# Credible Interval Format?

I'm reading a datasheet that has listed a "credible interval". By my novice understanding, it is similar to a confidence interval. It is formatted like this: (-8.13, -2.69). So would the -8.13 be the 97.5% CI and the -2.69 be the 2.5% CI? The reason I'm using 97.5/2.5 is because there's an older dataset of this data with those percentages outlined as the "confidence intervals", so I'm making a giant assumption and assuming the credible intervals are using the same percentages.

Is this a standard format for credible intervals? If so, what does each number represent?

• All "credible" tells you is that the probability interpretation is Bayesian and not frequentist. If they don't state the level of the interval, it's usually assumed to be 95%. All we can say about the endpoints is that we have 95% belief the parameter lies between those values. They might be the 2.5th, 97.5th quantiles of the posterior. But there are other ways to come up with such values like the half density interval (HDI). – AdamO Mar 29 '18 at 16:18

Confidence intervals and credible intervals are obtained by different means and they have different interpretations.

Intuitively, confidence intervals are obtained solely from the information encapsulated in the data about the unknown quantity to be estimated and are a frequentist statistics concept. In contrast, credible intervals are a Bayesian statistics concept and can incorporate not only the information present in the data but also any information about the unknown quantity to be estimated available before the data are collected (e.g., information available in the literature or from experts).

Both types of intervals are reported as a range of values (a, b), where a is the lower end point of the range and b is the upper end point.

Credible intervals have a more natural interpretation than confidence intervals.

For a 95% credible interval (a,b), the unknown quantity of interest is expected to lie in the range a to b with a 95% probability.

For a 95% confidence interval (a,b), if we were to repeat our study many, many times under similar conditions, we would expect that 95% of the resulting confidence intervals for the unknown quantity of interest would include this quantity. For this reason, we say that we are 95% confident that the unknown quantity of interest ranges from a to b. (Each repetition of the study would produce a sample of the same size as the one used to obtain the 95% confidence interval, but the samples would be different from one repetition to another, implying that the repeated confidence intervals would also be different. This repetition idea is crucial to understanding frequentist statistics concepts, but is usually something hypothetic we do in theory - in practice, we draw a single sample and use that sample to construct the confidence interval (a,b).)

I want to just add three things about credible intervals that are not in the excellent post by Isabella Ghement. The first is that credible intervals do not need to be connected, the second is that there are an infinite number of possible ones as is also true with confidence intervals, the third is that the confidence interval and the credible interval need not overlap each other.

You could have a case where the credible interval is $\{(-3,-1);(5,7)\}$. There are two principal causes for this. The first is that there is an omitted variable in the model and the data is clustering around two masses. To imagine this case, consider a business whose customers behave differently on rainy days and sunny days but your model did not include weather. The effect of the weather could possibly be seen in the disjoint interval. The second source would be a statistical run, where your sample is not representative of the answer.

The above case then highlights the difference between a Bayesian solution and a Frequentist solution. The Frequentist confidence interval could be $(1,4)$ because the Frequentist solution would average the two cases. This would be the case even if no data actually was in the interval itself. You can have a Frequentist solution with no representation of the parameter supported by the data. The Frequentist interval need not overlap the Bayesian one.

Finally, there is no such thing as a unique credible or unique confidence interval. Any function that covers the parameter in $1-\alpha$ percent of the time is a valid $1-\alpha$ percent confidence interval. On the Bayesian side, any sum of the area that adds to $1-\alpha$ percent is a valid credible interval. Usually, the credible region reported is the highest density credible region or the HDR. As it is almost always the HDR, it should be reported as the HDR as there are cases, though exceedingly uncommon, where it could be something else.

• Very insightful comment, Dave! – Isabella Ghement Mar 29 '18 at 18:46

An interval usually has two limits. So a credible interval or a confidence interval of (-8.13, -2.69) would span from -8.13 to -2.69. If nothing of the percentage is said, a 95 % credible interval for Bayesian statistics, or a 95 % confidence interval for Frequentist statistics, is the most likely confidence level. But that is not the only possible interval. Considering those limits to be the 2.5 % and 97.5 % quantiles is reasonable, but it is not necessarily so. You need to read more about their method and results. If there is a different percentage mentioned somewhere in the text that goes along with these data.

A confidence interval and a credible interval are very often close to each other in values, but there are very different theories behind them. They stem from two very different ways of doing statistics and a total beginner should probably first get accustomed with one oft them. When you feel advanced enough, you'll find lots of good texts on the differences on CrossValidated as well as on the web.