Understanding the Difference Between Frequentist Confidence Intervals and Bayesian Credible Intervals

Often students encounter two different approaches to statistical inference: frequentism and Bayesianism. These schools of thought offer different perspectives on how to interpret data and estimate parameters. In this article, we will explore the differences between frequentist confidence intervals and Bayesian credible intervals, helping to illuminate key concepts in a manner accessible to those not deeply versed in mathematical statistics.

Frequentist Approach

Frequentism is a foundational approach in statistics that relies on the idea of performing an experiment multiple times. Common statistical tests, such as the t-test, z-test, and likelihood ratio test, are rooted in frequentist principles. At its core, frequentism assumes that values or proportions we are trying to measure are fixed. For example, if you have a coin that might be unfair, a frequentist perspective would view the coin's bias as a fixed but unknown quantity, and the goal is to estimate that bias and construct a confidence interval around the estimate.

Bayesian Approach

Bayesianism, in contrast, takes a different approach by considering the things we are trying to estimate as random variables. The Bayesian framework requires us to assign a probability distribution to these parameters. This means that the parameters are not fixed but can vary according to a distribution. For instance, in the case of the coin, we might assign a prior probability distribution to the coin's bias and use this to construct a posterior distribution after observing data from our experiment. The credible interval, then, is derived from this posterior distribution.

Confidence Intervals vs. Credible Intervals

The interpretation of a frequentist confidence interval is that if we were to repeat the experiment many times and create confidence intervals from each dataset, approximately 95% of these intervals would contain the true parameter. Importantly, the confidence interval is about the process of repeated sampling rather than a fixed true parameter.

A Bayesian credible interval, on the other hand, is based on the posterior distribution of the parameter. This interval represents the probability that the true parameter lies within a certain range given the data and our prior beliefs. The Bayesian credible interval reflects the uncertainty around the parameter estimate, considering the parameter as a random variable. Thus, the Bayesian approach provides a probabilistic interpretation of the uncertainty.

Interpreting the Intervals

Students often mistakenly interpret the boundaries of confidence and credible intervals as precise and definitive. For example, if the 95% confidence interval for the proportion of Quora users who are left-handed is given as 5-12, many might think it means "We are 95% confident that the true proportion lies between 5 and 12, and any value within this range is equally plausible." However, this is a misunderstanding.

Instead, a Bayesian credible interval can be interpreted as providing a more nuanced understanding. For instance, a 95% credible interval might indicate that the true proportion is most likely between 7 and 10, with some probability for values between 5-7 or 10-13, and very low probabilities for values outside this range. This reflects the probabilistic nature of the Bayesian approach.

One of the challenges in explaining these concepts is that people tend to think in terms of sharp, definitive intervals rather than probabilistic ranges. Statistical inference requires a more nuanced and probabilistic mindset.

Practical Application

To further understand the differences, consider an example of estimating the proportion of left-handed users on Quora:

Frequentist Approach

Collect data and perform a hypothesis test or construct a confidence interval. Interpret the result as: "If we repeated the experiment many times, about 95% of the confidence intervals we construct would contain the true proportion."

Bayesian Approach

Choose a prior distribution and update it with the data to obtain a posterior distribution. Construct a credible interval from the posterior distribution. Interpret the result as: "Given the data and our prior beliefs, the true proportion lies within the interval with 95% probability."

The key difference lies in the interpretation of the interval: a confidence interval reflects uncertainty due to sampling, while a credible interval reflects uncertainty due to the random nature of the parameter itself.

Conclusion

While the frequentist and Bayesian approaches offer different perspectives on statistical inference, both are valuable tools in the statistician's toolkit. Understanding the differences between confidence intervals and credible intervals can help clarify these concepts and enhance one's ability to interpret and communicate statistical results effectively.

Keywords: frequentist confidence intervals, bayesian credible intervals, statistical inference