## 3 Non-Gaussian Reversible Jump Markov Chain Monte Carlo (NG-RJMCMC) Algorithm for Mixture Model
### Ng-RJMCMC Algorithm  
In the mixture model analysis, one method is known, namely the reversible jump Markov chain Monte Carlo (RJMCMC). The RJMCMC method can be used to estimate an unknown quantity, such as the number of mixture components, the weights of mixture components, and the distribution parameters of mixture components. For its usefulness in determining the number of mixture components in the indicated multimodal data, RJMCMC has been extensively used. For the Gaussian distribution, this was initially carried out by [PERSON] and [PERSON] [16]. Over time, RJMCMC can be used for distributions other than Gaussian. RJMCMC for general beta distribution was carried out in two studies by [PERSON] and [PERSON] [29; 58]. In their research, the general beta distribution consists of four parameters, namely the lower limit, the upper limit, and two shape parameters. Then, to use the RJMCMC algorithm, they obtained the location-scale parameterization for this distribution. Another study followed the location-scale parameterization for the RJMCMC method, namely research on the symmetry gamma distribution [59]. Not only in the RJMCMC method but also in other methods, the location-scale parameterization is carried out in the mixture analysis. First, research on the exponential and Gaussian distribution using the Dirichlet process mixture was carried out by [PERSON] et al. [60]. Secondly, research on the asymmetric Laplace error distribution using the likelihood-based approach was carried out by [PERSON] and [PERSON] [61]. Finally, research on the exponential distribution using the Gibbs sampler was carried out by [PERSON] et al. [62]. Based on the studies mentioned above, in the mixture analysis using any method, it will be easier if the distribution used in the research follows the location-scale (family) parameterization. Thus, we given Algorithm 1 as a modification of the RJMCMC algorithm. ```
1. Modify the distribution to be analyzed into a member of the location-scale family, determine: (a) form of transformation, and (b) location-scale parameter
2. Determine the appropriate priors for: (a) component weights (\(w\)), (b) location-scale parameter (\(\mu,\sigma\)), and (c) latent allocation variable (\(z\))
3. Do all six types of RJMCMC moves by [PERSON] and [PERSON] [16]. A sweep is defined as a complete run on these six moves [16]: (a) updating the component weights (\(w\)), (b) updating the location-scale parameter (\(\mu,\sigma\)), (c) updating the latent allocation variable (\(z\)), (d) updating the hyperparameters (\(\gamma\), \(\delta\), and \(\bar{\zeta}\)). Moves (a), (b), (c), and (d) can therefore be performed in parallel (e) do the splitting or combining mixture component (f) do the birth or death of an empty component. ```  
**Algorithm 1** NG-RJMCMC Algorithm  
Letting \(\Delta\) denote the state variable (in this study, \(\Delta\) be the complete set of unknowns (\(\mu\), \(\sigma\), \(k\), \(w\), \(z\))), and \(p(\Delta)\) be the target probability measure (the posterior distribution), we consider a countable family of move types, indexed by \(m=1,2,\cdots\). When the current state is \(\Delta\), a move type \(m\) and destination \(\Delta^{*}\) are proposed, with joint distribution given by \(q_{m}(\Delta,\Delta^{*})\). The move is accepted with probability  
\[\alpha_{m}(\Delta,\Delta^{*})=\min\biggl{\{}1,\frac{p(\Delta^{*})q_{m}(\Delta ^{*},\Delta)}{p(\Delta)q_{m}(\Delta,\Delta^{*})}\biggr{\}}. \tag{8}\]  
If \(\Delta^{*}\) has a higher dimensional space than \(\Delta\), it is possible to create a vector of continuous random variables \(\mathbf{u}\), independent of \(\Delta\)[16]. Then, the new state \(\Delta^{*}\) is set by an invertible deterministic function of \(\Delta\) and \(\mathbf{u}\): \(f(\Delta,\mathbf{u})\).