jMEF.UnivariateGaussianFixedVariance Class Reference

List of all members.

Public Member Functions

 UnivariateGaussianFixedVariance ()
 Class constructor with $\sigma^2=1$.
 UnivariateGaussianFixedVariance (double variance)
 Class constructor.
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,\sigma^2) $.
PVector drawRandomPoint (PVector L)
 Draws a point from the considered distribution.
double KLD (PVector LP, PVector LQ)
 Computes the Kullback-Leibler divergence between two univariate Gaussian distributions.

Detailed Description

Author:
Vincent Garcia

Frank Nielsen

Version:
1.0

License

See file LICENSE.txt

Description

The univariate Gaussian distribution, with fixed variance $ \sigma^2 $, 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 univariate Gaussian distribution as a member of an exponential family.

Parameters

The parameters of a given distribution are:

Constructor & Destructor Documentation

jMEF.UnivariateGaussianFixedVariance.UnivariateGaussianFixedVariance (  ) 

Class constructor with $\sigma^2=1$.

jMEF.UnivariateGaussianFixedVariance.UnivariateGaussianFixedVariance ( double  variance  ) 

Class constructor.

Parameters:
variance variance $ \sigma^2$

Member Function Documentation

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

Computes the density value $ f(x;\mu,\sigma^2) $.

Parameters:
x a point
param parameters (source, natural, or expectation)
Returns:
$ f(x;\mu,\sigma^2) = \frac{1}{ \sqrt{2\pi \sigma^2} } \exp \left( - \frac{(x-\mu)^2}{ 2 \sigma^2} \right) $

PVector jMEF.UnivariateGaussianFixedVariance.drawRandomPoint ( PVector  L  ) 

Draws a point from the considered distribution.

Parameters:
L source parameters $ \mathbf{\Lambda} = \lambda $
Returns:
a point

PVector jMEF.UnivariateGaussianFixedVariance.Eta2Lambda ( PVector  H  ) 

Converts expectation parameters to source parameters.

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

double jMEF.UnivariateGaussianFixedVariance.F ( PVector  T  ) 

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

Parameters:
T natural parameters $ \mathbf{\Theta} = \theta $
Returns:
$ F(\mathbf{\Theta}) = \frac{\sigma^2 \theta^2 + \log(2 \pi \sigma^2)}{2} $

double jMEF.UnivariateGaussianFixedVariance.G ( PVector  H  ) 

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

Parameters:
H expectation parameters $ \mathbf{H} = \eta $
Returns:
$ G(\mathbf{H}) = \frac{\eta^2}{2 \sigma^2} $

PVector jMEF.UnivariateGaussianFixedVariance.gradF ( PVector  T  ) 

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

Parameters:
T natural parameters $ \mathbf{\Theta} = \theta $
Returns:
$ \nabla F( \mathbf{\Theta} ) = \sigma^2 \theta $

PVector jMEF.UnivariateGaussianFixedVariance.gradG ( PVector  H  ) 

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

Parameters:
H expectation parameters $ \mathbf{H} = \eta $
Returns:
$ \nabla G( \mathbf{H} ) = \frac{\eta}{\sigma^2} $

double jMEF.UnivariateGaussianFixedVariance.k ( PVector  x  ) 

Computes the carrier measure $ k(x) $.

Parameters:
x a point
Returns:
$ k(x) = -\frac{x^2}{2 \sigma^2} $

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

Computes the Kullback-Leibler divergence between two univariate Gaussian distributions.

Parameters:
LP source parameters $ \mathbf{\Lambda}_P $
LQ source parameters $ \mathbf{\Lambda}_Q $
Returns:
$ D_{\mathrm{KL}}(f_P\|f_Q) = \frac{(\mu_Q-\mu_P)^2}{2\sigma^2} $

PVector jMEF.UnivariateGaussianFixedVariance.Lambda2Eta ( PVector  L  ) 

Converts source parameters to expectation parameters.

Parameters:
L source parameters $ \mathbf{\Lambda} = \lambda $
Returns:
expectation parameters $ \mathbf{H} = \lambda $

PVector jMEF.UnivariateGaussianFixedVariance.Lambda2Theta ( PVector  L  ) 

Converts source parameters to natural parameters.

Parameters:
L source parameters $ \mathbf{\Lambda} = \mu $
Returns:
natural parameters $ \mathbf{\Theta} = \frac{\mu}{\sigma^2} $

PVector jMEF.UnivariateGaussianFixedVariance.t ( PVector  x  ) 

Computes the sufficient statistic $ t(x)$.

Parameters:
x a point
Returns:
$ t(x) = x $

PVector jMEF.UnivariateGaussianFixedVariance.Theta2Lambda ( PVector  T  ) 

Converts natural parameters to source parameters.

Parameters:
T natural parameters $ \mathbf{\Theta} = \theta $
Returns:
source parameters $ \mathbf{\Lambda} = \theta \sigma^2 $

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