bmixnorm {bmixture} | R Documentation |

## Sampling algorithm for mixture of Normal distributions

### Description

This function consists of several sampling algorithms for Bayesian estimation for finite a mixture of Normal distributions.

### Usage

```
bmixnorm( data, k = "unknown", iter = 1000, burnin = iter / 2, lambda = 1,
k.start = NULL, mu.start = NULL, sig.start = NULL, pi.start = NULL,
k.max = 30, trace = TRUE )
```

### Arguments

`data` |
vector of data with size |

`k` |
number of components of mixture distribution. It can take an integer values. |

`iter` |
number of iteration for the sampling algorithm. |

`burnin` |
number of burn-in iteration for the sampling algorithm. |

`lambda` |
For the case |

`k.start` |
For the case |

`mu.start` |
Initial value for parameter of mixture distribution. |

`sig.start` |
Initial value for parameter of mixture distribution. |

`pi.start` |
Initial value for parameter of mixture distribution. |

`k.max` |
For the case |

`trace` |
Logical: if TRUE (default), tracing information is printed. |

### Details

Sampling from finite mixture of Normal distribution, with density:

`Pr(x|k, \underline{\pi}, \underline{\mu}, \underline{\sigma}) = \sum_{i=1}^{k} \pi_{i} N(x|\mu_{i}, \sigma^2_{i}),`

where `k`

is the number of components of mixture distribution (as a defult we assume is `unknown`

).
The prior distributions are defined as below

` P(K=k) \propto \frac{\lambda^k}{k!}, \ \ \ k=1,...,k_{max},`

` \pi_{i} | k \sim Dirichlet( 1,..., 1 ), `

` \mu_{i} | k \sim N( \epsilon, \kappa ), `

` \sigma_i | k \sim IG( g, h ), `

where `IG`

denotes an inverted gamma distribution. For more details see for more details see Stephens, M. (2000), doi: 10.1214/aos/1016120364.

### Value

An object with `S3`

class `"bmixnorm"`

is returned:

`all_k` |
a vector which includes the waiting times for all iterations. It is needed for monitoring the convergence of the BD-MCMC algorithm. |

`all_weights` |
a vector which includes the waiting times for all iterations. It is needed for monitoring the convergence of the BD-MCMC algorithm. |

`pi_sample` |
a vector which includes the MCMC samples after burn-in from parameter |

`mu_sample` |
a vector which includes the MCMC samples after burn-in from parameter |

`sig_sample` |
a vector which includes the MCMC samples after burn-in from parameter |

`data` |
original data. |

### Author(s)

Reza Mohammadi a.mohammadi@uva.nl

### References

Stephens, M. (2000) Bayesian analysis of mixture models with an unknown number of components-an alternative to reversible jump methods. *Annals of statistics*, 28(1):40-74, doi: 10.1214/aos/1016120364

Richardson, S. and Green, P. J. (1997) On Bayesian analysis of mixtures with an unknown number of components. *Journal of the Royal Statistical Society: series B*, 59(4):731-792, doi: 10.1111/1467-9868.00095

Green, P. J. (1995) Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. *Biometrika*, 82(4):711-732, doi: 10.1093/biomet/82.4.711

Cappe, O., Christian P. R., and Tobias, R. (2003) Reversible jump, birth and death and more general continuous time Markov chain Monte Carlo samplers. *Journal of the Royal Statistical Society: Series B*, 65(3):679-700

Mohammadi, A., Salehi-Rad, M. R., and Wit, E. C. (2013) Using mixture of Gamma distributions for Bayesian analysis in an M/G/1 queue with optional second service. *Computational Statistics*, 28(2):683-700, doi: 10.1007/s00180-012-0323-3

Mohammadi, A., and Salehi-Rad, M. R. (2012) Bayesian inference and prediction in an M/G/1 with optional second service. *Communications in Statistics-Simulation and Computation*, 41(3):419-435, doi: 10.1080/03610918.2011.588358

Wade, S. and Ghahramani, Z. (2018) Bayesian Cluster Analysis: Point Estimation and Credible Balls (with Discussion). *Bayesian Analysis*, 13(2):559-626, doi: 10.1214/17-BA1073

### See Also

### Examples

```
## Not run:
data( galaxy )
set.seed( 70 )
# Runing bdmcmc algorithm for the galaxy dataset
mcmc_sample = bmixnorm( data = galaxy )
summary( mcmc_sample )
plot( mcmc_sample )
print( mcmc_sample)
# simulating data from mixture of Normal with 3 components
n = 500
weight = c( 0.3, 0.5, 0.2 )
mean = c( 0 , 10 , 3 )
sd = c( 1 , 1 , 1 )
data = rmixnorm( n = n, weight = weight, mean = mean, sd = sd )
# plot for simulation data
hist( data, prob = TRUE, nclass = 30, col = "gray" )
x = seq( -20, 20, 0.05 )
densmixnorm = dmixnorm( x, weight, mean, sd )
lines( x, densmixnorm, lwd = 2 )
# Runing bdmcmc algorithm for the above simulation data set
bmixnorm.obj = bmixnorm( data, k = 3, iter = 1000 )
summary( bmixnorm.obj )
## End(Not run)
```

*bmixture*version 1.7 Index]