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In Bayesian statistics, a credible interval is a posterior probability interval[1] which is used for interval estimation in contrast to point estimation. Credible intervals are used for purposes similar to those of confidence intervals in frequentist statistics and an alternative terminology is to use Bayesian confidence interval instead of "credible interval".[2] When dealing with more than one unknown quantity simultaneously, the term credible region is used.

For example, a statement such as "following the experiment, a 90% credible interval for the parameter t is 35-45" means that the posterior probability that t lies in the interval from 35 to 45 is 0.9.

There are several ways of defining a credible interval from a given probability distribution for the parameter. Examples include:

  • choosing the narrowest interval, which for a unimodal distribution will involve choosing those values of highest probability density including the mode.
  • choosing the interval where the probability of being below the interval is as likely as being above it; the interval will include the median.
  • choosing the interval which has the mean as its central point, assuming the mean exists.

It is possible to frame the choice of a credible interval within decision theory and, in that context, an optimal interval will always be a highest probability density set.[3]

Distinction between a Bayesian credible interval and a frequentist confidence interval

A frequentist 90% confidence interval of 35-45 means that with a large number of repeated samples, 90% of the calculated confidence intervals would include the true value of the parameter. The probability that the parameter is inside the given interval (say, 35-45) is either 0 or 1 (the non-random unknown parameter is either there or not). In frequentist terms, the parameter is fixed (cannot be considered to have a distribution of possible values) and the confidence interval is random (as it depends on the random sample). Antelman (1997, p. 375) summarizes a confidence interval as "... one interval generated by a procedure that will give correct intervals 95 % [resp. 90 %] of the time". [4]

In general, Bayesian credible intervals do not coincide with frequentist confidence intervals for two reasons:

  • credible intervals incorporate problem-specific contextual information from the prior distribution whereas confidence intervals are based only on the data;
  • credible intervals and confidence intervals treat nuisance parameters in radically different ways.

Many professional statisticians and decisions scientists as well as non-statisticians intuitively interpret confidence intervals in the Bayesian credible interval sense and hence "credible intervals" are sometimes called "confidence intervals". It is widely accepted, especially in the decision sciences, that "credible interval" is merely the subjective subset of "confidence intervals". In fact, much research in calibrated probability assessments never uses the term "credible interval" and it is common to simply use "confidence interval".

References

  1. ^ Edwards, W., Lindman, H., Savage, L.J. (1963) Bayesian statistical inference in statistical research. Psychological Research, 70, 193-242
  2. ^ Lee, P.M. (1997) Bayesian Statistics: An Introduction, Arnold. ISBN 0-340-67785-6
  3. ^ O'Hagan, A. (1994) Kendall's Advance Theory of Statistics, Vol 2B, Bayesian Inference, Section 2.51. Arnold, ISBN 0-340-52922-9
  4. ^ Antelman, G. (1997) Elementary Bayesian Statistics (Madansky, A. & McCulloch, R. eds.). Edward Elgar, Cheltenham, UK. ISBN 978-1-85898-504-6
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