User:Timothee Flutre/Notebook/Postdoc/2012/08/16
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  ==  +  ==Variational Bayes approach for the mixture of Normals== 
  *  +  
+  * '''Motivation''': I have described on [http://openwetware.org/wiki/User:Timothee_Flutre/Notebook/Postdoc/2011/12/14 another page] the basics of mixture models and the EM algorithm in a frequentist context. It is worth reading before continuing. Here I am interested in the Bayesian approach as well as in a specific variational method (nicknamed "Variational Bayes").  
+  
+  
+  * '''Data''': we have N univariate observations, <math>y_1, \ldots, y_N</math>, gathered into the vector <math>\mathbf{y}</math>.  
+  
+  
+  * '''Assumptions''': we assume the observations to be exchangeable and distributed according to a mixture of K Normal distributions. The parameters of this model are the mixture weights (<math>w_k</math>), the means (<math>\mu_k</math>) and the precisions (<math>\tau_k</math>) of each mixture components, all gathered into <math>\Theta = \{w_1,\ldots,w_K,\mu_1,\ldots,\mu_K,\tau_1,\ldots,\tau_K\}</math>. There are two constraints: <math>\sum_{k=1}^K w_k = 1</math> and <math>\forall k \; w_k > 0</math>.  
+  
+  
+  * '''Observed likelihood''': <math>p(\mathbf{y}  \Theta, K) = \prod_{n=1}^N p(y_n\Theta,K) = \prod_{n=1}^N \sum_{k=1}^K w_k Normal(y_n;\mu_k,\tau_k)</math>  
+  
+  
+  * '''Latent variables''': let's introduce N latent variables, <math>z_1,\ldots,z_N</math>, gathered into the vector <math>\mathbf{z}</math>. Each <math>z_n</math> is a vector of length K with a single 1 indicating the component to which the <math>n^{th}</math> observation belongs, and K1 zeroes.  
+  
+  
+  * '''Augmented likelihood''': <math>p(\mathbf{y},\mathbf{z}\Theta,K) = \prod_{n=1}^N p(y_n,z_n\Theta,K) = \prod_{n=1}^N p(z_n\Theta,K) p(y_nz_n,\Theta,K) = \prod_{n=1}^N \prod_{k=1}^K w_k^{z_{nk}} Normal(y_n;\mu_k,\tau_k)^{z_{nk}}</math>  
+  
+  
+  * '''Priors''': we choose conjuguate ones  
+  ** for the parameters: <math>\forall k \; \mu_k \sim Normal(\mu_0,\tau_0)</math> and <math>\forall k \; \tau_k \sim Gamma(\alpha,\beta)</math>  
+  ** for the latent variables: <math>\forall n \; z_n \sim Multinomial_K(1,\mathbf{w})</math> and <math>\mathbf{w} \sim Dirichlet(\gamma)</math>  
Revision as of 13:49, 16 August 2012
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Variational Bayes approach for the mixture of Normals
