Last updated: 2018-09-05

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\[ \begin{align*} \mathbf{y} &= \sum_{l=1}^{L} X \boldsymbol{\beta}_{l} + \boldsymbol{\epsilon} \quad \boldsymbol{\epsilon}\sim N_{n}(0, \frac{1}{\tau}I) \\ \boldsymbol{\beta}_{l} &= \boldsymbol{\gamma}_{l} \beta_{l} \\ \boldsymbol{\gamma}_{l} &\sim Multinomial(1,\boldsymbol{\pi}) \\ \beta_{l} &\sim N(0, \frac{1}{\tau_{l}}) \end{align*} \]

The model is a sum of single effect regression model.

Note that exactly one element of each \(\boldsymbol{\beta}_{l}\) is non-zero, and the different vectors \(\boldsymbol{\beta}_{1}, \cdots, \boldsymbol{\beta}_{L}\) are independent. Thus at most L covariates have non-zero coefficients in this model.

Note that if \(L \ll p\) then the SuSiE model is approximately equivalent to a standard Bayesian multiple regression model in which L randomly-chosen covariates have non-zero coefficients. The only difference is that with some (small) probability some of the \(\boldsymbol{\beta}_{l}\) in the SuSiE model may overlap in which coordinate is non-zero, and so the number of non-zero effects has some (small) probability to be less than L.

The different parameter structure of the SuSiE model leads to a very simple and intuitive fitting procedure. This procedure provide not only point estimates (approximate posterior means) for \(\boldsymbol{\beta}\), but a variational approximation to the posterior distribution. Crucially, this variational approximation is both easy to interpret, and capable of capturing important dependencies among elements of \(\boldsymbol{\beta}\) in the posterior distribution.

When L = 1, it is the single effect regression model. We know \[ \log p(\mathbf{y}) = L_{1}(q(\boldsymbol{\beta})) = \log \left[\sum_{j=1}^{p} \pi_{j} BF(\mathbf{y}, \mathbf{x}_{j})\right] + \log N(\mathbf{y}; 0, \frac{1}{\tau}I) \]

Moreover, \[ \begin{align*} L_{1}(q(\boldsymbol{\beta}); \mathbf{y}) &= \int q(\boldsymbol{\beta}) \log \frac{p(\boldsymbol{\beta}, \mathbf{y})}{q(\boldsymbol{\beta})} d \boldsymbol{\beta} \\ &= \mathbb{E}_{q}(\log p(\mathbf{y}|\boldsymbol{\beta})) + \mathbb{E}_{q}\left[\log \frac{p(\boldsymbol{\beta})}{q(\boldsymbol{\beta})}\right] \\ &= -\frac{n}{2}\log(2\pi) + \frac{n}{2}\log \tau - \frac{\tau \mathbf{y}^{T}\mathbf{y}}{2} - \frac{\tau}{2}\mathbb{E}_{q}\left[ \boldsymbol{\beta}^{T}X^{T}X\boldsymbol{\beta}\right] + \tau \mathbf{y}^{T}X\mathbb{E}_{q}\left[\boldsymbol{\beta}\right] + \mathbb{E}_{q}\left[\log \frac{p(\boldsymbol{\beta})}{q(\boldsymbol{\beta})}\right] \\ &= C_{1} - \frac{\tau}{2}\mathbb{E}_{q}\left[ \boldsymbol{\beta}^{T}X^{T}X\boldsymbol{\beta}\right] + \tau \mathbf{y}^{T}X\mathbb{E}_{q}\left[\boldsymbol{\beta}\right] + \mathbb{E}_{q}\left[\log \frac{p(\boldsymbol{\beta})}{q(\boldsymbol{\beta})}\right] \end{align*} \] which can be maximized by setting q to be the posterior.

