bbricks (version 0.1.4)

sufficientStatistics_Weighted: Get weighted sample sufficient statistics

Description

This is a generic function that will generate the weighted sufficient statistics of a given "BayesianBrick" object. That is, for the model structure: $$theta|gamma \sim H(gamma)$$ $$x|theta \sim F(theta)$$ get the weighted sufficient statistics T(x). For a given sample set x, each row of x is an observation, the sample weights w, and a Bayesian bricks object obj. sufficientStatistics_Weighted() return the weighted sufficient statistics for different model structures:

class(obj)="LinearGaussianGaussian"

$$x \sim Gaussian(A z + b, Sigma)$$ $$z \sim Gaussian(m,S)$$ The sufficient statistics are:

• SA = \(sum_{i=1:N} w_i A_i^T Sigma^{-1} A_i\)

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

See ?sufficientStatistics.LinearGaussianGaussian for details.

class(obj)="GaussianGaussian"

Where $$x \sim Gaussian(mu,Sigma)$$ $$mu \sim Gaussian(m,S)$$ Sigma is known. The sufficient statistics are:

• N: the effective number of samples.

• xsum: the row sums of the samples.

See ?sufficientStatistics_Weighted.GaussianGaussian for details.

class(obj)="GaussianInvWishart"

Where $$x \sim Gaussian(mu,Sigma)$$ $$Sigma \sim InvWishart(v,S)$$ 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_Weighted.GaussianInvWishart for details.

class(obj)="GaussianNIW"

Where $$x \sim Gaussian(mu,Sigma)$$ $$Sigma \sim InvWishart(v,S)$$ $$mu \sim Gaussian(m,Sigma/k)$$ 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_Weighted.GaussianNIW for details.

class(obj)="GaussianNIG"

Where $$x \sim Gaussian(X beta,sigma^2)$$ $$sigma^2 \sim InvGamma(a,b)$$ $$beta \sim Gaussian(m,sigma^2 V)$$ 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_Weighted.GaussianNIG for details.

class(obj)="CatDirichlet"

Where $$x \sim Categorical(pi)$$ $$pi \sim Dirichlet(alpha)$$ The sufficient statistics of CatDirichlet object can either be x itself, or the counts of the unique labels in x. See ?sufficientStatistics_Weighted.CatDirichlet for details.

class(obj)="CatDP"

Where $$x \sim Categorical(pi)$$ $$pi \sim DirichletProcess(alpha)$$ The sufficient statistics of CatDP object can either be x itself, or the counts of the unique labels in x. See ?sufficientStatistics_Weighted.CatDP for details.

class(obj)="DP"

Where $$pi|alpha \sim DP(alpha,U)$$ $$z|pi \sim Categorical(pi)$$ $$theta_z|psi \sim H0(psi)$$ $$x|theta_z,z \sim F(theta_z)$$ The sufficient statistics of "DP" object is the same sufficient statistics of the "BasicBayesian" inside the "DP". See ?sufficientStatistics_Weighted.DP for details.

class(obj)="HDP"

Where $$G|gamma \sim DP(gamma,U)$$ $$pi_j|G,alpha \sim DP(alpha,G), j = 1:J$$ $$z|pi_j \sim Categorical(pi_j)$$ $$k|z,G \sim Categorical(G),\textrm{ if z is a sample from the base measure G}$$ $$theta_k|psi \sim H0(psi)$$ The sufficient statistics of "HDP" object is the same sufficient statistics of the "BasicBayesian" inside the "HDP". See ?sufficientStatistics_Weighted.HDP for details.

class(obj)="HDP2"

Where $$G |eta \sim DP(eta,U)$$ $$G_m|gamma,G \sim DP(gamma,G), m = 1:M$$ $$pi_{mj}|G_m,alpha \sim DP(alpha,G_m), j = 1:J_m$$ $$z|pi_{mj} \sim Categorical(pi_{mj})$$ $$k|z,G_m \sim Categorical(G_m),\textrm{ if z is a sample from the base measure } G_m$$ $$u|k,G \sim Categorical(G),\textrm{ if k is a sample from the base measure } G_m$$ $$theta_u|psi \sim H0(psi)$$ $$x|theta_u,u \sim F(theta_u)$$ The sufficient statistics of "HDP2" object is the same sufficient statistics of the "BasicBayesian" inside the "HDP2". See ?sufficientStatistics_Weighted.HDP2 for details.

Usage

sufficientStatistics_Weighted(obj, x, w, ...)

Arguments

obj

a "BayesianBrick" object used to select a method.

x

a set of samples.

w

numeric, sample weights.

...

further arguments passed to or from other methods.

Value

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

See Also

sufficientStatistics_Weighted.LinearGaussianGaussian for Linear Gaussian and Gaussian conjugate structure, sufficientStatistics_Weighted.GaussianGaussian for Gaussian-Gaussian conjugate structure, sufficientStatistics_Weighted.GaussianInvWishart for Gaussian-Inverse-Wishart conjugate structure, sufficientStatistics_Weighted.GaussianNIW for Gaussian-NIW conjugate structure, sufficientStatistics_Weighted.GaussianNIG for Gaussian-NIG conjugate structure, sufficientStatistics_Weighted.CatDirichlet for Categorical-Dirichlet conjugate structure, sufficientStatistics_Weighted.CatDP for Categorical-DP conjugate structure ...

Examples

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