
$$({\mathfrak{g}})$$
$$p(z_{i}=z_{u s e d}|\mathbf{z}_{-i},\mathbf{x},\mathbf{t})={\frac{N_{r_{i},z_{-i}}}{N_{r_{i}}+\alpha-1}}\cdot{\frac{N_{z_{-i},x_{i},t_{i}}+\eta/N_{T}}{N_{z_{-i},x_{i}}+\eta}}$$
$$({\boldsymbol{8}})$$
$$p(z_{i}=z_{n e w}|\mathbf{z}_{-i},\mathbf{x},\mathbf{t})={\frac{\alpha}{N_{r_{i}}+\alpha-1}}\cdot{\frac{1}{N_{T}}}$$

From Eq. (8), we can perceive that the model only favors topic components that are only used within the resource ri. Meanwhile, for other components that are not used by that resource, Nz−i,xi,ti would equal zero and thus result in zero probability in choosing them. Consequently, the model only chooses topic components for a resource either from components that are currently used by that resource, or it instantiates a new component for that resource with probabilities according to Eq. (8) and Eq. (9) respectively. As more new components are instantiated, each resource tends to own its components exclusively.

From the previous section, we can also perceive that each resource profile is generated independently (using symmetric Dirichlet prior) - there is no mechanism to link the used components across different resources16. As mentioned in [Teh et al. 2006], this is an undesired characteristic, because, in our context, we would expect "similar" resources to be described by the same set of "similar" topics.

One possible way to handle this problem is to use Hierarchical Dirichlet Process (HDP)
[Teh et al. 2006] as the prior instead of the symmetric Dirichlet prior. The idea of HDP is to link components at group-specific level together by introducing global components across all groups. Each group is only allowed to use some (or all) of these global components and thus, some of them are expected to be shared across several groups. We adapt this idea by considering all tags of resource r to belong to the resource group r. Similarly, all tags of user u belong to the user group u. Each of the resource groups is assigned to some topic components selected from the global topic component pool. Similarly, each of the user groups is assigned to some interest components selected from the global interest component pool. This extension is depicted in Figure 7. Suppose that a number of all possible topic components is NZ (which will be set to approach ∞ later on) and that for interest components is NX, we can describe such extension as a stochastic process as follows.

At the global level, the weight distribution of components is sampled according to
. (β1, ..., βNX ) ∼ Dirichlet(γx/NX**, ..., γ**x/NX) (generating global interest component weight)
. (α1, ..., αNZ) ∼ Dirichlet(γz/NZ**, ..., γ**z/NZ) (generating global topic component weight)
where γx and γz are concentration parameter, which controls diversity of interests and topics at global level.

At the group specific level,