# SMC for Mixture Distribution

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## Mixture Model

data $y_1,\ldots, y_c$ are i.i.d. with distribution

\[y_i\mid \theta_r\sim \sum\limits_{j=1}^rw_j\cal N(\mu_j,\lambda_j^{-1})\]where $\theta_r=(\mu_{1:r},\lambda_{1:r},w_{1:r}), 2\le 2<\infty$.

### prior distributions

same for each component $j=1,…,r$

\[\mu_j\sim \cal N(\xi, \kappa^{-1})\] \[\gamma_j\sim \Gamma(\alpha, \beta)\] \[w_{1:r-1}\sim \cal D(\rho)\]