1. Multidimensional Bayesian Mixture (MBM) Model
The generative process of a network is summarized as follows:
1. Draw from ;
2. For each group k in K groups;
Draw from ;
3. For each new node i:
(a) Draw a latent group from ;
(b) For each link in :
Draw an end node j from .
The joint distribution of the observed and latent variables can be written as
(1)
The conditional distributions of the is,
, (2)
where is the multinomial beta function, which can be expressed using the gamma function,
(3)
Commonly, . And the conditional distributions of the is,
(4)
Due to the conjugacy between the Dirichlet and Multinomial distributions, we can marginalize over nuisance parameters as following,
(5)
For the probability , is generated from a Dirichlet distribution,
(6)
Commonly, . And the conditional distributions of the is,
(7)
Due to the conjugacy between the Dirichlet and Multinomial distributions, we can marginalize over nuisance parameters as following,
(8)
Thus formula (1) can be simplify as,
(9)
2. Inference with Gibbs sampling for MBM model
In the gibbs sampling, the sampler iterates as follows: for each node i, given the group assignment for all except the node i, the group probability of the left-out node i is
(10)
For the ,
(11)
For the ,
(12)
Thus,
(13)
3. Multidimensional Bayesian Nonparametric mixture (MBNPM) model
The generative process of a network for the BNPM model can be converted from the Bayesian mixture model by sampling from a CRP, shown below.
1. Draw from ;
Draw a latent group from ;
2. For each group k in groups;
Draw from ;
3. For each new node i:
(a) Draw a latent group from ;
(b) For each link in :
Draw an end node j from .
In the Bayesian mixture model, we have obtained . And we can further convert it to:
(14)
In the bayesian nonparametric mixture model, , but only a finite number of them is used to generate the observed data. Let denotes the number of groups whose and denotes the number of groups whose . The formula (10) can be described as:
(15)
(16)
The can be converted to:
(17)
Thus the probability distribution of model,
(18)
Where K denotes the number of groups whose .
4. Inference with Gibbs sampling for MBNPM model
In the gibbs sampling, the sampler iterates as follows: for each node i, given the group assignment for all except the node i, the group probability of the left-out node i is
(19)
According to the CRP, we can get,
(20)
Thus,
, (21)
where , if ; , otherwise.