Generate the the density value of the posterior predictive distribution of the following structure:
$$G |eta \sim DP(eta,U)$$
$$G_m|gamma,G \sim DP(gamma,G), m = 1:M$$
$$pi_{mj}|G_m,alpha \sim DP(alpha,G_m), j = 1:J_m$$
$$z|pi_{mj} \sim Categorical(pi_{mj})$$
$$k|z,G_m \sim Categorical(G_m),\textrm{ if z is a sample from the base measure }G_{mj}$$
$$u|k,G \sim Categorical(G),\textrm{ if k is a sample from the base measure G}$$
$$theta_u|psi \sim H0(psi)$$
$$x|theta_u,u \sim F(theta_u)$$
where DP(eta,U) is a Dirichlet Process on positive integers, eta is the "concentration parameter", U is the "base measure" of this Dirichlet process, U is an uniform distribution on all positive integers. DP(gamma,G) is a Dirichlet Process on integers with concentration parameter gamma and base measure G. DP(alpha,G_m) is a Dirichlet Process on integers with concentration parameter alpha and base measure G_m. 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 "HDP2" object is simply a combination of a "CatHDP2" object (see ?CatHDP2) and an object of any "BasicBayesian" type.
In the case of HDP2, u, z and k can only be positive integers.
The model structure and prior parameters are stored in a "HDP2" object.
Posterior predictive density = p(u,z,k,x|eta,gamma,alpha,psi) when x is not NULL, or p(u,z,k|eta,gamma,alpha,psi) when x is NULL.
# S3 method for HDP2
dPosteriorPredictive(obj, x = NULL, u, k, z, m, j, LOG = TRUE, ...)A "HDP2" object.
Random samples of the "BasicBayesian" object.
integer, the partition label of the parameter space where the observation x is drawn from.
integer.
integer.
integer, group label.
integer, subgroup label.
Return the log density if set to "TRUE".
Additional arguments to be passed to other inherited types.
A numeric vector, the posterior predictive density.
Teh, Yee W., et al. "Sharing clusters among related groups: Hierarchical Dirichlet processes." Advances in neural information processing systems. 2005.