#### Keywords

Bayes, statistics

#### Abstract

This article shows how to compute confidence intervals for mean, standard-deviation, and variance using Bayesian methods. The method is implemented in SciPy as scipy.stats.bayes_mvs After reviewing some classical estimators for mean, variance, and standard-deviation and showing that un-biased estimates are not usually desirable, a Bayesian perspective is employed to determine what is known about mean, variance, and standard deviation given only that a data set in-fact has a common mean and variance. Maximum-entropy is used to argue that the likelihood function in this situation should be the same as if the data were independent and identically distributed Gaussian. A non-informative prior is derived for the mean and variance and Bayes rule is used to compute the posterior Probability Density Function (PDF) of \left(\mu,\sigma\right) as well as \left(\mu,\sigma^{2}\right) in terms of the sufficient statistics \bar{x}=\frac{1}{n}\sum_{i}x_{i} and C=\frac{1}{n}\sum_{i}\left(x_{i}-\bar{x}\right)^{2}. From the joint distribution marginals are determined. It is shown that \left(\frac{\mu-\bar{x}}{\sqrt{C}}\right)\sqrt{n-1} is distributed as Student-t with n-1 degrees of freedom, \sigma\sqrt{\frac{2}{nC}} is distributed as generalized-gamma with c=-2 and a=\frac{n-1}{2}, and \sigma^{2}\frac{2}{nC} is distributed as inverted-gamma with a=\frac{n-1}{2}. It is suggested to report the mean of these distributions as the estimate (or the peak if n is too small for the mean to be defined) and a confidence interval surrounding the median.

#### Recommended Citation

Oliphant, Travis E., "A Bayesian perspective on estimating mean, variance, and standard-deviation from data" (2006). *All Faculty Publications.* Paper 278.

http://scholarsarchive.byu.edu/facpub/278

#### Document Type

Peer-Reviewed Article

#### Publication Date

2006-12-05

#### Permanent URL

http://hdl.lib.byu.edu/1877/438

#### Language

English

#### College

Ira A. Fulton College of Engineering and Technology

#### Department

Electrical and Computer Engineering

#### Copyright Status

© 2006 Travis E. Oliphant

#### Copyright Use Information

http://lib.byu.edu/about/copyright/generic.php