normal-inverse-gamma
Probability density function
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
2
π
σ
2
β
α
Γ
(
α
)
(
1
σ
2
)
α
+
1
exp
(
−
2
β
+
λ
(
x
−
μ
)
2
2
σ
2
)
{\displaystyle {\frac {\sqrt {\lambda }}{\sqrt {2\pi \sigma ^{2}}}}{\frac {\beta ^{\alpha }}{\Gamma (\alpha )}}\left({\frac {1}{\sigma ^{2}}}\right)^{\alpha +1}\exp \left(-{\frac {2\beta +\lambda (x-\mu )^{2}}{2\sigma ^{2}}}\right)}
Mean
E
[
x
]
=
μ
{\displaystyle \operatorname {E} [x]=\mu }
E
[
σ
2
]
=
β
α
−
1
{\displaystyle \operatorname {E} [\sigma ^{2}]={\frac {\beta }{\alpha -1}}}
, for
α
>
1
{\displaystyle \alpha >1}
Mode
x
=
μ
(univariate)
,
x
=
μ
(multivariate)
{\displaystyle x=\mu \;{\textrm {(univariate)}},x={\boldsymbol {\mu }}\;{\textrm {(multivariate)}}}
σ
2
=
β
α
+
1
+
1
/
2
(univariate)
,
σ
2
=
β
α
+
1
+
k
/
2
(multivariate)
{\displaystyle \sigma ^{2}={\frac {\beta }{\alpha +1+1/2}}\;{\textrm {(univariate)}},\sigma ^{2}={\frac {\beta }{\alpha +1+k/2}}\;{\textrm {(multivariate)}}}
Variance
Var
[
x
]
=
β
(
α
−
1
)
λ
{\displaystyle \operatorname {Var} [x]={\frac {\beta }{(\alpha -1)\lambda }}}
, for
α
>
1
{\displaystyle \alpha >1}
Var
[
σ
2
]
=
β
2
(
α
−
1
)
2
(
α
−
2
)
{\displaystyle \operatorname {Var} [\sigma ^{2}]={\frac {\beta ^{2}}{(\alpha -1)^{2}(\alpha -2)}}}
, for
α
>
2
{\displaystyle \alpha >2}
Cov
[
x
,
σ
2
]
=
0
{\displaystyle \operatorname {Cov} [x,\sigma ^{2}]=0}
, for
α
>
1
{\displaystyle \alpha >1}
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
)
{\displaystyle x\mid \sigma ^{2},\mu ,\lambda \sim \mathrm {N} (\mu ,\sigma ^{2}/\lambda )\,\!}
has a normal distribution with mean
μ
{\displaystyle \mu }
and variance
σ
2
/
λ
{\displaystyle \sigma ^{2}/\lambda }
, where
σ
2
∣
α
,
β
∼
Γ
−
1
(
α
,
β
)
{\displaystyle \sigma ^{2}\mid \alpha ,\beta \sim \Gamma ^{-1}(\alpha ,\beta )\!}
has an inverse gamma distribution . Then
(
x
,
σ
2
)
{\displaystyle (x,\sigma ^{2})}
has a normal-inverse-gamma distribution, denoted as
(
x
,
σ
2
)
∼
N-
Γ
−
1
(
μ
,
λ
,
α
,
β
)
.
{\displaystyle (x,\sigma ^{2})\sim {\text{N-}}\Gamma ^{-1}(\mu ,\lambda ,\alpha ,\beta )\!.}
(
NIG
{\displaystyle {\text{NIG}}}
is also used instead of
N-
Γ
−
1
.
{\displaystyle {\text{N-}}\Gamma ^{-1}.}
)
The normal-inverse-Wishart distribution is a generalization of the normal-inverse-gamma distribution that is defined over multivariate random variables.
Characterization [ ]
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
)
{\displaystyle f(x,\sigma ^{2}\mid \mu ,\lambda ,\alpha ,\beta )={\frac {\sqrt {\lambda }}{\sigma {\sqrt {2\pi }}}}\,{\frac {\beta ^{\alpha }}{\Gamma (\alpha )}}\,\left({\frac {1}{\sigma ^{2}}}\right)^{\alpha +1}\exp \left(-{\frac {2\beta +\lambda (x-\mu )^{2}}{2\sigma ^{2}}}\right)}
For the multivariate form where
x
{\displaystyle \mathbf {x} }
is a
k
×
1
{\displaystyle k\times 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
)
α
+
1
+
k
/
2
exp
(
−
2
β
+
(
x
−
μ
)
T
V
−
1
(
x
−
μ
)
2
σ
2
)
.
