Last updated: 2018-10-30

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In SUSIE, we need individual level data. However, individual level data is hard to obtain in GWAS. To implement SUSIE using summary statistics, we only need a simple trick.

Estimates

\[ \beta_{lj}|\gamma_{lj}=1 \sim_{q_{l}} N(\mu_{lj}, s_{lj}^{2}) \quad q(\gamma_{l}) = \alpha_{lj} \]

\[ s_{lj}^{2} = \left(\tau_{l} + \tau (X^{T}X)_{jj}\right)^{-1} \]

\[ \mu_{lj} = \frac{\tau(X^{T}\mathbf{y}_{res,-l})_{j}}{\tau_{l} + \tau (X^{T}X)_{jj}} = s_{lj}^{2}\tau (X^{T}\mathbf{y}_{res,-l})_{j} = s_{lj}^{2}\tau [(X^{T}\mathbf{y})_{j} - \sum_{k\neq j}(X^{T}X)_{kj}\alpha_{lk}\mu_{lk}] \]

\[ \alpha_{lj} \propto \pi_{j} N(\frac{(X^{T}\mathbf{y}_{res,-l})_{j}}{(X^{T}X)_{jj}}; 0, \frac{1}{\tau(X^{T}X)_{jj}} + \frac{1}{\tau_{l}} ) \]

\[ \frac{\alpha_{lj}}{1-\alpha_{lj}} = \frac{\pi_{j}}{1-\pi_{j}} \sqrt{\tau_{l}s_{lj}^2} \exp(\frac{\mu_{lj}^{2}}{2s_{lj}^{2}}) = \frac{\pi_{j}}{1-\pi_{j}} BF \]

Summary STAT

We have \(X^{T}X\), so the only problematic part is \(X^{T}\mathbf{y}_{res,-j}\).

In susie code, \(\mathbf{y}_{res,-l}\) is the R (update_each_effect).

\[ \mathbf{y}_{res,-l} = R = Y - X\mathbf{r}_{l} \] where \(\mathbf{r}_{l} = \alpha_{l\cdot}\mu_{l\cdot}\).

We don’t have X and Y here. But we actually need \(X^{T}\mathbf{y}_{res,-l}\). \[ X^{T}\mathbf{y}_{res,-l} = X^{T}R = X^{T}Y - X^{T}X\mathbf{r}_{l} \] which can be easily computed using \(X^{T}Y\) and \(X^{T}X\).

ELBO

\(\mathbf{r} = \sum_{l=1}^{L} \mathbf{r}_{l}\)

\[ \begin{align*} L(q(\boldsymbol{\beta})) &= -\frac{n}{2}\log(2\pi) + \frac{n}{2}\log \tau - \frac{\tau}{2}\left[\|\mathbf{y} - X\mathbf{r}\|^{2} - \sum_{l=1}^{L}\|X\mathbf{r}_{l}\|^{2} + \sum_{l=1}^{L}\sum_{j=1}^{p}(X^{T}X)_{jj}\mathbb{E}(\beta_{lj}^2)\right] + \sum_{l=1}^{L} \left[ \log p(\mathbf{y}_{resid,-l}) + \frac{n}{2}\log(2\pi) - \frac{n}{2}\log \tau + \frac{\tau}{2}\left(\mathbf{y}_{resid,-l}^{T}\mathbf{y}_{resid,-l} - 2\mathbf{y}_{resid,-l}^{T}X\mathbb{E}_{q}\left[\boldsymbol{\beta}_{l}\right] + (X^{2})^{T}\mathbb{E}_{q}\left[\boldsymbol{\beta}_{l}\boldsymbol{\beta}_{l}^{T}\right] \right) \right] \\ &= -\frac{n}{2}\log(2\pi) + \frac{n}{2}\log \tau - \frac{\tau}{2}\left[\|\mathbf{y} - X\mathbf{r}\|^{2} - \sum_{l=1}^{L}\|X\mathbf{r}_{l}\|^{2} + \sum_{l=1}^{L}\sum_{j=1}^{p}(X^{T}X)_{jj}\mathbb{E}(\beta_{lj}^2)\right] + \sum_{l=1}^{L} \left[ \log \left[\sum_{j=1}^{p} \pi_{j} BF(\mathbf{y}_{resid,-l}, \mathbf{x}_{j})\right] + \log N(\mathbf{y}_{resid,-l}; 0, \frac{1}{\tau}I) + \frac{n}{2}\log(2\pi) - \frac{n}{2}\log \tau + \frac{\tau}{2}\left(\mathbf{y}_{resid,-l}^{T}\mathbf{y}_{resid,-l} - 2\mathbf{y}_{resid,-l}^{T}X\mathbb{E}_{q}\left[\boldsymbol{\beta}_{l}\right] + (X^{2})^{T}\mathbb{E}_{q}\left[\boldsymbol{\beta}_{l}\boldsymbol{\beta}_{l}^{T}\right] \right) \right] \\ &= -\frac{n}{2}\log(2\pi) + \frac{n}{2}\log \tau - \frac{\tau}{2}\left[\|\mathbf{y} - X\mathbf{r}\|^{2} - \sum_{l=1}^{L}\|X\mathbf{r}_{l}\|^{2} + \sum_{l=1}^{L}\sum_{j=1}^{p}(X^{T}X)_{jj}\mathbb{E}(\beta_{lj}^2)\right] + \sum_{l=1}^{L} \left[ \log \left[\sum_{j=1}^{p} \pi_{j} BF(\mathbf{y}_{resid,-l}, \mathbf{x}_{j})\right] - \tau \mathbf{y}_{resid,-l}^{T}X\mathbb{E}_{q}\left[\boldsymbol{\beta}_{l}\right] + \frac{\tau}{2}(X^{2})^{T}\mathbb{E}_{q}\left[\boldsymbol{\beta}_{l}\boldsymbol{\beta}_{l}^{T}\right] \right] \\ \end{align*} \]

\[ \|\mathbf{y} - X\mathbf{r}\|^2 = \mathbf{y}^{T}\mathbf{y} + \mathbf{r}^{T}X^{T}X\mathbf{r} - 2\mathbf{y}^{T}X\mathbf{r} = (n-1)Var(y) + \mathbf{r}^{T}X^{T}X\mathbf{r} - 2\mathbf{y}^{T}X\mathbf{r} \]

\[ \|X\mathbf{r}_{l}\|^{2} = \mathbf{r}_{l}^{T}X^{T}X\mathbf{r}_{l} \]

Session information

sessionInfo()
R version 3.5.1 (2018-07-02)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS  10.14

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_0.12.19      digest_0.6.18    
 [4] rprojroot_1.3-2   R.methodsS3_1.7.1 backports_1.1.2  
 [7] magrittr_1.5      git2r_0.23.0      evaluate_0.12    
[10] stringi_1.2.4     whisker_0.3-2     R.oo_1.22.0      
[13] R.utils_2.7.0     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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