81.152.193.100: /* Spherical Harmonics */ Removed the calculator, as it is no longer available at the given link

2014-09-26T11:21:44Z

Spherical Harmonics: Removed the calculator, as it is no longer available at the given link

en>Fgnievinski: /* External links */

2013-09-13T16:26:32Z

External links

← Older revision		Revision as of 18:26, 13 September 2013
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	~~I am 24 years old~~ and ~~my name~~ is ~~Darlene Loader~~. ~~I life~~ in ~~Neumunster~~ (~~Germany~~).<br><br>~~My blog~~ ... [~~http~~://~~tinyurl~~.~~com~~/~~lcazmbs nike free sko~~]		{{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]]

en>RockMagnetist: I fixed the redirect so "magnetic scalar potential" can be used.

2012-05-14T18:03:23Z

I fixed the redirect so "magnetic scalar potential" can be used.

New page

I am 24 years old and my name is Darlene Loader. I life in Neumunster (Germany).<br><br>My blog ... [http://tinyurl.com/lcazmbs nike free sko]

International Geomagnetic Reference Field - Revision history

81.152.193.100: /* Spherical Harmonics */ Removed the calculator, as it is no longer available at the given link

en>Fgnievinski: /* External links */

en>RockMagnetist: I fixed the redirect so "magnetic scalar potential" can be used.