Probability Intervals vs. Confidence Intervals

In Bayesian inference, a probability interval is a probabilistic region around a posterior moment, and is similar in use to a frequentist confidence interval. For example, if a statistical method estimates the mean height of a population to be 70 inches, with a 95% credible interval of 65-75, then the estimated posterior probability is that 95% of that population is between 65 and 75 inches, with a mean of 70.

A Bayesian probability interval, also called a credible interval, incorporates information from the prior distribution into the estimate, while confidence intervals are based solely on the data.

In contrast, a frequentist confidence interval, for the same example, must be interpreted as a range in which the mean would occur 95% of the time with repeated sampling, with no indication as to whether or not the current interval estimated from the current sample contains the mean. The probability that the mean is within the confidence interval is either 0 or 1. Frequentist methods consider the parameter, the mean in this case, to be fixed. Therefore, a frequentist parameter is only a point estimate, and does not have a probability distribution. For these reasons, many people intuitively misinterpret a frequentist confidence interval as though it were a Bayesian probability interval, perceiving the interval to be a probabilistic region around the parameter.

Frequentist confidence intervals may be associated with other problems. To continue with the previous example, also suppose that the sample population is the intended universe, and that extrapolation is not desired or possible. Suppose that the heights of all males were measured at a small company of 100 individuals. The Bayesian probability interval remains easy to interpret: the posterior probability indicates that 95% of the population is within the interval of 65-75 inches. The frequentist confidence interval becomes technically meaningless, because additional samples cannot be gathered, since the universe was measured to begin with. Therefore, in this example, a confidence interval should not be estimated, only the mean.

Bayesian probability intervals are preferable to frequentist confidence intervals.