Ever found yourself scratching your head over statistical intervals? Don't worry, you're not alone.

Understanding these concepts is crucial for making sense of data, especially when you're trying to quantify uncertainty around your estimates.

In this blog, we'll break down confidence intervals and credible intervals—the go-to tools from the worlds of frequentist and Bayesian statistics. We'll explore what they mean, how they're constructed, and when to use each one. By the end, you'll have a clearer picture of these intervals and how they can help you make better decisions with your data. Let's get started!

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

Understanding statistical intervals: confidence intervals and credible intervals

When we're diving into data analysis, statistical intervals are our allies for quantifying uncertainty around parameter estimates derived from sample data. They give us a range where the true parameter values likely reside based on the observed data. Two key types of intervals you'll encounter are confidence intervals and credible intervals, stemming from frequentist and Bayesian statistics, respectively.

Confidence intervals, rooted in frequentist statistics, are built solely from sample data without incorporating prior information. They indicate the range within which the true parameter would fall a certain percentage of times if we repeated the experiment many times. However, they don't provide a direct probability statement about the parameter for a single interval.

On the flip side, Bayesian credible intervals combine prior beliefs with observed data to estimate a parameter's plausible range. They offer a probability statement about the parameter falling within the interval, given the data and prior information. This interpretation can be more intuitive, as it directly addresses the probability of the parameter's value.

Credible intervals can be especially informative with smaller sample sizes, particularly when prior knowledge is crucial. However, they can be more computationally intensive due to the Bayesian framework. The choice between confidence and credible intervals depends on your statistical perspective, the prior information you have, and the context of your analysis.

Confidence intervals: the frequentist perspective

In the frequentist world, confidence intervals are all about repeated sampling. They reflect the long-run frequency properties of estimators, meaning that if we repeated the same experiment many times, the true parameter value would fall within the interval a certain percentage of the time. For example, a 95% confidence interval indicates that in 95 out of 100 repetitions, the true parameter would be captured within the intervals calculated.

It's important to note that a 95% confidence interval doesn't imply a 95% probability that the specific interval contains the true parameter value. This is a common misconception about confidence intervals. In the frequentist perspective, the parameter is considered a fixed, unknown quantity; it's the intervals that would vary if we repeated the sampling process.

How confidence intervals are constructed

Building confidence intervals typically involves these steps:

1. Determine the sample statistic (e.g., mean, proportion) that estimates the population parameter.

2. Calculate the standard error of the sample statistic, measuring the variability across different samples.

3. Choose a confidence level (like 95% or 99%) reflecting your desired level of certainty.

4. Use the appropriate formula or statistical table to compute the confidence interval based on the sample statistic, standard error, and confidence level.

The resulting confidence interval provides a range of plausible values for the population parameter based on your observed sample data. Remember, though, the confidence interval is a statement about the method's performance over repeated sampling—not a direct probability statement about the parameter itself.

Common misconceptions about confidence intervals

A big misconception is interpreting confidence intervals as direct probability statements about the parameter. A 95% confidence interval doesn't mean there's a 95% chance the true parameter lies within the interval. Instead, it means that if we repeated the sampling process many times, 95% of the resulting intervals would contain the true parameter value.

Another common mix-up is thinking that the parameter itself is a random variable that falls within the interval with a certain probability. In the frequentist framework, parameters are considered fixed, unknown quantities, and it's the intervals that are random and vary across different samples.

Credible intervals: a Bayesian approach

In the Bayesian framework, we treat parameters as random variables with prior probability distributions. These priors represent our beliefs about the parameter before observing data. Bayes' theorem is then used to update these priors with the likelihood of the observed data, resulting in the posterior distribution.

A credible interval is derived from this posterior distribution. It provides a direct probability statement about the parameter given the data. For instance, a 95% Bayesian credible interval means there's a 95% probability that the true parameter value lies within that range.

This interpretation is often more intuitive since it allows us to make probabilistic statements about the parameter itself. Bayesian credible intervals incorporate both prior information and the observed data, making them particularly useful when we have strong prior knowledge.

However, specifying priors can be challenging and somewhat subjective. The choice of prior can significantly impact the resulting credible interval, especially with small sample sizes. Therefore, it's crucial to carefully consider and justify your prior beliefs when using Bayesian methods.

Comparing confidence and credible intervals: philosophical and practical differences

Philosophical differences between frequentist and Bayesian methods

Frequentists and Bayesians have different takes on probability. For frequentists, probability is the long-term frequency of events; for Bayesians, it's a degree of belief. This philosophical difference leads to distinct interpretations of intervals and probability statements.

Practical implications in real-world data analysis

Choosing between confidence and credible intervals depends on the context, prior information, and the need for intuitive interpretations. When prior knowledge is crucial, Bayesian credible intervals may provide more meaningful insights. They can handle situations with small sample sizes or when incorporating prior information is beneficial.

However, frequentist confidence intervals remain widely used due to their simplicity and well-established properties. They're particularly useful when prior information is limited or when the focus is on long-term frequency properties.

At Statsig, we often utilize Bayesian methods in our A/B testing framework because they can be more informative and intuitive for decision-making. For example, Bayesian credible intervals directly address the probability of parameter values given the data, making them easier to interpret for non-statisticians.

It's crucial, though, to carefully consider the implications of practices like optional stopping in Bayesian A/B testing and to thoroughly evaluate the effectiveness of Bayesian methods through simulations tailored to your specific use cases.

Closing thoughts

Grasping the differences between confidence intervals and credible intervals is key to making informed decisions in data analysis. Whether you lean toward the frequentist or Bayesian approach, understanding when and how to use these intervals will enhance your statistical toolkit.

If you're eager to dive deeper, there are plenty of resources out there to explore these topics further. At Statsig, we're passionate about helping you make sense of data, and we incorporate these statistical concepts into our platform to empower better decision-making.

Hope you found this helpful!