Parameters μ {displaystyle mu ,} location (real) λ > 0 {displaystyle lambda >0,} (real) α > 0 {displaystyle alpha >0,} (real) β > 0 {displaystyle eta >0,} (real) Support x ∈ ( − ∞ , ∞ ) , σ 2 ∈ ( 0 , ∞ ) {displaystyle xin (-infty ,infty ),!,;sigma ^{2}in (0,infty )} PDF λ σ 2 π β α Γ ( α ) ( 1 σ 2 ) α + 1 e − 2 β + λ ( x − μ ) 2 2 σ 2 {displaystyle {rac {sqrt {lambda }}{sigma {sqrt {2pi }}}}{rac {eta ^{alpha }}{Gamma (alpha )}},left({rac {1}{sigma ^{2}}}
ight)^{alpha +1}e^{-{rac {2eta +lambda (x-mu )^{2}}{2sigma ^{2}}}}} |
In probability theory and statistics, the normal-inverse-gamma distribution (or Gaussian-inverse-gamma distribution) is a four-parameter family of multivariate continuous probability distributions. It is the conjugate prior of a normal distribution with unknown mean and variance.
Contents
Definition
Suppose
has a normal distribution with mean
has an inverse gamma distribution. Then
(
In a multivariate form of the normal-inverse-gamma distribution,
Probability density function
For the multivariate form where
where
Alternative parameterization
It is also possible to let
In the multivariate form, the corresponding change would be to regard the covariance matrix
Cumulative distribution function
Differential equation
The probability density function of the normal-inverse-gamma distribution is a solution to the following differential equation:
Marginal distributions
Given
while
In the multivariate case, the marginal distribution of
Posterior distribution of the parameters
See the articles on normal-gamma distribution and conjugate prior.
Interpretation of the parameters
See the articles on normal-gamma distribution and conjugate prior.
Generating normal-inverse-gamma random variates
Generation of random variates is straightforward:
- Sample
σ 2 α andβ - Sample
x from a normal distribution with meanμ and varianceσ 2 / λ