Credible interval

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Credible interval is a range of values within which an unknown parameter falls with a particular probability. It is a concept used in Bayesian statistics, a branch of statistics which interprets probability as a measure of belief or certainty rather than a frequency. This makes the credible interval fundamentally different from the confidence interval used in frequentist statistics, which instead describes how often a calculated interval would contain the parameter if the experiment were repeated many times.

Definition

A credible interval for a parameter is an interval within a probability distribution that spans a range of plausible values for that parameter. The interval is defined by two numbers, the lower and upper bounds, between which the parameter is said to lie with a certain probability. The probability associated with a credible interval is known as the credibility level or confidence level in frequentist terms, but these two concepts should not be confused.

Calculation

The calculation of a credible interval depends on the posterior distribution of the parameter of interest, which is obtained using Bayes' theorem. Bayes' theorem combines the prior distribution, which represents the initial beliefs about the parameter before considering the current data, with the likelihood, which is the probability of observing the current data given different values of the parameter. The result is the posterior distribution, a new probability distribution that reflects updated beliefs about the parameter after taking the data into account.

There are several methods to calculate credible intervals, depending on the form of the posterior distribution and the desired properties of the interval. The most common is the highest posterior density (HPD) interval, which contains the range of values with the highest probability density and thus represents the most plausible values for the parameter.

Interpretation

The interpretation of a credible interval is more intuitive than that of a confidence interval. A 95% credible interval means that there is a 95% probability that the parameter lies within the specified range, given the observed data and the prior information. This direct probabilistic interpretation is one of the reasons why Bayesian methods, and credible intervals in particular, are popular in many fields, including medicine, ecology, and machine learning.

Comparison with Confidence Intervals

While both credible intervals and confidence intervals aim to provide an estimate of where a parameter lies, their interpretations and calculations are fundamentally different. Confidence intervals are based on the concept of long-run frequency properties and do not allow for the direct probability statements about parameters that credible intervals do. This difference often leads to confusion and misinterpretation of the intervals' meanings.

Applications

Credible intervals are widely used in fields that employ Bayesian statistics. In medicine, they are used to estimate the effectiveness of treatments or the likelihood of disease presence. In environmental science, they help in predicting future climate conditions or assessing the risk of natural disasters. Their application is also growing in the field of artificial intelligence, particularly in machine learning and neural networks, where they can provide a measure of uncertainty in predictions and inferences.

Conclusion

Credible intervals offer a powerful tool for statistical inference in the Bayesian framework, providing a direct and intuitive way to express uncertainty about a parameter. Their application across various fields underscores the growing appreciation of Bayesian methods for their ability to incorporate prior knowledge and provide interpretable results.

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