Normal inverse gamma distribution - Alchetron, the free social encyclopedia


Parameters μ {\displaystyle \mu \,} location (real) λ > 0 {\displaystyle \lambda >0\,} (real) α > 0 {\displaystyle \alpha >0\,} (real) β > 0 {\displaystyle \beta >0\,} (real) Support x ∈ ( − ∞ , ∞ ) , σ 2 ∈ ( 0 , ∞ ) {\displaystyle x\in (-\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 {\frac {\sqrt {\lambda }}{\sigma {\sqrt {2\pi }}}}{\frac {\beta ^{\alpha }}{\Gamma (\alpha )}}\,\left({\frac {1}{\sigma ^{2}}}\right)^{\alpha +1}e^{-{\frac {2\beta +\lambda (x-\mu )^{2}}{2\sigma ^{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.

Definition

Suppose

x | σ 2 , μ , λ ∼ N ( μ , σ 2 / λ )

has a normal distribution with mean μ and variance σ 2 / λ , where

σ 2 | α , β ∼ Γ − 1 ( α , β )

has an inverse gamma distribution. Then ( x , σ 2 ) has a normal-inverse-gamma distribution, denoted as

( x , σ 2 ) ∼ N- Γ − 1 ( μ , λ , α , β ) .

( NIG is also used instead of N- Γ − 1 . )

In a multivariate form of the normal-inverse-gamma distribution, x | σ 2 , μ , V − 1 ∼ N ( μ , σ 2 V − 1 ) -- that is, conditional on σ 2 , x is a k × 1 random vector that follows the multivariate normal distribution with mean μ and covariance σ 2 V − 1 -- while, as in the univariate case, σ 2 | α , β ∼ Γ − 1 ( α , β ) .

Probability density function

f ( x , σ 2 | μ , λ , α , β ) = λ σ 2 π β α Γ ( α ) ( 1 σ 2 ) α + 1 exp ⁡ ( − 2 β + λ ( x − μ ) 2 2 σ 2 )

For the multivariate form where x is a k × 1 random vector,

f ( x , σ 2 | μ , V − 1 , α , β ) = | V | − 1 / 2 ( 2 π ) − k / 2 β α Γ ( α ) ( 1 σ 2 ) k / 2 + α + 1 exp ⁡ ( − 2 β + ( x − μ ) ′ V − 1 ( x − μ ) 2 σ 2 ) .

where | V | is the determinant of the k × k matrix V . Note how this last equation reduces to the first form if k = 1 so that x , V , μ are scalars.

Alternative parameterization

It is also possible to let γ = 1 / λ in which case the pdf becomes

f ( x , σ 2 | μ , γ , α , β ) = 1 σ 2 π γ β α Γ ( α ) ( 1 σ 2 ) α + 1 exp ⁡ ( − 2 γ β + ( x − μ ) 2 2 γ σ 2 )

In the multivariate form, the corresponding change would be to regard the covariance matrix V instead of its inverse V − 1 as a parameter.

Cumulative distribution function

F ( x , σ 2 | μ , λ , α , β ) = e − β σ 2 ( β σ 2 ) α ( erf ( λ ( x − μ ) 2 σ ) + 1 ) 2 σ 2 Γ ( α )

Differential equation

The probability density function of the normal-inverse-gamma distribution is a solution to the following differential equation:

{ σ 2 f ′ ( x ) + λ f ( x ) ( x − μ ) = 0 , f ( 0 ) = λ β α ( 1 σ 2 ) α − 1 e − 2 β − λ μ 2 2 σ 2 2 π σ Γ ( α ) }

Marginal distributions

Given ( x , σ 2 ) ∼ N- Γ − 1 ( μ , λ , α , β ) . as above, σ 2 by itself follows an inverse gamma distribution:

σ 2 ∼ Γ − 1 ( α , β )

while α λ β ( x − μ ) follows a t distribution with 2 α degrees of freedom.

In the multivariate case, the marginal distribution of x is a multivariate t distribution:

x ∼ t 2 α ( μ , β α V − 1 )

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 from an inverse gamma distribution with parameters α and β
Sample x from a normal distribution with mean μ and variance σ 2 / λ

The normal-gamma distribution is the same distribution parameterized by precision rather than variance

A generalization of this distribution which allows for a multivariate mean and a completely unknown positive-definite covariance matrix σ 2 V (whereas in the multivariate inverse-gamma distribution the covariance matrix is regarded as known up to the scale factor σ 2 ) is the normal-inverse-Wishart distribution