sufficientStatistics.LinearGaussianGaussian: Sufficient statistics of a "LinearGaussianGaussian" object

In bbricks: Bayesian Methods and Graphical Model Structures for Statistical Modeling

Description

For following model structure:

x \sim Gaussian(A z + b, Sigma)

z \sim Gaussian(m,S)

Where Sigma is known. A is a dimx x dimz matrix, x is a dimx x 1 random vector, z is a dimz x 1 random vector, b is a dimm x 1 vector. Gaussian() is the Gaussian distribution. See ?dGaussian for the definition of Gaussian distribution.
For one dimensional observations: x is a vector of length N, or a N x 1 matrix, each row is an observation; A is a N x dimz matrix; b is a length N vector. The sufficient statistics are:

• SA = A^T A / Sigma

• SAx = A^T (x-b) / Sigma

For dimx dimensional observations: x must be a N x m matrix, each row is an observation; A can be either a list or a matrix. When A is a list, A = {A_1,A_2,...A_N} is a list of dimx x dimz matrices. If A is a single dimx x dimz matrix, it will be replicated N times into a length N list; b can be either a matrix or a vector. When b is a matrix, b={b_1^T,...,b_N^T} is a N x dimx matrix, each row is a transposed vector. When b is a length dimx vector, it will be transposed into a row vector and replicated N times into a N x dimx matrix. The sufficient statistics are:

• SA = sum_{i=1:N} A_i^T Sigma^{-1} A_i

• SAx = sum_{i=1:N} A_i^T Sigma^{-1} (x_i-b_i)

Usage

## S3 method for class 'LinearGaussianGaussian'
sufficientStatistics(obj, x, A, b = NULL, foreach = FALSE, ...)

Arguments

obj	A "LinearGaussianGaussian" object.
x	matrix, Gaussian samples, when x is a matrix, each row is a sample of dimension ncol(x). when x is a vector, x is length(x) samples of dimension 1.
A	matrix or list. when x is a N x 1 matrix, A must be a matrix of N x dimz, dimz is the dimension of z; When x is a N x dimx matrix, where dimx > 1, A can be either a list or a matrix. When A is a list, A = {A_1,A_2,...A_N} is a list of dimx x dimz matrices. If A is a single dimx x dimz matrix, it will be replicated N times into a length N list
b	matrix, when x is a N x 1 matrix, b must also be a N x 1 matrix or length N vector; When x is a N x dimx matrix, where dimx > 1, b can be either a matrix or a vector. When b is a matrix, b={b_1^T,...,b_N^T} is a N x dimx matrix, each row is a transposed vector. When b is a length dimx vector, it will be transposed into a row vector and replicated N times into a N x dimx matrix. When b = NULL, it will be treated as a vector of zeros. Default NULL.
foreach	logical, specifying whether to return the sufficient statistics for each observation. Default FALSE.
...	Additional arguments to be passed to other inherited types.

Value

If foreach=TRUE, will return a list of sufficient statistics for each row of x, otherwise will return the sufficient statistics of x as a whole.

References

Murphy, Kevin P. Machine learning: a probabilistic perspective. MIT press, 2012.

See Also

LinearGaussianGaussian, sufficientStatistics_Weighted.LinearGaussianGaussian

Examples

## create a LinearGaussianGaussian object
## where x is 2 dimensional, z is 3 dimensional
obj <- LinearGaussianGaussian(gamma=list(Sigma=matrix(c(2,1,1,2),2,2),
                              m=c(0.2,0.5,0.6),S=diag(3)))
x <- rGaussian(100,mu = runif(2),Sigma = diag(2))
A <- matrix(runif(6),2,3)
b <- runif(2)
sufficientStatistics(obj,x=x,A=A,b=b)
Alist <- replicate(100,A,simplify=FALSE)
## should print the same thing as above:
sufficientStatistics(obj,x=x,A=Alist,b=b)

bbricks documentation built on July 8, 2020, 7:29 p.m.