credible interval
/ KRED-uh-bul IN-tur-vul /
A credible interval is the Bayesian way of saying 'I believe the true value lies between here and here.' After you've done a Bayesian analysis and have a posterior — a full distribution of belief over the unknown quantity — a 95% credible interval is simply the range that holds 95% of that belief. It lets you compress 'my whole distribution of belief' down to a clean, reportable bracket: 'I'm 95% sure the true value is between 12 and 18.'
Its great virtue is that it means exactly what people naturally want it to mean. A 95% credible interval really does carry the interpretation 'there's a 95% probability the true value is in this range, given my data and assumptions.' This is famously different from its frequentist cousin, the confidence interval, which despite the similar name does not let you say that about any single interval — a subtlety that trips up nearly everyone. The credible interval simply says the intuitive thing.
Why it matters: it is the honest, communicable summary of Bayesian uncertainty, ideal for reports and decisions. But carry its caveats. First, its meaning is conditional on your prior and your model — change those and the interval changes, so it inherits any bias they carry. Second, there's no single credible interval: the same 95% can be carved many ways (the central one, or the narrowest possible 'highest-density' one), and they can differ for lopsided distributions. And a 95% interval is, by design, wrong about 5% of the time — it is a statement of calibrated belief, not a guarantee of capture.
After testing a new drug, a Bayesian analysis gives a posterior for its effect, and the 95% credible interval for the average improvement is 3 to 9 points. You can correctly say: 'Given the data and our prior, there's a 95% probability the true average improvement lies between 3 and 9 points.' Try to say that exact sentence about a frequentist confidence interval and a statistician will wince — it isn't licensed there.
A credible interval licenses the natural statement 'there's a 95% chance the value is in here' — a confidence interval does not.
Don't confuse a credible interval with a frequentist confidence interval — they answer different questions, and only the credible interval supports the everyday phrase 'a 95% chance the value lies in this range.' But that statement is always conditional on your prior and model, which it silently inherits.