# sufficientStatistics.HDP: Sufficient statistics of a "HDP" object In bbricks: Bayesian Methods and Graphical Model Structures for Statistical Modeling

## Description

For following model structure:

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)

x|theta_k,k \sim F(theta_k)

where DP(gamma,U) is a Dirichlet Process on positive integers, gamma is the "concentration parameter", U is the "base measure" of this Dirichlet process, U is an uniform distribution on all positive integers. DP(alpha,G) is a Dirichlet Process on integers with concentration parameter alpha and base measure G. The choice of F() and H0() can be described by an arbitrary "BasicBayesian" object such as "GaussianGaussian","GaussianInvWishart","GaussianNIW", "GaussianNIG", "CatDirichlet", and "CatDP". See `?BasicBayesian` for definition of "BasicBayesian" objects, and see for example `?GaussianGaussian` for specific "BasicBayesian" instances. As a summary, An "HDP" object is simply a combination of a "CatHDP" object (see `?CatHDP`) and an object of any "BasicBayesian" type.
In the case of HDP, z and k can only be positive integers.
The sufficient statistics of a set of samples x in a "HDP" object is the same sufficient statistics of the "BasicBayesian" inside the "HDP", see examples.

## Usage

 ```1 2``` ```## S3 method for class 'HDP' sufficientStatistics(obj, x, ...) ```

## Arguments

 `obj` A "HDP" object. `x` Random samples of the "BasicBayesian" object. `...` further arguments passed to the corresponding sufficientStatistics method of the "BasicBayesian" object.

## Value

Return the sufficient statistics of the corresponding BasicBayesian type, see examples.

## References

Teh, Yee W., et al. "Sharing clusters among related groups: Hierarchical Dirichlet processes." Advances in neural information processing systems. 2005.

`HDP`, `sufficientStatistics_Weighted.HDP`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13``` ```## a HDP with Gaussian NIW observations obj1 <- HDP(gamma=list(gamma=1,alpha=1,j=2, H0aF="GaussianNIW", parH0=list(m=0,k=1,v=2,S=1))) ## a HDP with Categorical-Dirichlet observations obj2 <- HDP(gamma=list(gamma=1,alpha=1,j=2, H0aF="CatDirichlet", parH0=list(alpha=1,uniqueLabels=letters[1:3]))) x1 <- rnorm(100) x2 <- sample(letters[1:3],100,replace = TRUE) sufficientStatistics(obj = obj1,x=x1,foreach = TRUE) sufficientStatistics(obj = obj1,x=x1,foreach = FALSE) sufficientStatistics(obj = obj2,x=x2,foreach = FALSE) ```