p ( zk|tk, xk, ti, xi, zi;i = 1 : n)
=Γ(Nr=rk + α)
Γ(Nr=rk + α + 1) ·
Γ(Pi
δr=rk
(zi, z) + α/NZ + 1)
Γ(Pi
δr=rk
(zi, z) + α/NZ)
·Γ(Nx=xk,z=zk + η)
Γ(Nx=xk,z=zk + η + 1) ·
Γ(Pi
δx=xk,z=zk
(ti, tk) + η/NT + 1)
Γ(Pi
δx=xk,z=zk
(ti, tk) + η/NT )
=
Pi
δr=rk
(zi, zk) + α/NZ
Nr=rk + α.
Pi
δx=xk,z=zk
(ti, tk) + η/NT
Nx=xk,z=zk + η
=
Nr=rk,z=zk + α/NZ
Nr=rk + α·
Nx=xk,z=zk,t=tk + η/NT
Nx=xk,z=zk + η
(18)
Intuitively, we can perceive from Eq. (18) that Nr=rk,z=zk+α/NZ
Nr=rk +αtell us how resource r
is likely to be described by the topic z; as the later part, Nx=xk,z=zk,t=tk +η/NT
Nx=**xk,z**=zk+ηtell us how
tag t is likely to be chosen given interest x and z.
Similarly, we can obtain the posterior probability of xk as we did for zk.
$$p(x_{k}|t_{k},z_{k},t_{i},x_{i},z_{i};i=1:n)\,=\,{\frac{N_{u=u_{k},x=x_{k}}+\beta/N_{X}}{N_{u=u_{k}}+\beta}}$$ $$\cdot\,\,\,{\frac{N_{x=x_{k},z=z_{k},t=t_{k}}+\eta/N_{T}}{N_{x=x_{k},z=z_{k}}+\eta}}$$
$$\quad(19)$$
We can now generalize Eq. (18) and Eq. (19) for sampling posterior probabilities of topic z and interest x of a present tuple i given all other tuples. We define Nri,z−i as the number of all toples having r = ri and z but excluding the present tuple i. Similarly, Nz−i,xi,ti is a number of all tuples having x = xi, t = ti and z but excluding the present tuple i. As z−i represents all topic assignments except that of the tuple i.

$$p(z_{i}|\mathbf{z}_{-i},\mathbf{x},\mathbf{t})=\frac{N_{r_{i},z_{-i}}+\alpha/N_{Z}}{N_{r_{i}}+\alpha-1}.\frac{N_{z_{-i},x_{i},t_{i}}+\eta/N_{T}}{N_{z_{-i},x_{i}}+\eta}\tag{20}$$  $$p(x_{i}|\mathbf{x}_{-i},\mathbf{z},\mathbf{t})=\frac{N_{u_{i},x_{-i}}+\beta/N_{X}}{N_{u_{i}}+\beta-1}\cdot\frac{N_{x_{-i},z_{i},t_{i}}+\eta/N_{T}}{N_{x_{-i},z_{i}}+\eta}\tag{21}$$

## Acknowledgments

We would like to thank anonymous reviewers for providing useful comments and suggestions to improve the manuscript. This material is based in part upon work supported by the National Science Foundation under Grant Numbers CMMI-0753124 and IIS-0812677. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

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