# 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

 ```1 2``` ```## 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

 ``` 1 2 3 4 5 6 7 8 9 10 11``` ```## 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.