When L = 2, \[ \mathbf{y} = X \boldsymbol{\beta}_{1} + X \boldsymbol{\beta}_{2} +\boldsymbol{\epsilon} \]

\[ L_{2}(q(\boldsymbol{\beta}); \mathbf{y}) = \mathbb{E}_{q}(\log p(\mathbf{y}|\boldsymbol{\beta}_{1}, \boldsymbol{\beta}_{1})) + \mathbb{E}_{q_{1}}\left[\log \frac{p(\boldsymbol{\beta}_{1})}{q(\boldsymbol{\beta}_{1})}\right] + \mathbb{E}_{q_{2}}\left[\log \frac{p(\boldsymbol{\beta}_{2})}{q(\boldsymbol{\beta}_{2})}\right] \]

We first conditional on the first effect by treating \(q_{1}\) fixed and maximize over \(q_{2}\), \[ \begin{align*} L_{2}(q_{2}(\boldsymbol{\beta}); \mathbf{y}) &= -\frac{n}{2}\log(2\pi) + \frac{n}{2}\log \tau - \frac{\tau \mathbf{y}_{2}^{T}\mathbf{y}_{2}}{2} + \mathbb{E}_{q_{1}}\left[\log \frac{p(\boldsymbol{\beta}_{1})}{q_{1}(\boldsymbol{\beta}_{1})}\right] - \frac{\tau}{2}\mathbb{E}_{q_{2}}\left[ \boldsymbol{\beta}_{2}^{T}X^{T}X\boldsymbol{\beta}_{2}\right] + \tau \mathbf{y}_{2}^{T}X\mathbb{E}_{q_{2}}\left[\boldsymbol{\beta}_{2}\right] + \mathbb{E}_{q_{2}}\left[\log \frac{p(\boldsymbol{\beta}_{2})}{q_{2}(\boldsymbol{\beta}_{2})}\right] \\ &= C_{2} - \frac{\tau}{2}\mathbb{E}_{q_{2}}\left[ \boldsymbol{\beta}_{2}^{T}X^{T}X\boldsymbol{\beta}_{2}\right] + \tau \mathbf{y}_{2}^{T}X\mathbb{E}_{q_{2}}\left[\boldsymbol{\beta}_{2}\right] + \mathbb{E}_{q_{2}}\left[\log \frac{p(\boldsymbol{\beta}_{2})}{q_{2}(\boldsymbol{\beta}_{2})}\right] \end{align*} \]

\(\mathbf{y}_{2} = \mathbf{y} - X\mathbb{E}_{q_{1}}\left[\boldsymbol{\beta}_{1}\right]\)

\[ L_{2}(q_{2}(\boldsymbol{\beta}); \mathbf{y}) \propto L_{1}(q_{2}(\boldsymbol{\beta}); \mathbf{y}_{2}) \] which can be maximized by setting \(q_{2}\) to be the posterior.

In general, the lower bound is \[ L(q(\boldsymbol{\beta})) = -\frac{n}{2}\log(2\pi) + \frac{n}{2}\log \tau - \mathbb{E}_{q}\left[\frac{\tau}{2}\|\mathbf{y} - \sum_{l=1}^{L}X\boldsymbol{\beta}_{l}\|^{2}\right] + \sum_{l=1}^{L} \mathbb{E}_{q_{l}}\left[\log \frac{p(\boldsymbol{\beta}_{l})}{q_{l}(\boldsymbol{\beta}_{l})}\right] \]

For each l single effect regression model, \[ \begin{align*} \log p(\mathbf{y}_{resid,-l}) &= L_{l}(q_{l}(\boldsymbol{\beta})) = -\frac{n}{2}\log(2\pi) + \frac{n}{2}\log \tau - \mathbb{E}_{q_{l}}\left[\frac{\tau}{2}\|\mathbf{y} - X\boldsymbol{\beta}_{l}\|^{2}\right] + \mathbb{E}_{q_{l}}\left[\log \frac{p(\boldsymbol{\beta}_{l})}{q_{l}(\boldsymbol{\beta}_{l})}\right] \\ \mathbb{E}_{q_{l}}\left[\log \frac{p(\boldsymbol{\beta}_{l})}{q_{l}(\boldsymbol{\beta}_{l})}\right] &= \log p(\mathbf{y}_{resid,-l}) + \frac{n}{2}\log(2\pi) - \frac{n}{2}\log \tau + \mathbb{E}_{q_{l}}\left[\frac{\tau}{2}\|\mathbf{y}_{resid,-l} - X\boldsymbol{\beta}_{l}\|^{2}\right] \end{align*} \]

