A credible interval is a Bayesian way to describe uncertainty about an unknown parameter, such as a population mean, a probability, or a physical constant. Instead of giving only one estimate, it gives a range of parameter values that are most plausible after seeing the data. A 95% credible interval means that, according to the posterior distribution, there is 95% probability that the parameter lies inside the interval.
This direct probability statement is one reason credible intervals are useful in science, engineering, and data analysis.
Credible intervals come from the posterior distribution, which combines prior beliefs with the likelihood of the observed data. The shaded area under a posterior curve represents probability assigned to parameter values, so an interval is chosen to contain a specified amount of that area. This differs from a frequentist confidence interval, where the parameter is treated as fixed and the interval is random across repeated samples.
In practice, credible intervals help communicate both the best estimate and the remaining uncertainty in a way that matches many people's natural interpretation of probability.
Understanding Statistics: Credible Intervals
To build an interval, statisticians first choose a model for how the observations arise. For example, a survey response may be treated as a success or failure, while repeated measurements of a length may be treated as values scattered around a true value. The model tells us which parameter matters and how likely different data sets are for each possible parameter value.
A prior distribution then supplies starting information. It can come from earlier experiments, a well tested physical theory, expert knowledge, or a deliberately broad assumption when little is known. New data update this starting position.
With a small data set, the prior can noticeably affect the result. With a large, informative data set, the observations usually have much more influence.
There is more than one sensible way to select the range. An equal tailed interval leaves the same probability outside each end. This is convenient and familiar, especially when the posterior curve is roughly symmetric.
A highest density interval instead collects the parameter values with the greatest posterior density until the desired probability is reached. It is often shorter than an equal tailed interval. For a skewed distribution, the two kinds of intervals can have quite different endpoints.
A posterior with two separate peaks can even produce two separated ranges. Students should not assume every credible interval is centered on the most likely value or has a symmetric shape.
In real work, the posterior is often too complicated to calculate exactly. Researchers may use computer methods that generate many plausible parameter values from the posterior. These values are called posterior samples.
To form a credible interval, they sort the samples and locate suitable cutoff values, or find the shortest region containing the required fraction. This method appears in weather forecasting, medical studies, election modelling, quality control, and physics experiments. A lab group estimating a detector efficiency, for instance, can report a central estimate with a credible interval to show how much the data still leave uncertain.
The interpretation depends on the full set of assumptions used to create the posterior. A narrow interval is not automatically trustworthy. It may be narrow because the data are precise, but it may be falsely narrow if the model ignores measurement error, selection bias, or variation between groups.
Priors deserve attention too. A sensible check is to repeat the analysis using several reasonable priors and see whether the conclusion changes greatly. When learning this topic, separate uncertainty about the parameter from ordinary variation in future observations.
Check which interval definition was used, inspect the shape of the posterior, and state the model assumptions clearly. These habits matter more than memorising a single calculation rule.
Key Facts
- Bayes' theorem: posterior proportional to likelihood times prior, written p(theta | data) ∝ p(data | theta)p(theta).
- A 95% credible interval contains 95% of the posterior probability for the parameter theta.
- For an equal-tailed 95% credible interval, P(theta < L | data) = 0.025 and P(theta > U | data) = 0.025.
- For a normal posterior theta | data ~ N(mu, sigma^2), an approximate 95% credible interval is mu ± 1.96sigma.
- A credible interval is interpreted as P(L ≤ theta ≤ U | data) = 0.95, given the model and prior.
- A confidence interval does not mean there is a 95% probability the fixed parameter is inside one computed interval.
Vocabulary
- Credible interval
- A range of parameter values that contains a chosen amount of posterior probability, such as 95%.
- Posterior distribution
- The probability distribution for a parameter after combining the prior distribution with the observed data.
- Prior distribution
- A probability distribution that represents what is believed about a parameter before the current data are used.
- Likelihood
- A function that measures how compatible the observed data are with different possible parameter values.
- Confidence interval
- A frequentist interval procedure that captures the true fixed parameter in a specified percentage of repeated samples.
Common Mistakes to Avoid
- Saying a 95% confidence interval means a 95% probability that the parameter is inside it is wrong because frequentist parameters are treated as fixed, not random.
- Ignoring the prior is wrong because a Bayesian credible interval depends on both the likelihood and the prior distribution.
- Assuming all 95% credible intervals are symmetric is wrong because skewed posterior distributions can produce asymmetric intervals.
- Treating the shaded posterior area as data frequency is wrong because it represents probability about parameter values after conditioning on the observed data.
Practice Questions
- 1 A posterior distribution for theta is approximately normal with mean 12 and standard deviation 2. Find the approximate 95% credible interval using theta = mu ± 1.96sigma.
- 2 An equal-tailed 90% credible interval is built from a posterior distribution. What posterior probability is left below the lower bound, and what probability is left above the upper bound?
- 3 A study reports a 95% credible interval of 0.62 to 0.78 for a treatment success probability. Explain in words what this interval means, and state how the interpretation differs from a 95% confidence interval.