该论文注释21中，提及并简化了论文《Topic and role discovery in social networks with experiments on Enron and academic email》- A. McCallum, X. Wang & A. Corrada-Emmanuel.的式子：
P ( x d i , z d i ∣ x − d i , z − d i , w , α , β , a , r ) = P ( x d i , z d i , w d i ∣ x − d i , z − d i , w − d i , α , β , a , r ) P ( w d i ∣ x − d i , z − d i , w − d i , α , β , a , r ) ∝ P ( x , z , w ∣ α , β , a , r ) P ( x − d i , z − d i , w − d i ∣ α , β , a , r ) ∝ Γ ( α z d i + n a d x d i z d i ) Γ ( α z d i + n a d x d i z d i − 1 ) Γ ( β w d i + m z d i w d i ) Γ ( β w d i + m z d i w d i − 1 ) Γ ( ∑ t = 1 T ( α t + n a d x d i t ) ) Γ ( ∑ t = 1 T ( α t + n a d x d i t ) − 1 ) Γ ( ∑ v = 1 V ( β v + m z d i v ) ) Γ ( ∑ v = 1 V ( β v + m z d i v ) − 1 ) ∝ α z d i + n a d x d i z d i − 1 ∑ t = 1 T ( α t + n a d x d i ) − 1 β w d i + m z d i w d i − 1 ∑ v = 1 ( β v + m z d i v ) − 1 \begin{alignedat}{3} &P\left(x_{d i}, z_{d i} \mid \mathbf{x}_{-d i}, \mathbf{z}_{-d i}, \mathbf{w}, \alpha, \beta, \mathbf{a}, \mathbf{r}\right) \\ &= \frac{P\left(x_{d i}, z_{d i}, w_{d i} \mid \mathbf{x}_{-d i}, \mathbf{z}_{-d i}, \mathbf{w}_{-d i}, \alpha, \beta, \mathbf{a}, \mathbf{r}\right)}{P\left(w_{d i} \mid \mathbf{x}_{-d i}, \mathbf{z}_{-d i}, \mathbf{w}_{-d i}, \alpha, \beta, \mathbf{a}, \mathbf{r}\right)} \propto \frac{P(\mathbf{x}, \mathbf{z}, \mathbf{w} \mid \alpha, \beta, \mathbf{a}, \mathbf{r})}{P\left(\mathbf{x}_{-d i}, \mathbf{z}_{-d i}, \mathbf{w}_{-d i} \mid \alpha, \beta, \mathbf{a}, \mathbf{r}\right)} \\ &\propto\frac{\frac{\Gamma\left(\alpha_{z_{d i}}+n_{a_{d} x_{d i} z_{d i}}\right)}{\Gamma\left(\alpha_{z_{d i}}+n_{\left.a_{d} x_{d i} z_{d i}-1\right)}\right.} \frac{\Gamma\left(\beta_{w_{d i}}+m_{z_{d i} w_{d i}}\right)}{\Gamma\left(\beta_{w_{d i}}+m_{\left.z_{d i} w_{d i}-1\right)}\right.} }{ \frac{\Gamma\left(\sum_{t=1}^{T}\left(\alpha_{t}+n_{a_{d} x_{d i} t}\right)\right)}{\Gamma\left(\sum_{t=1}^{T}\left(\alpha_{t}+n_{a_{d} x_{d i} t}\right)-1\right)} \frac{\Gamma\left(\sum_{v=1}^{V}\left(\beta_{v}+m_{z_{d i} v}\right)\right)}{\Gamma\left(\sum_{v=1}^{V}\left(\beta_{v}+m_{z_{d i} v}\right)-1\right)} }\propto\frac{\alpha_{z_{d i}}+n_{a_{d} x_{d i} z_{d i}}-1}{\sum_{t=1}^{T}\left(\alpha_{t}+n_{a_{d} x_{d i}}\right)-1} \frac{\beta_{w_{d i}}+m_{z_{d i} w_{d i}}-1}{\sum_{v=1}^{}\left(\beta_{v}+m_{z_{d i} v}\right)-1} \end{alignedat} P(xdi,zdi∣x−di,z−di,w,α,β,a,r)=P(wdi∣x−di,z−di,w−di,α,β,a,r)P(xdi,zdi,wdi∣x−di,z−di,w−di,α,β,a,r)∝P(x−di,z−di,w−di∣α,β,a,r)P(x,z,w∣α,β,a,r)∝Γ(∑t=1T(αt+nadxdit)−1)Γ(∑t=1T(αt+nadxdit))Γ(∑v=1V(βv+mzdiv)−1)Γ(∑v=1V(βv+mzdiv))Γ(αzdi+nadxdizdi−1)Γ(αzdi+nadxdizdi)Γ(βwdi+mzdiwdi−1)Γ(βwdi+mzdiwdi)∝∑t=1T(αt+nadxdi)−1αzdi+nadxdizdi−1∑v=1(βv+mzdiv)−1βwdi+mzdiwdi−1
具体是将其简化成:
p ( z i ∣ z ⃗ ¬ i , w ⃗ ¬ i ) = p ( z i , w i ∣ z ⃗ ¬ i , w ⃗ ¬ i ) / p ( w i ∣ z ⃗ ¬ i , w ⃗ ¬ i ) ∝ p ( z ⃗ , w ⃗ ) / p ( z ⃗ ¬ i , w ⃗ ¬ i ) p(z_{i} \mid \vec{z}_{\neg i}, \vec{w}_{\neg i})=p(z_{i}, w_{i} \mid \vec{z}_{\neg i}, \vec{w}_{\neg i}) / p(w_{i} \mid \vec{z}_{\neg i}, \vec{w}_{\neg i}) \propto p(\vec{z}, \vec{w}) / p(\vec{z}_{\neg i}, \vec{w}_{\neg i}) p(zi∣z¬i,w¬i)=p(zi,wi∣z¬i,w¬i)/p(wi∣z¬i,w¬i)∝p(z,w)/p(z¬i,w¬i)
但这个简化写法似乎有误，应为：
p ( z i ∣ z ⃗ ¬ i , w ⃗ ) = p ( z i , w i ∣ z ⃗ ¬ i , w ⃗ ¬ i ) / p ( w i ∣ z ⃗ ¬ i , w ⃗ ¬ i ) ∝ p ( z ⃗ , w ⃗ ) / p ( z ⃗ ¬ i , w ⃗ ¬ i ) p(z_{i} \mid \vec{z}_{\neg i}, \vec{w})=p(z_{i}, w_{i} \mid \vec{z}_{\neg i}, \vec{w}_{\neg i}) / p(w_{i} \mid \vec{z}_{\neg i}, \vec{w}_{\neg i}) \propto p(\vec{z}, \vec{w}) / p(\vec{z}_{\neg i}, \vec{w}_{\neg i}) p(zi∣z¬i,w)=p(zi,wi∣z¬i,w¬i)/p(wi∣z¬i,w¬i)∝p(z,w)/p(z¬i,w¬i)
以及，本人对相关公式的推演草稿（未做整理）暂存如下：
