Credible interval

Imagine you are guessing a number.

You can pick a small group of numbers. This group is a credible interval. It shows where the right answer likely lives. Some groups are very small. Others might be split into parts.

This helps us see what is most likely. It is a way to show what we know. It makes our guesses much better.

Imagine you are guessing a number. You want to be right. A credible interval is a group of numbers. It shows where the answer likely lives.

This idea comes from Bayesian statistics. This is a way to use math to show what we know. In this way, the number we want is seen as a random thing. It might change. The interval we pick stays fixed. This is different from a confidence interval. In a confidence interval, the number stays fixed. Only the interval changes.

There are many ways to make these groups. One way is the smallest credible interval. This is also called the highest density interval. It is the smallest group that holds a certain amount of probability.

Another way uses quantiles. These are parts of a group. A median credible interval puts the same chance above and below. This group always includes the median. You can also find these groups using computer tests. One way is called Markov chain Monte Carlo. This helps us estimate the best group of numbers.

Imagine you are trying to guess a secret number. You might not know the exact number, but you can guess a range. A credible interval is a special way to pick that range. It uses math to show where an answer likely lives. This tool is used in Bayesian statistics. In this way of thinking, the unknown number is seen as a random variable. This means the number itself could be many different values. The interval we pick stays fixed to show our best guess.

There are many ways to build these groups of numbers. One way is the smallest credible interval. People also call this the highest density interval. This method finds the narrowest range that holds a certain amount of probability. If a group has only one peak, this interval will include the mode. The mode is the most likely value in the group. Another way is to use the smallest credible set. This is also known as the highest density region. For some groups, this might not be one single line. It could be several disconnected parts.

Other people use a method called a quantile-based interval. This method splits the group into equal parts. For example, a median credible interval is very common. It puts the same amount of probability above and below the range. This type of interval always contains the median. You can also find a lowest credible interval or a highest credible interval. These are helpful when your numbers have limits. Some people even pick a range where the mean is in the center. The mean is the average of all the values.

It is helpful to compare this to a different idea called a confidence interval. These two ideas come from different ways of looking at the world. In a confidence interval, the true number is fixed and never changes. Instead, the interval itself is what changes with every new sample. Bayesian credible intervals work the opposite way. They look at the uncertainty of the target number itself. One famous book on this is by P.M. Lee from 1997. Another important work is by A. O'Hagan from 1994.

Sometimes these two different ideas can actually give the same answer. This happens in very special cases. For example, they might match if you use a uniform flat distribution. This is a type of starting guess used in Bayesian math. It can also happen with a scale parameter using a Jeffreys' prior. Most of the time, however, the two methods stay different. Scientists also use computer tests to find these ranges. One method is called Markov chain Monte Carlo. This uses simulations to help estimate the best group of numbers.

A credible interval is a mathematical tool used in Bayesian statistics. It helps researchers describe a probability distribution by providing a specific range of values. This range represents where an unobserved parameter is likely to fall. For example, if a researcher calculates a 95% credible interval between 35 and 45, they are stating there is a 0.95 probability the parameter lies within those bounds. These intervals are essential for characterizing posterior probability distributions or predictive probability distributions. When these sets are expanded to include disconnected or multivariate sets, they are called credible sets or credible regions.

There are several distinct ways to define these intervals. One method is the Smallest Credible Interval (SCI). This is also known as the highest density interval. The SCI is the narrowest possible range that contains a specific amount of probability mass. If a distribution is unimodal, meaning it has only one peak, the SCI will always include the mode. The mode is the most frequent or likely value in the distribution. If the probability is at least 0.5, the SCI must also contain the median.

Another approach involves the Smallest Credible Set (SCS), often called the highest density region. This method is useful for multimodal distributions, which have multiple peaks. Unlike a single interval, an SCS can be disconnected, meaning it consists of several separate ranges. This set always contains the mode of the distribution. For multivariate cases, these sets are often bounded by probability density contour lines. These sets always contain the mode, but they do not necessarily include the mean, the geometric median, or the coordinate-wise median.

Researchers also use quantile-based credible intervals to define these ranges. These are calculated by taking an inter-quantile interval for a predefined probability. A common version is the Median Credible Interval (MCI), also called the equal-tailed interval. In an MCI, the probability of being below the interval is equal to the probability of being above it. This specific type of interval always contains the median. Other variations include the Lowest Credible Interval (LCI) and the Highest Credible Interval (HCI). These specific types are often more useful when dealing with bounded variables.

It is important to distinguish credible intervals from frequentist confidence intervals. These two concepts arise from fundamentally different statistical philosophies. In frequentist statistics, the parameter is treated as a fixed value that does not change. The confidence interval is the random variable because it depends on the specific sample collected. A 95% confidence interval means that if you took many samples, 95% of the resulting intervals would contain the true parameter. In contrast, Bayesian credible intervals treat the parameter itself as a random variable. The bounds of a credible interval are considered fixed based on the data and the prior distribution.

Because they follow different logic, these two methods treat nuisance parameters differently. However, they can occasionally coincide in very specific mathematical scenarios. For a single parameter and data that can be summarized by a single sufficient statistic, the two might match. This happens if the unknown parameter is a location parameter and the researcher uses a uniform flat prior distribution. It can also occur if the parameter is a scale parameter and the researcher uses a Jeffreys' prior. In these special cases, the Bayesian and frequentist results align perfectly.

Beyond manual calculation, scientists often use advanced computer techniques to estimate these intervals. One widely used method is called Markov chain Monte Carlo, or MCMC. This simulation technique allows researchers to explore complex distributions that are difficult to solve with standard formulas. By using these simulations, they can find the most accurate credible intervals for their data. This ability to model uncertainty makes Bayesian statistics a powerful tool in modern scientific research.