## 2 Proposed Method
### The Infinite Mixture Model
#### 2.3.1 Priors and Conditional Posterior Distributions  
In contrast to deterministic approximation, which is generally based on the EM algorithm--where the missing data noted by \(\mathcal{Z}=(\bar{Z}_{1},\ldots,\bar{Z}_{N})\) where \(\bar{Z}_{i}=(Z_{i1},\ldots,Z_{iK})\) for \(i=1,\ldots\)--the \(N\). Bayesian approach to mixture learning is more efficient. Indeed, it is able to overcome the drawbacks related to the deterministic methods (maximum likelihood-based technique) such as its convergence to local maxima and their tendency to overfit noisy and sparse data. In the following, we develop our fully nonparametric Bayesian approach for learning the infinite multivariate generalized Normal mixture model (inMGNMM). Thus, the likelihood of complete data (both observed and missing) is expressed as:  
\[p(\mathcal{Y},\mathcal{Z}|\Theta)=\prod_{n=1}^{N}\prod_{k=1}^{K}(w_{k}p(\bar{ Y}_{n}|\theta_{k}))^{Z_{nk}}. \tag{3}\]  
The role of the likelihood is to update the prior knowledge as regards the parameters of the model (\(p(\Theta)\)). Then, the priors on the model parameters will be updated and expressed in terms of posterior knowledge \(p(\Theta|\mathcal{Y},\mathcal{Z})\). By exploiting the Bayes' rule, we have the following relationship between priors, likelihood and posteriors [4]:  
\[p(\Theta|\mathcal{Z},\mathcal{Y})=\frac{p(\mathcal{Z},\mathcal{Y}|\Theta)p( \Theta)}{\int p(\mathcal{Z},\mathcal{Y}|\Theta)p(\Theta)}\propto p(\mathcal{ Z},\mathcal{Y}|\Theta)p(\Theta), \tag{4}\]  
where \((\mathcal{Z},\mathcal{Y})\), \(p(\Theta)\) and \(p(\Theta|\mathcal{Z},\mathcal{Y})\) are the complete data, the prior information about the model's parameters and the posterior, respectively. It is important to note that the choice of the priors is one of the crucial aspects of Bayesian modeling and directly influences the final results [31]. The provided observed data will certainly improve these priors. In what follows, selecting and specifying the best choice for priors is covered as well as the finding of the underlying equations for the posteriors. After that, we will simulate a value very close to the truth for \(\Theta\sim p(\Theta|\mathcal{Z},\mathcal{Y})\) by applying the efficient Gibbs sampler algorithm. In the current work, we choose priors for the parameters of the proposed model as follows:  
* We start by specifying a common prior for \(\vec{\mu}_{j}\) as a Normal distribution, thus we have \(\vec{\mu}_{j}\sim\mathcal{N}(\vec{\mu}_{0},\Sigma_{0})\). * For the shape parameter \(,\beta_{j}\), an appropriate prior is the Gamma distribution (\(\beta_{j}\sim\mathcal{G}(\alpha_{\beta},\beta_{\beta})\)). * For the covariance matrix \(\Sigma_{j}\), a natural prior choice is the Inverted Wishart (\(\Sigma_{j}\sim IW(\psi,\
u)\)). Having all these prior distributions, we move now to calculate all conditional posterior distributions based on the [PERSON] theorem. The conditional posterior distribution for the mean parameter is determined as:  
\[ \begin{split}& p(\vec{\mu}_{j}|\mathcal{Z},\mathcal{Y})\propto p(\vec{\mu}_{j}) \prod_{Z_{ij}=1}p(\vec{Y}_{i}|\vec{\mu}_{j},\beta_{j},\Sigma_{j})\\ &\propto\mathcal{N}(\vec{\mu}_{0},\Sigma_{0})\prod_{Z_{ij}=1}p( \vec{Y}_{i}|\vec{\mu}_{j},\beta_{j},\Sigma_{j})\\ &\propto\exp(-\frac{1}{2}(\vec{\mu}_{j}-\vec{\mu}_{0})^{T}\Sigma_ {0}^{-1}(\vec{\mu}_{j}-\vec{\mu}_{0}))\ \ \times\\ &\exp\Big{(}\sum_{Z_{ij}=1}(\frac{-1}{2m^{\beta_{j}}}((\vec{Y}_{i }-\vec{\mu}_{j})^{T}\Sigma_{j}^{-1}(\vec{Y}_{i}-\vec{\mu}_{j}))^{\beta_{j}}) \Big{)}.\end{split}