Gaussian Mixture Covariance

Last updated: 2018-09-24

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We have a Gaussian mixture model \[ X \sim \sum_{k=1}^{K} \pi_{k} N_{p}(\mu_{k}, \Sigma_{k}) \]

We can write \[ X = Y + E \\ Y = \mu_{k} \quad \text{with probability} \ \pi_{k} \\ E|Y \sim N(0, \Sigma_{k}) \] The mean of X is \[ \mathbb{E}(X) = \mathbb{E}(\mathbb{E}(X|Y)) = \mathbb{E}(Y) = \sum_{k=1}^{K} \pi_{k} \mu_{k} = \tilde{\mu} \]

The covariance of X is \[ \begin{align*} \mathbb{C}ov(X) &= \mathbb{E}(\mathbb{V}ar(X∣Y))+\mathbb{V}ar(\mathbb{E}(X∣Y)) \\ &= \mathbb{E}(\Sigma_{k} \ \text{with probability} \ \pi_{k})+\mathbb{V}ar(Y) \\ &= \sum_{k=1}^{K} \pi_{k}\Sigma_{k} + \sum_{k=1}^{K}\pi_{k}(\mu_{k} - \tilde{\mu})(\mu_{k} - \tilde{\mu})^{T} \\ &= \sum_{k=1}^{K} \pi_{k}\left(\Sigma_{k} + \mu_{k}\mu_{k}^{T}\right) - \tilde{\mu}\tilde{\mu}^{T} \end{align*} \]

sessionInfo()

R version 3.5.1 (2018-07-02)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS High Sierra 10.13.6

Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/3.5/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.5/Resources/lib/libRlapack.dylib

locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

loaded via a namespace (and not attached):
 [1] workflowr_1.1.1   Rcpp_0.12.18      digest_0.6.15    
 [4] rprojroot_1.3-2   R.methodsS3_1.7.1 backports_1.1.2  
 [7] git2r_0.23.0      magrittr_1.5      evaluate_0.11    
[10] stringi_1.2.4     whisker_0.3-2     R.oo_1.22.0      
[13] R.utils_2.6.0     rmarkdown_1.10    tools_3.5.1      
[16] stringr_1.3.1     yaml_2.2.0        compiler_3.5.1   
[19] htmltools_0.3.6   knitr_1.20       

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