jMEF.MultivariateIsotropicGaussian Class Reference

List of all members.

Public Member Functions

double F (PVector T)

Computes the log normalizer $F( \mathbf{\Theta} )$ .

PVector gradF (PVector T)

Computes $\nabla F ( \mathbf{\Theta} )$ .

double G (PVector H)

Computes $G(\mathbf{H})$ .

PVector gradG (PVector H)

Computes $\nabla G (\mathbf{H})$ .

PVector t (PVector x)

Computes the sufficient statistic $ t(x)$ .

double k (PVector x)

Computes the carrier measure $ k(x) $ .

PVector Lambda2Theta (PVector L)

Converts source parameters to natural parameters.

PVector Theta2Lambda (PVector T)

Converts natural parameters to source parameters.

PVector Lambda2Eta (PVector L)

Converts source parameters to expectation parameters.

PVector Eta2Lambda (PVector H)

Converts expectation parameters to source parameters.

double density (PVector x, PVector param)

Computes the density value $f(x;\mu)$ .

PVector drawRandomPoint (PVector L)

Draws a point from the considered distribution.

double KLD (PVector LP, PVector LQ)

Computes the Kullback-Leibler divergence between two multivariate isotropic Gaussian distributions.

Detailed Description

Author:: Vincent Garcia
Frank Nielsen

Version:: 1.0

License

See file LICENSE.txt

Description

The multivariate isotropic Gaussian distribution ( $\Sigma=$ Id) is an exponential family and, as a consequence, the probability density function is given by

$f(x; \mathbf{\Theta}) = \exp \left( \langle t(x), \mathbf{\Theta} \rangle - F(\mathbf{\Theta}) + k(x) \right)$

where $\mathbf{\Theta}$ are the natural parameters. This class implements the different functions allowing to express a multivariate Gaussian distribution as a member of an exponential family.

Parameters

The parameters of a given distribution are:

Source parameters $\mathbf{\Lambda} = \mu \mbox{ with } \mu \in \mathds{R}^d$
Natural parameters $\mathbf{\Theta} = \theta$
Expectation parameters $\mathbf{H} = \eta$

Member Function Documentation

double jMEF.MultivariateIsotropicGaussian.density	(	PVector	x,
		PVector	param
	)

Computes the density value $f(x;\mu)$ .

Parameters:

	x	point
	param	parameters (source, natural, or expectation)

Returns:: $f(x;\mu) = \frac{1}{ (2\pi)^{d/2} } \exp \left( - \frac{(x-\mu)^T (x-\mu)}{2} \right) \mbox{ for } x \in \mathds{R}^d$

PVector jMEF.MultivariateIsotropicGaussian.drawRandomPoint ( PVector L )

Draws a point from the considered distribution.

Parameters:

L

source parameters $\mathbf{\Lambda} = \mu$

Returns:: a point.

PVector jMEF.MultivariateIsotropicGaussian.Eta2Lambda ( PVector H )

Converts expectation parameters to source parameters.

Parameters:

H

expectation parameters $\mathbf{H} = \eta$

Returns:: source parameters $\mathbf{\Lambda} = \eta$

double jMEF.MultivariateIsotropicGaussian.F ( PVector T )

Computes the log normalizer $F( \mathbf{\Theta} )$ .

Parameters:

T

natural parameters $\mathbf{\Theta} = \theta$

Returns:: $F(\mathbf{\theta}) = \frac{1}{2} \theta^\top\theta + \frac{d}{2}\log 2\pi$

double jMEF.MultivariateIsotropicGaussian.G ( PVector H )

Computes $G(\mathbf{H})$ .

Parameters:

H

expectation parameters $\mathbf{H} = \eta$

Returns:: $F(\mathbf{\theta})= \frac{1}{2} \eta^\top\eta + \frac{d}{2}\log 2\pi$

PVector jMEF.MultivariateIsotropicGaussian.gradF ( PVector T )

Computes $\nabla F ( \mathbf{\Theta} )$ .

Parameters:

T

natural $\mathbf{\Theta} = \theta$

Returns:: $\nabla F( \mathbf{\Theta} ) = \theta$

PVector jMEF.MultivariateIsotropicGaussian.gradG ( PVector H )

Computes $\nabla G (\mathbf{H})$ .

Parameters:

H

expectation parameters $\mathbf{H} = \eta$

Returns:: $\nabla G(\mathbf{H}) = \eta$

double jMEF.MultivariateIsotropicGaussian.k ( PVector x )

Computes the carrier measure $ k(x) $ .

Parameters:

x

a point

Returns:: $k(x) = -\frac{1}{2}x^\top x$

double jMEF.MultivariateIsotropicGaussian.KLD	(	PVector	LP,
		PVector	LQ
	)

Computes the Kullback-Leibler divergence between two multivariate isotropic Gaussian distributions.

Parameters:

	LP	source parameters $\mathbf{\Lambda}_P$
	LQ	source parameters $\mathbf{\Lambda}_Q$

Returns:: $D_{\mathrm{KL}}(f_P \| f_Q) = \frac{1}{2} ( \mu_Q - \mu_P )^\top( \mu_Q - \mu_P )$

PVector jMEF.MultivariateIsotropicGaussian.Lambda2Eta ( PVector L )

Converts source parameters to expectation parameters.

Parameters:

L

source parameters $\mathbf{\Lambda} = \mu$

Returns:: expectation parameters $\mathbf{H} = \mu$

PVector jMEF.MultivariateIsotropicGaussian.Lambda2Theta ( PVector L )

Converts source parameters to natural parameters.

Parameters:

L

source parameters $\mathbf{\Lambda} = \mu$

Returns:: natural parameters $\mathbf{\Theta} = \mu$

PVector jMEF.MultivariateIsotropicGaussian.t ( PVector x )

Computes the sufficient statistic $ t(x)$ .

Parameters:

x

a point

Returns:

PVector jMEF.MultivariateIsotropicGaussian.Theta2Lambda ( PVector T )

Converts natural parameters to source parameters.

Parameters:

T

natural parameters $\mathbf{\Theta} = \theta$

Returns:: source parameters $\mathbf{\Lambda} = \theta$

The documentation for this class was generated from the following file:

MultivariateIsotropicGaussian.java


Public Member Functions
double	F (PVector T)
	Computes the log normalizer $F( \mathbf{\Theta} )$ .
PVector	gradF (PVector T)
	Computes $\nabla F ( \mathbf{\Theta} )$ .
double	G (PVector H)
	Computes $G(\mathbf{H})$ .
PVector	gradG (PVector H)
	Computes $\nabla G (\mathbf{H})$ .
PVector	t (PVector x)
	Computes the sufficient statistic .
double	k (PVector x)
	Computes the carrier measure .
PVector	Lambda2Theta (PVector L)
	Converts source parameters to natural parameters.
PVector	Theta2Lambda (PVector T)
	Converts natural parameters to source parameters.
PVector	Lambda2Eta (PVector L)
	Converts source parameters to expectation parameters.
PVector	Eta2Lambda (PVector H)
	Converts expectation parameters to source parameters.
double	density (PVector x, PVector param)
	Computes the density value $f(x;\mu)$ .
PVector	drawRandomPoint (PVector L)
	Draws a point from the considered distribution.
double	KLD (PVector LP, PVector LQ)
	Computes the Kullback-Leibler divergence between two multivariate isotropic Gaussian distributions.