sufficientStatistics: Get sample sufficient statistics

Description

This is a generic function that will generate the sufficient statistics of a given Bayesian bricks object. i.e. for the model structure: $t h e t a | g a m m a \sim H (g a m m a)$ $x | t h e t a \sim F (t h e t a)$ get the sufficient statistics T(x). For a given sample set x, each row of x is an observation, and a Bayesian bricks object obj. sufficientStatistics() return the sufficient statistics for different model structures:

class(obj)="LinearGaussianGaussian"

$x \sim G a u s s i a n (A z + b, S i g m a)$ $z \sim G a u s s i a n (m, S)$ The sufficient statistics are:

SA = $s u m_{i = 1 : N} A_{i}^{T} S i g m a^{- 1} A_{i}$
SAx = $s u m_{i = 1 : N} A_{i}^{T} S i g m a^{- 1} (x_{i} - b_{i})$

See ?sufficientStatistics.LinearGaussianGaussian for details.

class(obj)="GaussianGaussian"

Where $x \sim G a u s s i a n (m u, S i g m a)$ $m u \sim G a u s s i a n (m, S)$ Sigma is known. The sufficient statistics are:

N: the effective number of samples.
xsum: the row sums of the samples.

See ?sufficientStatistics.GaussianGaussian for details.

class(obj)="GaussianInvWishart"

Where $x \sim G a u s s i a n (m u, S i g m a)$ $S i g m a \sim I n v W i s h a r t (v, S)$ mu is known. The sufficient statistics are:

N: the effective number of samples.
xsum: the sample scatter matrix centered on the mean vector.

See ?sufficientStatistics.GaussianInvWishart for details.

class(obj)="GaussianNIW"

Where $x \sim G a u s s i a n (m u, S i g m a)$ $S i g m a \sim I n v W i s h a r t (v, S)$ $m u \sim G a u s s i a n (m, S i g m a / k)$ The sufficient statistics are:

N: the effective number of samples.
xsum: the row sums of the samples.
S: the uncentered sample scatter matrix.

See ?sufficientStatistics.GaussianNIW for details.

class(obj)="GaussianNIG"

Where $x \sim G a u s s i a n (X b e t a, s i g m a^{2})$ $s i g m a^{2} \sim I n v G a m m a (a, b)$ $b e t a \sim G a u s s i a n (m, s i g m a^{2} V)$ X is a row vector, or a design matrix where each row is an obervation. The sufficient statistics are:

N: the effective number of samples.
SXx: covariance of X and x
SX: the uncentered sample scatter matrix.
Sx: the variance of x

See ?sufficientStatistics.GaussianNIG for details.

class(obj)="CatDirichlet"

Where $x \sim C a t e g o r i c a l (p i)$ $p i \sim D i r i c h l e t (a l p h a)$ The sufficient statistics of CatDirichlet object can either be x itself, or the counts of the unique labels in x. See ?sufficientStatistics.CatDirichlet for details.

class(obj)="CatDP"

Where $x \sim C a t e g o r i c a l (p i)$ $p i \sim D i r i c h l e t P r o c e s s (a l p h a)$ The sufficient statistics of CatDP object can either be x itself, or the counts of the unique labels in x. See ?sufficientStatistics.CatDP for details.

class(obj)="DP"

Where $p i | a l p h a \sim D P (a l p h a, U)$ $z | p i \sim C a t e g o r i c a l (p i)$ $t h e t a_{z} | p s i \sim H 0 (p s i)$ $x | t h e t a_{z}, z \sim F (t h e t a_{z})$ The sufficient statistics of "DP" object is the same sufficient statistics of the "BasicBayesian" inside the "DP". See ?sufficientStatistics.DP for details.

class(obj)="HDP"

Where $G | g a m m a \sim D P (g a m m a, U)$ $p i_{j} | G, a l p h a \sim D P (a l p h a, G), j = 1 : J$ $z | p i_{j} \sim C a t e g o r i c a l (p i_{j})$ $k | z, G \sim C a t e g o r i c a l (G), if z is a sample from the base measure G$ $t h e t a_{k} | p s i \sim H 0 (p s i)$ The sufficient statistics of "HDP" object is the same sufficient statistics of the "BasicBayesian" inside the "HDP". See ?sufficientStatistics.HDP for details.

class(obj)="HDP2"

Where $G | e t a \sim D P (e t a, U)$ $G_{m} | g a m m a, G \sim D P (g a m m a, G), m = 1 : M$ $p i_{m j} | G_{m}, a l p h a \sim D P (a l p h a, G_{m}), j = 1 : J_{m}$ $z | p i_{m j} \sim C a t e g o r i c a l (p i_{m j})$ $k | z, G_{m} \sim C a t e g o r i c a l (G_{m}), if z is a sample from the base measure G_{m}$ $u | k, G \sim C a t e g o r i c a l (G), if k is a sample from the base measure G$ $t h e t a_{u} | p s i \sim H 0 (p s i)$ $x | t h e t a_{u}, u \sim F (t h e t a_{u})$ The sufficient statistics of "HDP2" object is the same sufficient statistics of the "BasicBayesian" inside the "HDP2". See ?sufficientStatistics.HDP2 for details.

Usage

sufficientStatistics(obj, x, ...)

Arguments

obj

a "BayesianBrick" object used to select a method.

a set of samples.

...

further arguments passed to or from other methods.

Value

An object of corresponding sufficient statistics class, such as "ssGaussian"

Examples

Run this code

# NOT RUN {
x <- rGaussian(10,mu = 1,Sigma = 1)
obj <- GaussianNIW()                    #an GaussianNIW object
sufficientStatistics(obj=obj,x=x)
# }

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