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Summary of SuSiE (Sum of Single Effects)

Reference: Wang et al, 2012

I. Model

\(\textbf y=\textbf X\textbf b+\textbf e,\textbf e\sim N(0,\sigma^2 I_n),\textbf X,\textbf y\in \mathbb R^n,\textbf b\in \mathbb R^p\).

\(\textbf b=\sum_{l=1}^L b_l\boldsymbol \gamma_l,\boldsymbol \gamma_l\sim Mult(1,\pi),b_l\sim N(0,\sigma_{bl}^2)\).

Here \(L,\pi\) are given and fixed.

We want to estimate the posterior inclusion probability (PIP)

\[ \alpha_j:=\mathbb P(\beta_j\neq 0|\textbf X,\textbf y), \]

and the posterior mean \(\mu_{1j}\) and variance \(\sigma_{1j}^2\) for all \(1\leq j\leq p\).

II. Simple version: single effect regression model (SER)

Here we assume \(L=1\). We introduce SER model because fitting SER model is an important step in fitting SuSiE.

Model

\(\textbf y=\textbf X\beta+\textbf e,\textbf e\sim N(0,\sigma^2 I_n),\textbf X,\textbf y\in \mathbb R^n,\beta\in\mathbb R\).

\(\beta=b\boldsymbol \gamma,\gamma\sim Mult(1,\pi),b\sim N(0,\sigma_b^2)\).

Here \(\pi\) is given and fixed.

Goal: find

\[ PIP_k=\mathbb P(\gamma_k=1|X,y). \]

Method

Assume the variance \(\sigma^2\) and \(\sigma_b^2\) are known. Calculate the Bayes Factor

\[ BF(y,X;\sigma^2,\sigma_b^2)=... \]

Also, the posterior distribution

\[ \beta_k|X_k,y,\sigma^2,\sigma_b^2,\gamma_k=1\sim N(\mu_{1k},\sigma_{1k}^2). \]

The bayes factor, posterior mean \(\mu_{1k}\) and variance \(\sigma_{1k}^2\) all have a close form, and are easy to compute.

Therefore, for given \(\sigma^2,\sigma_b^2\), we have

\[ \alpha_k=\mathbb P(\gamma_k=1|X,y,\sigma^2,\sigma_b^2)=\frac{BF(y,X_k;\sigma^2,\sigma_b^2)\cdot\pi_k}{\sum_{j=1}^p BF(y,X_j;\sigma^2,\sigma_b^2)\cdot\pi_j}. \]

This is also easy to compute. By putting everything above together, we have a function SER with input \((X,y,\sigma^2,\sigma^2_b)\), whic outputs the important parameters of the posterior distribution \((\boldsymbol\alpha=(\alpha_1,\alpha_2,\dots,\alpha_k),\boldsymbol\mu_1=(\mu_{11},\mu_{12},\dots,\mu_{1p}),\boldsymbol\sigma_1^2=(\sigma^2_{11},\sigma^2_{12},\dots,\sigma^2_{1p}))\).

III. Fitting SuSiE: Iterative Bayesian stepwise selection (IBSS)

Algorithm

For given data \(\textbf X,\textbf y\), hyperparameters \(\sigma^2,\boldsymbol \sigma_0^2\), number of effects \(L\).

  • Initialize \(\bar{\textbf b_l}=0\) for all \(1\leq l\leq L\) (or any other initialization)
  • Repeat until converge
    • For i in 1,\(\dots\),L do
      • \(\textbf r_l\leftarrow \textbf y-\sum_{l^\prime\neq l}\textbf X\bar{\textbf b_{l^\prime}}\)
      • \((\boldsymbol\alpha_l,\boldsymbol \mu_{1l},\boldsymbol \sigma_{1l})\leftarrow SER(\textbf r_l,\textbf X;\sigma^2,\sigma_{0l}^2)\)
      • \(\bar{\textbf b_l}\leftarrow \boldsymbol{\alpha}_l\circ \boldsymbol{\mu}_{1l}\)
  • Return \((\boldsymbol\alpha_l,\boldsymbol \mu_{1l},\boldsymbol \sigma_{1l})\) for all \(1\leq l\leq L\).

A variational inference explanation

Variational inference finds an approximation \(q(\textbf b_1,\dots,\textbf b_L)\) to the posterior distribution \(p_{post}:=p(\textbf b_1,\dots,\textbf b_L|\textbf y)\), which minimizes the KL-divergence from \(q\) to \(p_post\), \(D_{KL}(q,p_{post})\).

It can be hard to compute, but we can write it as

\[ D_{KL}(q,p_{post})=\log p(\textbf y|\sigma^2,\boldsymbol \sigma_0^2)-F(q;\sigma^2,\boldsymbol\sigma_0^2), \]

where \(F\) is known as the evidence lower bound (ELBO), and it is easy to compute.