Statistics How To

Credible Interval: Simple Definition

Bayesian Statistics >

A credible interval is the interval in which an (unobserved) parameter has a given probability.

It’s the Bayesian equivalent of the confidence interval you’ve probably encountered before. However, unlike a confidence interval, it is dependent on the prior distribution (specific to the situation). In confidence intervals we also treat the parameter as a fixed value, and the bounds are random variables; in credible intervals, the estimated parameter is treated as a random variable while the bounds are considered fixed.

Examples of Credible Intervals

Suppose we are running an experiment on the distribution of birth weights of children born in a given town. If the subjective probability that the birthweight β is somewhere between 2.8 kgs and 3.5 is 90 %, we can say that 2.8 ≤ β ≤ 3.5 is a 90% credible interval.

If you find out that the 95% credible interval for your statistics final score is 70 to 90, this means you have a 95% chance of having a score between those two numbers.

Technical Definition of a Credible Interval

Remember that an alpha level, α, defines the amount of uncertainty; the probability that we support the alternate hypothesis when really, the null hypothesis is true. We can say that our 100 (1 – α) % credible interval C defines the subset on the parameter space, which we’ll call θ, such that the integral




π here is the prior probability distribution. So, for instance, if you needed a 95 percent credible interval, you would be working to find the interval over which the integral of the prior probability distribution sums to 0.95.

There’s one shortcut we can sometimes take. If the parameter space θ is a discrete space — that is, if we’re working with a set of discrete data points rather than a continuous distribution– we can sum over our interval rather than take an integral.

References

Hitchcock, David. Bayesian Inference: Posterior Intervals (Stat 535 Lecture Notes)
Retrieved from http://people.stat.sc.edu/Hitchcock/stat535slidesday3.pdf on February 16, 2018
Jaynes, E. T. (1976). “Confidence Intervals vs Bayesian Intervals”, in Foundations of Probability Theory, Statistical Inference, and Statistical Theories of Science, pp. 175 et seq. Retrieved from http://bayes.wustl.edu/etj/articles/confidence.pdf on February 17, 2018

------------------------------------------------------------------------------

Need help with a specific statistics question? Chegg offers 30 minutes of free tutoring, so you can try them out before committing to a subscription. Click here for more details.

If you prefer an online interactive environment to learn R and statistics, this free R Tutorial by Datacamp is a great way to get started. If you're are somewhat comfortable with R and are interested in going deeper into Statistics, try this Statistics with R track.

Comments are now closed for this post. Need to post a correction? Please post a comment on our Facebook page.
Credible Interval: Simple Definition was last modified: February 26th, 2018 by Stephanie