bbricks (version 0.1.4)

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

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 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

# NOT RUN {
## 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)
# }