## 2 Methodology
### Posterior Distribution Based on Gibbs Sampling Method  
Posterior distribution, which is augmented with the component indicator Z, is estimated by likelihood and prior distribution. The posterior distribution is in the same probability distribution family as the prior probability distribution since this paper used conjugate form of prior distribution; the details of derivations are expressed as follows:  
\[\text{p}(w|y)\propto\text{L}(y|w)\cdot\pi(w)\propto\frac{n!}{\prod_{i}n!} \prod_{i=1}^{K}w_{i}^{n_{i}}\cdot\frac{1}{B(e)}\prod_{i=1}^{K}w_{i}^{n_{i}-1} =\prod_{i=1}^{K}w_{i}^{n_{i}+n_{i}-1} \tag{9}\]  
\[B(e)=\frac{\prod_{i=1}^{K}\Gamma(e_{i})}{\Gamma\left(\sum_{i=1}^{K}e_{i} \right)},e=(e_{1},\cdots,e_{K})\]  
where \(\Gamma\) is the gamma function as a normalizing constant, \(e_{i}\) is the hyper-parameter of prior distribution, and \(n_{i}\) is the number of observations allocated to component k.  
\[\text{p}(\tau,\ \mu|y)\propto\text{L}(y|\tau,\mu)\cdot\pi(\mu|\tau)\cdot\pi(\tau) \tag{10}\]\[\propto\tau^{n/2}\exp\left(-\frac{\tau}{2}\sum(y_{i}-\mu)^{2}\right)\cdot\tau^{1/ 2}\exp\left(-\frac{n_{0}\tau}{2}(\mu-\mu_{0})^{2}\right)\cdot\tau^{\alpha-1}e^{- \beta\tau}\]  
which leads to a gamma posterior for \(\tau\):  
\[\mathrm{P}(\tau|y)\propto\tau^{\alpha+\frac{n}{2}-1}\exp\left(-\tau\left( \beta+\frac{1}{2}\sum(y_{i}-\ \overline{y})^{2}+\frac{nn_{0}}{2(n+n_{0})}(\ \overline{y}-\mu_{0})^{2}\right)\right)\]  
\[\tau|y\ \sim\ \mathrm{Gamma}\left(\alpha+\frac{n}{2},\ \beta+\frac{1}{2}\sum(y_{i}- \ \overline{y})^{2}+\frac{nn_{0}}{2(n+n_{0})}(\ \overline{y}-\mu_{0})^{2}\right)\]  
where \(\overline{y}\) is the mean of the observations in component k, \(\mu_{0},n_{0}\) are parameters of the prior distribution for the mean, and \(\alpha,\ \beta\) are parameters of the prior distribution for precision. \[\mathrm{p}(\mu|y)\ \propto\ \mathrm{L}(y|\mu)\cdot\pi(\mu) \tag{11}\]  
\[\propto\ \exp(-\frac{\tau}{2}\sum_{i=1}^{n}(y_{i}-\mu)^{2})\ \exp(-\frac{n_{0}\tau}{2}(\mu-\mu_{0})^{2})\]  
\[\propto\ \exp(-\frac{n\tau}{2}(\mu^{2}-2\ \overline{y}\ \mu)-\frac{n_{0}\tau}{2}(\mu^{2}-2 \mu_{0}\mu)^{2})\]  
\[\propto\ \exp(-\frac{1}{2}(n\tau+n_{0}\tau)(\mu^{2}-2\frac{(n\tau\ \overline{y}+n_{0}\tau\mu_{0})}{(n\tau+n_{0}\tau)}\mu))\]  
\[B(e)=\frac{\prod_{i=1}^{K}\Gamma(e_{i})}{\Gamma\left(\sum_{i=1}^{K}e_{i} \right)},e=(e_{1},\cdots,e_{K})\]0  
\[B(e)=\frac{\prod_{i=1}^{K}\Gamma(e_{i})}{\Gamma\left(\sum_{i=1}^{K}e_{i} \right)},e=(e_{1},\cdots,e_{K})\]1  
where \(\ \overline{y}\) is the mean of the observations in component k, n is the number of observations allocated to component k, and \(\mu_{0},n_{0}\) are parameters of the prior distribution for mean. The posterior distributions of the proposed Bayesian mixture model are expressed as in \[B(e)=\frac{\prod_{i=1}^{K}\Gamma(e_{i})}{\Gamma\left(\sum_{i=1}^{K}e_{i} \right)},e=(e_{1},\cdots,e_{K})\]2  
Although the proposed mixture model is based on a standard distribution, the inference of the posterior distribution of the model parameters is analytically intractable ([PERSON] et al., 2000; [PERSON], et al., 2003). Fortunately, due to new computational techniques, it has recently become possible to estimate mixture models using the Markov Chain Monte Carlo (MCMC) algorithms. The Gibbs sampling method is a special MCMC algorithm that is widely used to generate sample draw sequences from an unknown joint posterior distribution. During each iteration of the algorithm, samples of each parameter are alternately drawn from the conditional posterior distribution, given the other parameters drawn recently. If the sequence is long enough, it can be used to estimate the joint distribution. This paper used the Gibbs sampling method including the burn-in period to estimate the posterior distributions of model parameters.