In probability theory

Probability theory

Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

 and statistics

Statistics

Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

, the normal-scaled inverse gamma distribution is a four-parameter family of multivariate continuous probability distribution

Probability distribution

In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....

s. It is the conjugate prior

Conjugate prior

In Bayesian probability theory, if the posterior distributions p are in the same family as the prior probability distribution p, the prior and posterior are then called conjugate distributions, and the prior is called a conjugate prior for the likelihood...

 of a normal distribution with unknown mean

Mean

In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....

 and variance

Variance

In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...

.

Definition

Suppose

has a normal distribution with mean

Mean

In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....

  and variance

Variance

In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...

 , where

has an inverse gamma distribution. Then
has a normal-scaled inverse gamma distribution, denoted as

Probability density function

Alternative Parameterization

It is also possible to let in which case the pdf becomes

Generating normal-gamma random variates

Generation of random variates is straightforward:

1. Sample from an inverse gamma distribution with parameters and
2. Sample from a normal distribution with mean and variance

Related distributions

• The normal-gamma distribution is the same distribution parameterized by precision
  Precision (statistics)
  In statistics, the term precision can mean a quantity defined in a specific way. This is in addition to its more general meaning in the contexts of accuracy and precision and of precision and recall....
  
   rather than variance
  Variance
  In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...
  
  
• A generalization of this distribution which allows for a multivariate mean and a positive-definite covariance matrix is the Multivariate normal-inverse Wishart distribution