## 2 Proposed Method
### The Infinite Mixture Model
#### 2.3.1 Priors and Conditional Posterior Distributions
 \tag{5}\]The conditional posterior distribution for the shape parameter is determined as: \[p(\beta_{j}|\mathcal{Z},\mathcal{Y})\propto p(\beta_{j})\prod_{Z_{ ij}=1}p(\vec{\gamma}_{i}|\vec{\mu}_{j},\beta_{j},\Sigma_{j})\] (6) \[\propto\mathcal{G}(\alpha_{\beta},\beta_{\beta})\prod_{Z_{ij}=1}p( \vec{\gamma}_{i}|\vec{\mu}_{j},\beta_{j},\Sigma_{j})\] \[\propto\beta_{j}^{\alpha_{\beta}-1}e^{-\beta_{\beta}\beta_{j}} \Big{(}\frac{\beta_{j}}{\Gamma(d/2\beta_{j})2^{d/2\beta_{j}}}\Big{)}^{n_{j}} \times\] \[\exp\Big{(}\sum_{Z_{ij}=1}(\frac{-1}{2m^{\beta_{j}}}((\vec{\gamma }_{i}-\vec{\mu}_{j})^{T}\Sigma_{j}^{-1}(\vec{\gamma}_{i}-\vec{\mu}_{j}))^{ \beta_{j}})\Big{)}.\]  
The conditional posterior distribution of covariance matrix is calculated as:  
\[p(\Sigma_{j}|\mathcal{Z},\mathcal{Y})\propto p(\Sigma_{j})\prod_ {Z_{ij}=1}p(\vec{\gamma}_{i}|\vec{\mu}_{j},\beta_{j},\Sigma_{j}) \tag{7}\] \[\propto IW(\psi,\
u)\prod_{Z_{ij}=1}p(\vec{\gamma}_{i}|\vec{\mu}_ {j},\beta_{j},\Sigma_{j})\] \[\propto|\Sigma_{j}|^{-(v+d+1)/2}e^{-1/2 tr(\psi\vec{\Sigma}_{j}^{- 1})}\times\] \[\exp\Big{(}\sum_{Z_{ij}=1}(\frac{-1}{2m^{\beta_{j}}}((\vec{\gamma }_{i}-\vec{\mu}_{j})^{T}\Sigma_{j}^{-1}(\vec{\gamma}_{i}-\vec{\mu}_{j}))^{\beta _{j}})\Big{)}.\]  
As for the conditional posterior distribution of the mixing weight \(\vec{w}\), we know that \(\vec{w}\) is determined on the simplex \(\{(w_{1},\ldots,w_{K}):\sum_{j=1}^{K-1}w_{k}<1\}\), then, in order to account for the infinite mixture principle, a natural prior that we have to state is the Dirichlet distribution with parameters \(\frac{\delta}{M}\)[20]. \[p(\vec{w}|\delta)=\frac{\Gamma(\delta)}{\prod_{k=1}^{K}\Gamma(\frac{\delta}{K })}\prod_{k=1}^{K}w_{k}^{\frac{\delta}{K}-1}. \tag{8}\]  
The inference of \(\vec{w}\) is performed via the inference of the membership vectors \(\vec{Z}_{n}\). Thus, we can deduce \(p(\mathcal{Z}|\vec{w})\) as follows:  
\[p(\mathcal{Z}|\vec{w}) = \prod_{n=1}^{N}p(\vec{Z}_{n}|\vec{w})=\prod_{n=1}^{N}w_{1}^{Z_{1 }}\ldots w_{M}^{Z_{nM}}\] \[\propto\mathcal{G}(\alpha_{\beta},\beta_{\beta})\prod_{Z_{ij}=1}p( \vec{\gamma}_{i}|\vec{\mu}_{j},\beta_{j},\Sigma_{j})\]0  
where \(a_{k}\) represents the total elements in cluster \(k\). When considering the Dirichlet integral, we will obtain \(p(\mathcal{Z}|\eta)=\prod_{n=1}^{N}\frac{(\Sigma_{<n}\mathbb{I}_{Z_{n}=Z_{j}}) ^{\frac{\delta}{K}}}{((n-1)+\delta)}\). For \(\vec{\gamma}_{l},l=1,\ldots,n-1\), the conditional prior that \(\vec{X}_{n}\) to be in component \(k\) is given as [32]:  
\[\propto\mathcal{G}(\alpha_{\beta},\beta_{\beta})\prod_{Z_{ij}=1}p( \vec{\gamma}_{i}|\vec{\mu}_{j},\beta_{j},\Sigma_{j})\]1  
In order to \(N\) vectors, Equation (10) is updated as follows [32]:  
\[\propto\mathcal{G}(\alpha_{\beta},\beta_{\beta})\prod_{Z_{ij}=1}p( \vec{\gamma}_{i}|\vec{\mu}_{j},\beta_{j},\Sigma_{j})\]2  
where \(\mathcal{Z}_{-n}=\{\vec{Z}_{1},\ldots,\vec{Z}_{n-1},\vec{Z}_{n+1},\ldots,\vec{ Z}_{N}\}\), \(a_{-nk}\) is the number of vectors, excluding \(\vec{\gamma}_{i}\), in cluster \(j\). It is then necessary to calculate the conditional posterior by multiplying the prior (Equation (11)) by the likelihood \(\vec{\gamma}_{j}\). It is worth noting that, during the initialization step, the algorithm starts by supposing that all the vectors are in the same cluster. Moreover, the initial values for the model's parameters are supposedly produced from their prior distribution. To sample the vectors \(Z_{i}\), it is required to compute the integral in Equation (13); however, this quantity is not analytically tractable. To deal with this issue, we propose to apply an efficient algorithm named \"Metropolis-Hastings algorithm (M-H)\". The latter has the advantage of making the right decision to accept or reject the new samples--at iteration (t)--using an acceptance rate defined as follows:  
\[\propto\mathcal{G}(\alpha_{\beta},\beta_{\beta})\prod_{Z_{ij}=1}p( \vec{\gamma}_{i}|\vec{\mu}_{j},\beta_{j},\Sigma_{j})\]3  
where \(q\) is the target distribution.