Bayesian Estimation & Credible Intervals
Bayesian updating hands you a whole posterior distribution. That's wonderfully complete, but eventually someone asks "so what's your estimate, and how sure are you?" Boiling the posterior down to a number and a range is Bayesian estimation, and the range it produces (the credible interval) finally means what everyone always wished a confidence interval meant.
Two summaries of a posterior
- A point estimate: usually the posterior mean (the balance point), but sometimes the median or the mode (the peak, called the MAP, maximum a posteriori).
- A credible interval: a range that contains a stated share of the posterior probability, say 95%. The most common version is the central interval, chopping 2.5% off each tail.
🎮 Read the Credible Interval Off the Posterior
Estimating a proportion with a flat prior. The purple curve is your posterior; the shaded band holds the central credibility you choose. Add data and watch it tighten.
The interpretation everyone wants
A 95% credible interval means exactly what it sounds like: given your data and prior, there is a 95% probability the parameter lies in this range. Compare that to a frequentist confidence interval, where the 95% refers to the long-run behavior of the procedure, not this particular interval. The Bayesian statement is the direct probability claim people instinctively (but wrongly) read into confidence intervals.
Central vs. highest-density intervals
The shaded band above is a central interval: equal probability trimmed from each tail. An alternative is the highest posterior density (HPD) interval: the shortest range containing the target probability. For a symmetric posterior they're identical; for a skewed one the HPD hugs the peak and is a little narrower. Both are legitimate. Central intervals are simpler, HPD intervals are tightest.
Watch it tighten
Slide the trials up and the posterior narrows, so the credible interval shrinks. More data, more certainty, exactly as it should. Crank the credibility from 90% to 99% and the band widens: demanding more certainty means admitting a wider range, the same precision-vs-confidence trade-off you saw with confidence intervals.
Why it matters: credible intervals are how Bayesian results get reported — in trials, forecasts, and industry experiments. They deliver an honest, directly-interpretable measure of uncertainty, which is a big part of why Bayesian methods have surged in popularity.
Problem 32 of the practice problems takes a Beta(8, 12) prior from last quarter, updates it on 33 completions out of 60 users, and asks for the posterior's mean, mode and standard deviation, then why that mean sits below the observed completion rate. To watch the Beta distribution itself change shape as its two parameters move, the distribution playground has it alongside eight others.
Common questions
What is the difference between a credible interval and a confidence interval?
A 95% credible interval means what everyone wants: "given the data (and prior), the parameter is 95% likely to be in here." A 95% confidence interval promises only that the procedure captures the truth 95% of the time across repeated studies. With flat priors and decent data they're often numerically similar — but only the credible interval licenses the direct probability sentence.
How do I test a hypothesis with a credible interval?
You can simply read the probability off the posterior, which is the whole advantage: P(θ > 0 | data) is an ordinary number you can quote, with no null hypothesis anywhere. When you do want a decision rule, the usual one is a ROPE, a region of practical equivalence: state in advance the range of values you would count as "no meaningful effect", say −0.1 to 0.1 on your outcome's scale, and compare it with the credible interval. If the interval falls entirely inside the ROPE you can accept the null for practical purposes, which no p-value will ever let you do; if it falls entirely outside, you reject it; if it straddles the boundary, the honest answer is that the data have not decided. Choosing the ROPE is a judgment about what matters in your field, so state it before you look, exactly as you would a smallest effect size of interest.
Do Bayesian and frequentist results ever agree?
Constantly: with flat/weak priors and reasonable sample sizes, credible and confidence intervals often match to two decimals, since the likelihood dominates both. They part ways when priors carry real information, when data is thin, and in interpretation always. The practical upshot: the frameworks usually corroborate each other, and genuine disagreement is itself diagnostic — it means your prior is doing heavy lifting.