International Geomagnetic Reference Field: Difference between revisions
en>RockMagnetist I fixed the redirect so "magnetic scalar potential" can be used. |
en>Fgnievinski |
||
Line 1: | Line 1: | ||
{{Probability distribution | | |||
name =normal-inverse-gamma| | |||
type =density| | |||
pdf_image =| | |||
cdf_image =| | |||
parameters =<math>\mu\,</math> [[location parameter|location]] ([[real number|real]])<br /><math>\lambda > 0\,</math> (real)<br /><math>\alpha > 0\,</math> (real)<br /><math>\beta > 0\,</math> (real)| | |||
support =<math>x \in (-\infty, \infty)\,\!, \; \sigma^2 \in (0,\infty)</math>| | |||
pdf =<math>\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} } </math> | |||
| | |||
cdf =| | |||
mean =| | |||
median =| | |||
mode =| | |||
variance =| | |||
skewness =| | |||
kurtosis =| | |||
entropy =| | |||
mgf =| | |||
char =| | |||
}} | |||
In [[probability theory]] and [[statistics]], the '''normal-inverse-gamma distribution''' (or '''Gaussian-inverse-gamma distribution''') is a four-parameter family of multivariate continuous [[probability distribution]]s. It is the [[conjugate prior]] of a [[normal distribution]] with unknown [[mean]] and [[variance]]. | |||
==Definition== | |||
Suppose | |||
:<math> x | \sigma^2, \mu, \lambda\sim \mathrm{N}(\mu,\sigma^2 / \lambda) \,\! </math> | |||
has a [[normal distribution]] with [[mean]] <math> \mu</math> and [[variance]] <math> \sigma^2 / \lambda</math>, where | |||
:<math>\sigma^2|\alpha, \beta \sim \Gamma^{-1}(\alpha,\beta) \!</math> | |||
has an [[inverse gamma distribution]]. Then <math>(x,\sigma^2) </math> | |||
has a normal-inverse-gamma distribution, denoted as | |||
:<math> (x,\sigma^2) \sim \text{N-}\Gamma^{-1}(\mu,\lambda,\alpha,\beta) \! . | |||
</math> | |||
(<math>\text{NIG}</math> is also used instead of <math>\text{N-}\Gamma^{-1}.</math>) | |||
In a multivariate form of the normal-inverse-gamma distribution, <math> \mathbf{x} | \sigma^2, \boldsymbol{\mu}, \mathbf{V}^{-1}\sim \mathrm{N}(\boldsymbol{\mu},\sigma^2 \mathbf{V}) \,\! </math> -- that is, conditional on <math> \sigma^2 </math>, <math> \mathbf{x} </math> is a <math> k \times 1 </math> random vector that follows the [[multivariate normal distribution]] with mean <math> \boldsymbol{\mu} </math> and [[covariance matrix | covariance]] <math> \sigma^2\mathbf{V}</math> -- while, as in the univariate case, <math>\sigma^2|\alpha, \beta \sim \Gamma^{-1}(\alpha,\beta) \!</math>. | |||
==Characterization== | |||
===Probability density function=== | |||
: <math>f(x,\sigma^2|\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) </math> | |||
For the multivariate form where <math> \mathbf{x} </math> is a <math> k \times 1 </math> random vector, | |||
: <math>f(\mathbf{x},\sigma^2|\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)^{k/2 + \alpha + 1} \exp \left( -\frac { 2\beta + (\mathbf{x} - \boldsymbol{\mu})' \mathbf{V}^{-1} (\mathbf{x} - \boldsymbol{\mu})} {2\sigma^2} \right). </math> | |||
where <math>|\mathbf{V}|</math> is the [[determinant]] of the <math> k \times k </math> [[matrix (mathematics) | matrix]] <math>\mathbf{V}</math>. Note how this last equation reduces to the first form if <math>k = 1</math> so that <math>\mathbf{x}, \mathbf{V}, \boldsymbol{\mu}</math> are [[scalar (mathematics)| scalars]]. | |||
==== Alternative parameterization ==== | |||
It is also possible to let <math> \gamma = 1 / \lambda</math> in which case the pdf becomes | |||
: <math>f(x,\sigma^2|\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)</math> | |||
In the multivariate form, the corresponding change would be to regard the covariance matrix <math>\mathbf{V}</math> instead of its [[invertible matrix | inverse]] <math>\mathbf{V}^{-1}</math> as a parameter. | |||
===Cumulative distribution function=== | |||
==Properties== | |||
===Marginal distributions=== | |||
