesian experiments or dive into David Robinson's insights. Happy experimenting—we hope you find this useful!