\[ \mathbb{E}_{q}\left[\|\mathbf{y} - X\boldsymbol{\beta}\|^{2}\right] = \mathbf{y}^{T}\mathbf{y} - 2\mathbf{y}^{T}X\mathbb{E}_{q}\left[\boldsymbol{\beta}\right] + \mathbb{E}_{q}\left[\boldsymbol{\beta}^{T}X^{T}X\boldsymbol{\beta} \right] = \mathbf{y}^{T}\mathbf{y} - 2\mathbf{y}^{T}X\mathbb{E}_{q}\left[\boldsymbol{\beta}\right] + (X^{2})^{T}\mathbb{E}_{q}\left[\boldsymbol{\beta}\boldsymbol{\beta}^{T}\right] \] Let \(\mathbf{r}_{l} = \mathbb{E}_{q}(\boldsymbol{\beta}_{l})\), \(\mathbf{r} = \sum_{l=1}^{L}\mathbf{r}_{l}\)

\[ \begin{align*} \mathbb{E}_{q}\left( (\sum_{l=1}^{L}X\boldsymbol{\beta}_{l})^{T}(\sum_{l=1}^{L}X\boldsymbol{\beta}_{l}) \right) &= \sum_{l,l'}\mathbb{E}(\boldsymbol{\beta}_{l}^{T}X^{T}X\boldsymbol{\beta}_{l'}) \\ &= \sum_{l,l'} \mathbf{r}_{l}^{T}X^{T}X\mathbf{r}_{l'} - \sum_{l}\mathbf{r}_{l}^{T}X^{T}X\mathbf{r}_{l} + \sum_{l}\mathbb{E}(\boldsymbol{\beta}_{l}^{T}X^{T}X\boldsymbol{\beta}_{l}) \\ &= \|X\mathbf{r}\|^2 - \sum_{l=1}^{L}\|X\mathbf{r}_{l}\|^{2} + \sum_{l}\mathbb{E}(\boldsymbol{\beta}_{l}^{T}X^{T}X\boldsymbol{\beta}_{l}) \\ &= \|X\mathbf{r}\|^2 - \sum_{l=1}^{L}\|X\mathbf{r}_{l}\|^{2} + \sum_{l}\sum_{j=1}^{p}(X^{T}X)_{jj}\mathbb{E}(\beta_{lj}^2)\\ \end{align*} \]

\[ \begin{align*} \mathbb{E}_{q}\left[\|\mathbf{y} - \sum_{l=1}^{L}X\boldsymbol{\beta}_{l}\|^{2}\right] &= \mathbf{y}^{T}\mathbf{y} - 2\mathbf{y}^{T}X\sum_{l=1}^{L}\mathbb{E}_{q}(\boldsymbol{\beta}_{l}) + \mathbb{E}_{q}\left( (\sum_{l=1}^{L}X\boldsymbol{\beta}_{l})^{T}(\sum_{l=1}^{L}X\boldsymbol{\beta}_{l}) \right) \\ &= \mathbf{y}^{T}\mathbf{y} - 2\mathbf{y}^{T}X\mathbf{r} + \|X\mathbf{r}\|^2 - \sum_{l=1}^{L}\|X\mathbf{r}_{l}\|^{2} + \sum_{l}\sum_{j=1}^{p}(X^{T}X)_{jj}\mathbb{E}(\beta_{lj}^2) \\ &= \|\mathbf{y} - X\mathbf{r}\|^{2} - \sum_{l=1}^{L}\|X\mathbf{r}_{l}\|^{2} + \sum_{l}\sum_{j=1}^{p}(X^{T}X)_{jj}\mathbb{E}(\beta_{lj}^2) \end{align*} \]

\[ L(q(\boldsymbol{\beta})) = -\frac{n}{2}\log(2\pi) + \frac{n}{2}\log \tau - \frac{\tau}{2}\left[\|\mathbf{y} - X\mathbf{r}\|^{2} - \sum_{l=1}^{L}\|X\mathbf{r}_{l}\|^{2} + \sum_{l=1}^{L}\sum_{j=1}^{p}(X^{T}X)_{jj}\mathbb{E}(\beta_{lj}^2)\right] + \sum_{l=1}^{L} \left[ \log p(\mathbf{y}_{resid,-l}) + \frac{n}{2}\log(2\pi) - \frac{n}{2}\log \tau + \frac{\tau}{2}\left(\mathbf{y}_{resid,-l}^{T}\mathbf{y}_{resid,-l} - 2\mathbf{y}_{resid,-l}^{T}X\mathbb{E}_{q}\left[\boldsymbol{\beta}_{l}\right] + (X^{2})^{T}\mathbb{E}_{q}\left[\boldsymbol{\beta}_{l}\boldsymbol{\beta}_{l}^{T}\right] \right) \right] \]

Updates in each iteration:

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

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

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

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

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

Session information

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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