A Bayesian average is a method of estimating the
mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set.
For a data set, the ''arithme ...
of a population using outside information, especially a pre-existing belief, which is factored into the calculation. This is a central feature of
Bayesian interpretation. This is useful when the available data set is small.
Calculating the Bayesian average uses the prior mean ''m'' and a constant ''C''. ''C'' is chosen based on the typical data set size required for a robust estimate of the sample mean. 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.
:
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.
When the
are binary values 0 or 1, ''m'' can be interpreted as the prior estimate of a binomial probability with the Bayesian average giving a posterior estimate for the observed data. In this case, ''C'' can be chosen based on the desired
binomial proportion confidence interval
In statistics, a binomial proportion confidence interval is a confidence interval for the probability of success calculated from the outcome of a series of success–failure experiments (Bernoulli trials). In other words, a binomial proportion conf ...
for the sample value. For example, for rare outcomes when ''m'' is small choosing
ensures a 99% confidence interval has width about ''2m''.
See also
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Additive smoothing
In statistics, additive smoothing, also called Laplace smoothing or Lidstone smoothing, is a technique used to smooth categorical data. Given a set of observation counts \textstyle from a \textstyle -dimensional multinomial distribution with ...
References
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Average
In ordinary language, an average is a single number taken as representative of a list of numbers, usually the sum of the numbers divided by how many numbers are in the list (the arithmetic mean). For example, the average of the numbers 2, 3, 4, 7, ...
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