Prior distributions: Incorporating historical knowledge

Ever tried explaining to your boss why you're not just using the data in front of you, but also incorporating what you learned from last quarter's experiments? That's essentially what Bayesian priors are all about - using what you already know to make better decisions with new data.

The thing is, picking the right prior can feel like choosing a Netflix show with your partner. You know what worked before, you have some preferences, but ultimately you need to make a choice that won't come back to haunt you. Let's talk about how to actually do this well.

Understanding prior distributions in Bayesian analysis

So what exactly are priors? They're basically your starting assumptions before you see any new data. Think of them as your educated guess based on everything you know so far.

There are three main types you'll run into:

• Non-informative priors: These are your "I know nothing" priors. You use them when you genuinely have no clue what to expect and want the data to do all the talking.

• Informative priors: The opposite - you actually know something useful from past experiments or domain expertise, so you bake that knowledge in.

• Conjugate priors: The math nerd's favorite. They make calculations way easier because the math works out nicely.

Choosing the right prior is where things get interesting (and where people often mess up). You need to balance what you know, what data you have, and how much computational power you're willing to burn.

The Reddit statistics community has some pretty heated debates about where priors actually come from in real-world applications. The short answer? It depends. Sometimes you pull from historical data or expert opinions, other times you go with software defaults and hope for the best.

Getting good informative priors from domain experts isn't easy. You can't just ask "what's your prior?" - you need to translate their knowledge into probability distributions. But when done right, it makes your analysis way more credible.

Leveraging historical knowledge to inform priors

Here's where things get practical. Your best source for priors is often sitting right in your historical data.

Take medical research. If you're studying life expectancy, you don't start from scratch. You already know smokers tend to live shorter lives, so you build that into your prior. Same goes for A/B testing at tech companies - you probably have months or years of conversion rate data that tells you what's reasonable to expect.

In social science research, teams often rely on meta-analyses or default priors in software like JASP to get started. It's not perfect, but it beats pulling numbers out of thin air.

The tricky part is actually extracting this knowledge from experts. You've got two approaches:

• Direct: "What range do you expect for this parameter?"

• Indirect: "How surprised would you be if we saw X?"

Both work, but indirect methods often get more honest answers. People are better at expressing surprise than pinpointing exact probability distributions.

The payoff for getting this right is huge. Good priors mean more accurate estimates, especially when you're dealing with small samples or complex relationships. It's like having a head start in a race - you still need to run, but you're not starting from the blocks.

Practical applications of informative priors

Let's get specific about where this actually helps.

In A/B testing, informative priors are game-changers. Instead of waiting weeks for statistical significance, you can incorporate what you learned from previous tests. Statsig's Bayesian calculators make this pretty straightforward - plug in your historical conversion rates, and you've got a solid starting point.

Clinical trials take this even further with meta-analytic predictive (MAP) priors. Here's the basic idea:

1. Collect data from multiple similar studies

2. Synthesize it into a prior distribution

3. Run your trial with a smaller sample size

This is huge for rare disease research where every patient counts. You're not throwing away knowledge from previous trials - you're building on it.

Education researchers face similar challenges. When setting priors for intervention studies, they pull from similar programs or pilot studies. If a reading intervention typically improves scores by 10-20%, that becomes your prior. Much better than assuming it could be anywhere from -100% to +1000%.

The reality check: Picking priors in practice isn't as clean as textbooks suggest. You'll rarely have perfect historical data. Sometimes you'll use Gaussian distributions because they're convenient, not because they're right. That's fine - just be honest about it.

Best practices and considerations in prior selection

Let's talk about how to avoid shooting yourself in the foot with bad priors.

First rule: transparency beats perfection. Document where your priors came from. Did you use historical data? Expert opinion? Software defaults from JASP? Say so. Your future self (and reviewers) will thank you.

Here's the good news - as you collect more data, priors matter less. This is the beauty of Bayesian methods. Start with a reasonable guess, and the data will pull you toward the truth. Eventually, Bayesian and frequentist approaches converge.

Sensitivity analysis is your friend. Try a few different priors and see if your conclusions change. If they don't, great - your results are robust. If they do, you know you need more data before making strong claims.

Real-world prior selection usually looks like this:

• Check historical data (if you have it)

• Talk to domain experts (if they'll talk to you)

• Look at published studies (if they exist)

• Run sensitivity analyses (always)

• Document everything (seriously, do this)

For medical data analysis, this might mean understanding baseline differences in populations. For tech companies, it's about historical conversion rates and seasonal patterns.

The bottom line: Yes, priors can be subjective. But subjectivity isn't the enemy - hidden subjectivity is. Make your assumptions explicit, test their impact, and let the data do its job.

Closing thoughts

Bayesian priors aren't magic - they're just a formal way of saying "let's not pretend we know nothing." The trick is being thoughtful about what you assume and honest about uncertainty.

Start simple. Use historical data when you have it. Ask experts when you don't. Run sensitivity analyses always. And remember: even a rough prior beats no prior when you're making decisions with limited data.

Want to dive deeper? Check out Andrew Gelman's blog for practical Bayesian wisdom, or play around with Statsig's Bayesian calculators to see how priors affect your A/B tests in real-time.

Hope you find this useful!