# Bayesian Analysis of Mixture Distribution

##### Posted on June 27, 2017 0 Comments

HAVE NOT FINISHED!!!

# Bayesian Models for Mixtures

## Basic Formulation

$y_i, i=1,2,…,n$ have i.i.d. distribution

the objective is to infer the number $k$ of components.

an alternative representation:

given the values of group label $z_i$, the observations are draw from their respective individual subpopulations:

where $\theta_r=(\mu_{1:r},\lambda_{1:r},w_{1:r}), 2\le 2<\infty$.

## Hierarchical Model and Priors

In a Bayesian framework, the unknown $k, w, \theta$ are regarded as drawn from appropriate prior distributions.

it is natural to **impose further conditional independences**, so

so we have the Bayesian hierarchical model

for full flexibility, add an extra layer to the hierarchy, and allow the priors for $k,w$ and $\theta$ to depend on hyperparameter $\lambda,\delta,\eta$ respectively.

## Normal Mixtures

$\theta_j=(\mu_j,\sigma_j^2),j=1,2,\ldots,k$

prior distribution are that the $\mu_j$ and $\sigma_j^{-2}$ are all drawn independently, with normal and gamma priors

so $\eta=(\xi,\kappa,\alpha,\beta)$

the prior on $w$ will always be taken as symetric Dirichlet,

a proper prior distribution for $k$ and a common choice is the Poisson distribution with hyperparameter $\lambda$

# Reversible Jump Markov Chain Monte Carlo Algorithm for Mixtures

TODO

## Weak Prior Information for Component Parameters

it seems restrictive to suppose that knowledge of the range of data implies much about the size of the $\sigma_j^2$, so introduce an additional hierarchical level by allowing $\beta$ to follow a gamma distribution with parameter $g$ and $h$. Generally take $\alpha >1>g$ to express the belief that $\sigma_j^2$ are similar, the scale parameter $h$ will be a small multiple of $1/R^2$.

## prior distributions

same for each component $j=1,…,r$