# dPosteriorPredictive.CatHDP: Posterior predictive density function of a "CatHDP" object In bbricks: Bayesian Methods and Graphical Model Structures for Statistical Modeling

## Description

Generate the the density value of the posterior predictive distribution of the following 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}

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. Categorical() is the Categorical distribution. See `dCategorical` for the definition of the Categorical distribution.
In the case of CatHDP, z and k can only be positive integers.
The model structure and prior parameters are stored in a "CatHDP" object.
Posterior predictive density = p(z,k|alpha,gamma,U,j)

## Usage

 ```1 2``` ```## S3 method for class 'CatHDP' dPosteriorPredictive(obj, z, k, j, LOG = TRUE, ...) ```

## Arguments

 `obj` A "CatHDP" object. `z` integer, the elements of the vector must all greater than 0, the samples of a Categorical distribution. `k` integer, the elements of the vector must all greater than 0, the samples of a Categorical distribution. `j` integer, group label. `LOG` Return the log density if set to "TRUE". `...` Additional arguments to be passed to other inherited types.

## Value

A numeric vector, the posterior predictive density.

## References

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

`CatHDP`, `dPosteriorPredictive.CatHDP`