# Bayesian average

A Bayesian average is a method of estimating the mean of a population using outside information, especially a pre-existing belief,[1] that is factored into the calculation. This is a central feature of Bayesian interpretation. This is relevant when the available data set is small.[2]

Calculating the Bayesian average uses the prior mean m and a constant C. C is assigned a value that is proportional to the typical data set size. The value is larger when the expected variation between data sets (within the larger population) is small. It is smaller when the data sets are expected to vary substantially from one another.

${\displaystyle {\bar {x}}={Cm+\sum _{i=1}^{n}x_{i} \over C+n}}$

This is equivalent to adding C data points of value m to the data set. It is a weighted average of a prior average m and the sample average.