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
\( {\displaystyle x\mid \sigma ^{2},\mu ,\lambda \sim \mathrm {N} (\mu ,\sigma ^{2}/\lambda )\,\!} \)
has a normal distribution with mean \( \mu \) and variance \( \sigma^2 / \lambda \) , where
\( {\displaystyle \sigma ^{2}\mid \alpha ,\beta \sim \Gamma ^{-1}(\alpha ,\beta )\!} \)
has an inverse gamma distribution. Then \( (x,\sigma^2) \) has a normal-inverse-gamma distribution, denoted as
\( (x,\sigma^2) \sim \text{N-}\Gamma^{-1}(\mu,\lambda,\alpha,\beta) \! . \)
( \( \text{NIG} \) is also used instead of \( \text{N-}\Gamma^{-1}.) \)
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
\( {\displaystyle f(x,\sigma ^{2}\mid \mu ,\lambda ,\alpha ,\beta )={\frac {\sqrt {\lambda }}{\sigma {\sqrt {2\pi }}}}\,{\frac {\beta ^{\alpha }}{\Gamma (\alpha )}}\,\left({\frac {1}{\sigma ^{2}}}\right)^{\alpha +1}\exp \left(-{\frac {2\beta +\lambda (x-\mu )^{2}}{2\sigma ^{2}}}\right)} \)
For the multivariate form where \( \mathbf{x} \) is \( k\times 1 \) random vector,
\( {\displaystyle f(\mathbf {x} ,\sigma ^{2}\mid \mu ,\mathbf {V} ^{-1},\alpha ,\beta )=|\mathbf {V} |^{-1/2}{(2\pi )^{-k/2}}\,{\frac {\beta ^{\alpha }}{\Gamma (\alpha )}}\,\left({\frac {1}{\sigma ^{2}}}\right)^{\alpha +1+k/2}\exp \left(-{\frac {2\beta +(\mathbf {x} -{\boldsymbol {\mu }})'\mathbf {V} ^{-1}(\mathbf {x} -{\boldsymbol {\mu }})}{2\sigma ^{2}}}\right).} \)
where \( |\mathbf{V}| \) is the determinant of the \( k \times k \) matrix \( \mathbf {V} \) . Note how this last equation reduces to the first form if k k=1 so that \( \mathbf{x}, \mathbf{V}, \boldsymbol{\mu} \) are scalars.
Alternative parameterization
It is also possible to let \( \gamma = 1 / \lambda \) in which case the pdf becomes
\( } {\displaystyle f(x,\sigma ^{2}\mid \mu ,\gamma ,\alpha ,\beta )={\frac {1}{\sigma {\sqrt {2\pi \gamma }}}}\,{\frac {\beta ^{\alpha }}{\Gamma (\alpha )}}\,\left({\frac {1}{\sigma ^{2}}}\right)^{\alpha +1}\exp \left(-{\frac {2\gamma \beta +(x-\mu )^{2}}{2\gamma \sigma ^{2}}}\right)} \)
In the multivariate form, the corresponding change would be to regard the covariance matrix \( \mathbf {V} \) instead of its inverse \( \mathbf {V} ^{-1} \) as a parameter.
Cumulative distribution function
\( {\displaystyle F(x,\sigma ^{2}\mid \mu ,\lambda ,\alpha ,\beta )={\frac {e^{-{\frac {\beta }{\sigma ^{2}}}}\left({\frac {\beta }{\sigma ^{2}}}\right)^{\alpha }\left(\operatorname {erf} \left({\frac {{\sqrt {\lambda }}(x-\mu )}{{\sqrt {2}}\sigma }}\right)+1\right)}{2\sigma ^{2}\Gamma (\alpha )}}} \)
Properties
Marginal distributions
Given \( (x,\sigma^2) \sim \text{N-}\Gamma^{-1}(\mu,\lambda,\alpha,\beta) \! . as above, σ 2 {\displaystyle \sigma ^{2}} \sigma ^{2} by itself follows an inverse gamma distribution:
\( \sigma^2 \sim \Gamma^{-1}(\alpha,\beta) \! \)
while \( \sqrt{\frac{\alpha\lambda}{\beta}} (x - \mu) \) follows a t distribution with \( 2 \alpha \) degrees of freedom.
In the multivariate case, the marginal distribution of \( \mathbf {x} \) is a multivariate t distribution:
\( {\displaystyle \mathbf {x} \sim t_{2\alpha }({\boldsymbol {\mu }},{\frac {\beta }{\alpha }}\mathbf {V} ^{-1})\!} \)
Summation
Scaling
Exponential family
Information entropy
Kullback–Leibler divergence
Maximum likelihood estimation
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}} \sigma ^{2} from an inverse gamma distribution with parameters α {\displaystyle \alpha } \alpha and β {\displaystyle \beta } \beta
Sample x {\displaystyle x} x from a normal distribution with mean μ {\displaystyle \mu } \mu and variance σ 2 / λ {\displaystyle \sigma ^{2}/\lambda } \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 \( \sigma^2 \mathbf{V} \) (whereas in the multivariate inverse-gamma distribution the covariance matrix is regarded as known up to the scale factor \( \sigma ^{2}) \) 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
Probability distributions (List)
Discrete univariate
with finite support
Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher soliton discrete uniform Zipf Zipf–Mandelbrot
Discrete univariate
with infinite support
beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Flory–Schulz Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta
Continuous univariate
supported on a bounded interval
arcsine ARGUS Balding–Nichols Bates beta beta rectangular continuous Bernoulli Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle
Continuous univariate
supported on a semi-infinite interval
Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Fréchet gamma gamma/Gompertz generalized gamma generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared
scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared noncentral F Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull
discrete Weibull Wilks's lambda
Continuous univariate
supported on the whole real line
Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's SU Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt
Continuous univariate
with support whose type varies
generalized chi-squared generalized extreme value generalized Pareto Marchenko–Pastur q-exponential q-Gaussian q-Weibull shifted log-logistic Tukey lambda
Mixed continuous-discrete univariate
rectified Gaussian
Multivariate (joint)
Discrete
Ewens
multinomial
Dirichlet-multinomial
negative multinomial
Continuous
Dirichlet
generalized Dirichlet
multivariate Laplace
multivariate normal
multivariate stable
multivariate t
normal-inverse-gamma
normal-gamma
Matrix-valued
inverse matrix gamma
inverse-Wishart
matrix normal
matrix t
matrix gamma
normal-inverse-Wishart
normal-Wishart
Wishart
Directional
Univariate (circular) directional
Circular uniform
univariate von Mises
wrapped normal
wrapped Cauchy
wrapped exponential
wrapped asymmetric Laplace
wrapped Lévy
Bivariate (spherical)
Kent
Bivariate (toroidal)
bivariate von Mises
Multivariate
von Mises–Fisher
Bingham
Degenerate and singular
Degenerate
Dirac delta function
Singular
Cantor
Families
Circular compound Poisson elliptical exponential natural exponential location–scale maximum entropy mixture Pearson Tweedie wrapped
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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