Given <math> (x,\sigma^2) \sim \text{N-}\Gamma^{-1}(\mu,\lambda,\alpha,\beta) \! . | |||
</math> as above, <math>\sigma^2</math> by itself follows an [[inverse gamma distribution]]: | |||
:<math>\sigma^2 \sim \Gamma^{-1}(\alpha,\beta) \!</math> | |||
while <math> \sqrt{\frac{\beta}{\alpha\lambda}} (x - \mu) </math> follows a [[Student's t-distribution | t distribution]] with <math> 2 \alpha </math> degrees of freedom. | |||
In the multivariate case, the marginal distribution of <math>\mathbf{x}</math> is a [[multivariate Student distribution | multivariate t distribution]]: | |||
:<math>\mathbf{x} \sim t_{2\alpha}(\boldsymbol{\mu}, \frac{\beta}{\alpha} \mathbf{V}) \!</math> | |||
===Summation=== | |||
===Scaling=== | |||
===Exponential family=== | |||
===Information entropy=== | |||
===Kullback-Leibler divergence=== | |||
== Maximum likelihood estimation == | |||
{{Empty section|date=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 <math>\sigma^2</math> from an inverse gamma distribution with parameters <math>\alpha</math> and <math>\beta</math> | |||
# Sample <math>x</math> from a normal distribution with mean <math>\mu</math> and variance <math>\sigma^2/\lambda</math> | |||
== Related distributions == | |||
* The [[normal-gamma distribution]] is the same distribution parameterized by [[precision (statistics)|precision]] rather than [[variance]] | |||
* A generalization of this distribution which allows for a multivariate mean and a completely unknown positive-definite covariance matrix <math>\sigma^2 \mathbf{V}</math> (whereas in the multivariate inverse-gamma distribution the covariance matrix is regarded as known up to the scale factor <math>\sigma^2</math>) is the [[normal-inverse-Wishart 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 | |||
{{ProbDistributions|multivariate}} | |||
[[Category:Continuous distributions]] | |||
[[Category:Multivariate continuous distributions]] | |||
[[Category:Normal distribution]] | |||
[[Category:Probability distributions]] |
Revision as of 17:26, 13 September 2013
Template:Probability distribution 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
has a normal distribution with mean and variance , where
has an inverse gamma distribution. Then has a normal-inverse-gamma distribution, denoted as
In a multivariate form of the normal-inverse-gamma distribution, -- that is, conditional on , is a random vector that follows the multivariate normal distribution with mean and covariance -- while, as in the univariate case, .
Characterization
Probability density function
For the multivariate form where is a random vector,
where is the determinant of the matrix . Note how this last equation reduces to the first form if so that are scalars.
Alternative parameterization
It is also possible to let in which case the pdf becomes
In the multivariate form, the corresponding change would be to regard the covariance matrix instead of its inverse as a parameter.
Cumulative distribution function
Properties
Marginal distributions
Given as above, by itself follows an inverse gamma distribution:
while follows a t distribution with degrees of freedom.
In the multivariate case, the marginal distribution of is a multivariate t distribution:
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 from an inverse gamma distribution with parameters and
- Sample from a normal distribution with mean and variance
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 (whereas in the multivariate inverse-gamma distribution the covariance matrix is regarded as known up to the scale factor ) is the normal-inverse-Wishart 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
55 yrs old Metal Polisher Records from Gypsumville, has interests which include owning an antique car, summoners war hack and spelunkering. Gets immense motivation from life by going to places such as Villa Adriana (Tivoli).
my web site - summoners war hack no survey ios