Credible vs confidence intervals: Bayesian approach

Ever tried explaining to your product manager why you're "95% confident" about something, only to watch their eyes glaze over when you mention it's not actually a 95% probability? Yeah, me too. This confusion sits at the heart of one of statistics' most misunderstood concepts: the difference between confidence intervals and credible intervals.

Here's the thing - both types of intervals try to answer the same basic question: "How uncertain are we about this number?" But they approach it from completely different philosophical angles, and understanding the distinction can save you from some seriously awkward conversations about what your data actually means.

The foundations of confidence intervals and credible intervals

Let's start with confidence intervals - the workhorse of traditional statistics. These bad boys tell you something that sounds simple but is actually kind of weird: if you repeated your experiment a bunch of times, 95% of those intervals would contain the true value. Notice I didn't say "there's a 95% chance the true value is in this interval." That's because confidence intervals don't work that way, and this trips up even experienced analysts.

Think of it like fishing with a net. A 95% confidence interval is like saying "if I throw this net 100 times, I'll catch the fish about 95 times." But once you've thrown it? You either caught the fish or you didn't - there's no probability about it.

Credible intervals take a different approach. These come from Bayesian statistics, where we treat uncertainty differently. With a credible interval, you can actually say "there's a 95% probability the parameter is in this range, given what we know." It's the interpretation everyone wants from confidence intervals but can't have.

The key difference? Credible intervals combine what you knew before (your prior beliefs) with what your data tells you. It's like updating your weather forecast as new data comes in, rather than just looking at today's measurements in isolation.

Both approaches help us quantify uncertainty, but they're answering subtly different questions. Choosing between them isn't just about math - it's about what kind of statement you want to make and what assumptions you're comfortable with.

Philosophical differences: Frequentist vs Bayesian approaches

The split between confidence and credible intervals comes down to a fundamental disagreement about what probability even means. Frequentists see probability as what happens in the long run - flip a coin enough times, and you'll get heads about half the time. The parameter you're estimating? It's fixed. It has one true value, and probability describes how your measurements dance around it.

Bayesians flip this on its head. To them, probability represents uncertainty about what's true. The parameter isn't fixed in some philosophical sense - it's unknown, so we describe our uncertainty about it with probability. This lets Bayesians do something frequentists can't: directly calculate the probability that a parameter falls in a specific range.

Here's where it gets practical. Say you're testing a new feature's conversion rate. A frequentist confidence interval tells you: "If we ran this test many times, 95% of our intervals would contain the true conversion rate." But a Bayesian credible interval says: "Given our data and what we knew before, there's a 95% chance the conversion rate is between X and Y."

Which one's more useful? Depends on what you're trying to do. If you need to make a decision right now based on this one test, the Bayesian interpretation is probably more helpful. But if you're setting up a system that needs to work reliably across many tests, the frequentist guarantee might be what you want.

The real kicker is that in many practical situations, the two intervals are pretty similar numerically. The philosophical differences matter more for interpretation than for the actual numbers you get.

Practical applications of credible intervals in Bayesian analysis

So when do credible intervals really shine? They're fantastic when you have useful prior information. David Robinson showed this beautifully in his analysis of baseball batting averages. Early in the season, when a player has only a few at-bats, their observed average might be .000 or 1.000 - obviously not their true skill level. By incorporating prior knowledge about typical batting averages, credible intervals give you a much more realistic range.

This advantage becomes huge when you're dealing with:

• Small sample sizes (like early product launches)

• Rare events (think conversion rates for expensive products)

• Sequential decisions (should we stop this A/B test early?)

Speaking of A/B testing, Bayesian methods are gaining ground in experimentation platforms. Companies like VWO and Google Optimize use Bayesian approaches because they let you answer questions like "What's the probability that variant A is better than B?" - exactly what stakeholders want to know. At Statsig, the platform supports both approaches, recognizing that different teams have different needs.

The real power of credible intervals shows up in decision-making. Instead of the binary "significant or not" world of confidence intervals, you get probability distributions. You can calculate expected losses, make risk-adjusted decisions, and actually answer questions like "What's the chance we'll lose more than $10,000 if we pick the wrong variant?"

But here's the catch - you need to specify priors, and that's where things get tricky. Bad priors lead to bad posteriors (as the saying goes). In practice, many teams use weakly informative priors that don't assume much but still help stabilize estimates when data is sparse.

Common misconceptions and choosing the appropriate interval

Let's clear up the biggest confusion: a 95% confidence interval does not mean there's a 95% chance the parameter is inside. I've seen senior data scientists get this wrong. The parameter is either in there or it's not - there's no probability about it from a frequentist view. The 95% refers to the long-run success rate of the method, not this specific interval.

Credible intervals have their own pitfalls. The big one? Your prior can hijack your results if you're not careful. I've seen teams use "informative" priors that were really just their hopes and dreams encoded as probability distributions. The results looked great until reality caught up.

So how do you choose? Here's my practical framework:

Use confidence intervals when:

• You have no useful prior information

• You need to satisfy regulatory requirements

• Your audience expects traditional p-values

• You're running many similar tests and need frequentist guarantees

Use credible intervals when:

• You have genuine prior knowledge to incorporate

• You need to make decisions based on probability statements

• You're dealing with small samples or rare events

• Your stakeholders want intuitive interpretations

The team at Statsig wrote a nice comparison that digs into the computational aspects too. In practice, many modern platforms give you both options, so you don't have to choose once and stick with it forever.

Remember, neither approach is inherently superior. They're tools designed for different jobs. The key is understanding what each one actually tells you and matching that to what you need to know. Don't let anyone tell you one is always better - that's like saying hammers are better than screwdrivers.

Closing thoughts

At the end of the day, both confidence and credible intervals are trying to help you express uncertainty in your estimates. The frequentist approach gives you guarantees about long-run behavior, while the Bayesian approach lets you make direct probability statements about parameters. Neither is wrong - they're just answering different questions.

The real skill is knowing when to use each one and, more importantly, how to explain what they mean to your stakeholders. Because let's be honest - most people just want to know if they should ship the feature or not. Your job is to translate the uncertainty in a way that helps them make better decisions.

Want to dive deeper? Start with Andrew Gelman's "Bayesian Data Analysis" for the full Bayesian treatment, or Casella and Berger's "Statistical Inference" for the frequentist perspective. And if you're working on A/B tests specifically, experiment with both approaches in your platform of choice to see which one fits your workflow better.

Hope you find this useful!