{\displaystyle f(\mathbf {x} ,\sigma ^{2}\mid \mu ,\mathbf {V} ^{-1},\alpha ,\beta )=|\mathbf {V} |^{-1/2}{(2\pi )^{-k/2}}\,{\frac {\beta ^{\alpha }}{\Gamma (\alpha )}}\,\left({\frac {1}{\sigma ^{2}}}\right)^{\alpha +1+k/2}\exp \left(-{\frac {2\beta +(\mathbf {x} -{\boldsymbol {\mu }})^{T}\mathbf {V} ^{-1}(\mathbf {x} -{\boldsymbol {\mu }})}{2\sigma ^{2}}}\right).}
where
|
V
|
{\displaystyle |\mathbf {V} |}
is the determinant of the
k
×
k
{\displaystyle k\times k}
matrix
V
{\displaystyle \mathbf {V} }
. Note how this last equation reduces to the first form if
k
=
1
{\displaystyle k=1}
so that
x
,
V
,
μ
{\displaystyle \mathbf {x} ,\mathbf {V} ,{\boldsymbol {\mu }}}
are scalars .
Alternative parameterization [ ]
It is also possible to let
γ
=
1
/
λ
{\displaystyle \gamma =1/\lambda }
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
)
{\displaystyle f(x,\sigma ^{2}\mid \mu ,\gamma ,\alpha ,\beta )={\frac {1}{\sigma {\sqrt {2\pi \gamma }}}}\,{\frac {\beta ^{\alpha }}{\Gamma (\alpha )}}\,\left({\frac {1}{\sigma ^{2}}}\right)^{\alpha +1}\exp \left(-{\frac {2\gamma \beta +(x-\mu )^{2}}{2\gamma \sigma ^{2}}}\right)}
In the multivariate form, the corresponding change would be to regard the covariance matrix
V
{\displaystyle \mathbf {V} }
instead of its inverse
V
−
1
{\displaystyle \mathbf {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
)
{\displaystyle F(x,\sigma ^{2}\mid \mu ,\lambda ,\alpha ,\beta )={\frac {e^{-{\frac {\beta }{\sigma ^{2}}}}\left({\frac {\beta }{\sigma ^{2}}}\right)^{\alpha }\left(\operatorname {erf} \left({\frac {{\sqrt {\lambda }}(x-\mu )}{{\sqrt {2}}\sigma }}\right)+1\right)}{2\sigma ^{2}\Gamma (\alpha )}}}
Properties [ ]
Marginal distributions [ ]
Given
(
x
,
σ
2
)
∼
N-
Γ
−
1
(
μ
,
λ
,
α
,
β
)
.
{\displaystyle (x,\sigma ^{2})\sim {\text{N-}}\Gamma ^{-1}(\mu ,\lambda ,\alpha ,\beta )\!.}
as above,
σ
2
{\displaystyle \sigma ^{2}}
by itself follows an inverse gamma distribution :
σ
2
∼
Γ
−
1
(
α
,
β
)
{\displaystyle \sigma ^{2}\sim \Gamma ^{-1}(\alpha ,\beta )\!}
while
α
λ
β
(
λ
+
1
)
(
x
−
μ
)
{\displaystyle {\sqrt {\frac {\alpha \lambda }{\beta (\lambda +1)}}}(x-\mu )}
follows a t distribution with
2
α
{\displaystyle 2\alpha }
degrees of freedom. [1]
In the multivariate case, the marginal distribution of
x
{\displaystyle \mathbf {x} }
is a multivariate t distribution :
x
∼
t
2
α
(
μ
,
β
α
V
−
1
)
{\displaystyle \mathbf {x} \sim t_{2\alpha }({\boldsymbol {\mu }},{\frac {\beta }{\alpha }}\mathbf {V} ^{-1})\!}
Summation [ ]
Scaling [ ]
Exponential family [ ]
Information entropy [ ]
Kullback–Leibler divergence [ ]
Maximum likelihood estimation [ ]
This section is empty. You can help by . (July 2010 )
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
{\displaystyle \sigma ^{2}}
from an inverse gamma distribution with parameters
α
{\displaystyle \alpha }
and
β
{\displaystyle \beta }
Sample
x
{\displaystyle x}
from a normal distribution with mean
μ
{\displaystyle \mu }
and variance
σ
2
/
λ
{\displaystyle \sigma ^{2}/\lambda }
Related distributions [ ]
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
{\displaystyle \sigma ^{2}\mathbf {V} }
(whereas in the multivariate inverse-gamma distribution the covariance matrix is regarded as known up to the scale factor
σ
2
{\displaystyle \sigma ^{2}}
) is the normal-inverse-Wishart distribution
See also [ ]
Compound probability distribution
References [ ]
Denison, David G. T. ; Holmes, Christopher C.; Mallick, Bani K.; Smith, Adrian F. M. (2002) Bayesian Methods for Nonlinear Classification and Regression , Wiley. ISBN 0471490369
Koch, Karl-Rudolf (2007) Introduction to Bayesian Statistics (2nd Edition), Springer. ISBN 354072723X
Discrete univariate
with finite support with infinite support
Continuous univariate
supported on a bounded interval supported on a semi-infinite interval supported on the whole real line with support whose type varies
Mixed univariate
Multivariate (joint) Directional Degenerate and singular Families