Midterm Report Nitin Yandong Ming Yanbo

Team members

Nitin Agarwal

Yandong Liu

Yanbo Xu

Ming Sun

Gibbs Sampling for Collaboration Influence Model

We want $P(Z,X,R|W)$, the posterior distribution of topic Z, (author, collaborator) pair X and which favor of collaboration over influence R given the words W in the corpus:

$P(Z,X,R|W)={\frac {P(Z,X,R,W)}{\sum _{Z,X,R}P(Z,X,R,W)}}$

We begin by calculating $P(W|Z,X,R)$ and $P(Z,X,R)$:

$P(W|Z,X,R)=P(W|Z)=\prod _{z=1}^{T}({\frac {\Gamma (\sum _{v=1}^{V}\beta _{v})}{\prod _{v=1}^{V}\Gamma (\beta _{v})}}({\frac {\prod _{v=1}^{V}\Gamma (n_{z}^{w_{v}}+\beta _{v})}{\Gamma (\sum _{v=1}^{V}\beta _{v}+\sum _{v=1}^{V}n_{z}^{w_{v}})}}))$

$P(Z,X,R)=(\prod _{i_{w}=1}^{W}{\frac {1}{n_{r_{i_{w}}}(a_{i_{w}})+\eta _{r_{i_{w}}}}})\prod _{p=1}^{P}({\frac {\Gamma (\sum _{z}\alpha _{z})}{\prod _{z=1}^{T}\Gamma (\alpha _{z})}}{\frac {\prod _{z}\Gamma (n_{p}^{z}+\alpha _{z})}{\Gamma (\sum _{z}\alpha _{z}+\sum _{z}n_{p}^{z})}})$,

where P is the number of all the different author-collaborator-favor of collaboration combination (a,a',r).

So the Gibbs sampling of $P(z_{i},x_{i},r_{i},w_{i}|Z_{-i},X_{-i},R_{-i},W_{-i})$ :

$P(z_{i},x_{i},r_{i},w_{i}|Z_{-i},X_{-i},R_{-i},W_{-i})$

$={\frac {P(Z,X,R,W)}{P(Z_{-i},X_{-i},R_{-i},W_{-i})}}$

$={\frac {1}{n_{r_{i}}+\eta _{r_{i}}}}{\frac {n_{p,-i}^{t}+\alpha _{t}}{\sum _{z}n_{p,-i}^{z}+\sum _{z}\alpha _{z}}}{\frac {n_{t,-i}^{w_{v}}+\beta _{v}}{\sum _{v}n_{t,-i}+\sum _{v}\beta _{v}}}$

Further manipulation can turn the above equation into update equations for the topic and author-collaboration of each corpus token:

$P(z_{i}|Z_{-i},X,W,R)\propto {\frac {n_{z_{i}}^{w_{v}}+\beta _{v}}{\sum _{v}n_{z_{i}}^{w_{v}}+\beta _{v}}}{\frac {n_{x_{i}}^{z_{i}}+\alpha _{z_{i}}}{\sum _{z'}n_{x_{i}}^{z'}+\alpha _{z'}}}{\frac {n_{r_{i}}+\eta _{r_{i}}}{\sum _{r_{i}}(n_{r_{i}}+\eta _{r_{i}})}}$

$P(x_{i},r_{i}|Z,X_{-i},W,R_{-i})\propto {\frac {n_{x_{i},r_{i}}^{z_{i}}+\alpha _{z_{i}}}{\sum _{z'}n_{x_{i},r_{i}}^{z'}+\alpha _{z'}}}{\frac {n_{r_{i}}+\eta _{r_{i}}}{\sum _{r_{i}}(n_{r_{i}}+\eta _{r_{i}})}}$