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Consider the linear problem
\[ Y=\beta_0+\sum_{k=1}^p\textbf X_k\beta_k+\epsilon,\epsilon\sim N(0,\sigma^2) \] where the range of \(p\) can be as large as \(10^5\sim 10^6\).
Under the Bayesian setting, we assume the prior distribution of \(\beta_k\) to be a spike-and-slab prior. And we introduce the indicator variable \(\gamma=(\gamma_1,\gamma_2,\dots,\gamma_k)^T\), \(\gamma_k=0\) indicates \(\beta_k=0~w.p.1\), i.e.\(\beta_k\) is not a true factor. And \(\gamma_k=1\) indicates \(\beta_k\neq 0\), i.e. \(\beta_k\) is a true factor. In the variable selection scenario, our goal is to estimate \(\gamma\), specifically posterior inclusion probabilities (PIP), which we will define later.
We also assume \(\mathbb P(\gamma_k=1)=\pi\). And for true factors we assume \[ \beta_k|\gamma_k=1\sim N(0,\sigma^2\sigma_\beta^2). \]
The parameters involved: \(\theta=(\sigma^2,\sigma_\beta^2,\pi)\). It turns out that the estimation of \(\gamma\) can be very sensitive to \(\theta\). Therefore, we assume an approriate prior for \(\theta\), and calculate
\[ PIP(k)=\mathbb P(\gamma_k=1|\textbf X,Y)=\frac{\sum_{\gamma_{-k}}\iint \mathbb P(Y,\beta,\gamma_k=1,\gamma_{-k}|\textbf X,\theta)d\beta d\theta}{\sum_{\gamma}\iint\mathbb P(Y,\beta,\gamma|\textbf X,\theta)p(\theta) d\beta d\theta}. \]
Computing PIP directly will be extremely time-consuming. To sum over \(\gamma\), there are \(2^{p}\) terms in total. It is impossible to compute when \(p\) is large. Not to mention each term is a complex high-dimensional integration (\(\beta\in\mathbb R^p\)).
Variational inference will provide a reasonable approximation to this, so that we don’t have to deal with such a complex problem. We will show the detail of this method in the following sections.
We do importance sampling on the parameters \(\theta=(\sigma^2,\sigma_\beta^2,\pi)\), for each sample:
Step 1. Estimate the variance \(Var(\beta_k|\gamma_k=1)=s_k^2\).
Step 2. Compute PIP and posterior means with a coordinate-descent method.
Step 3. Estimate the importance weights for each sample with a lowerbound.
The variational inference framework is to approximate the posterior distribution \(f(\beta,\gamma)=\mathbb P(\beta,\gamma|\textbf X,y,\theta)\) by searching for a distribution \(q(\beta,\gamma)\) to minimize the KL-divergence \[ D(q||f)=\int q(\beta,\gamma)\log\frac{q(\beta,\gamma)}{f(\beta,\gamma)}d\beta d\gamma. \] And restrict \(q(\beta,\gamma)\) to be of the form \[ q(\beta,\gamma;\phi)=\prod_{k=1}^p q(\beta_k,\gamma_k,\phi_k). \] (Independent factor assumption)
And the individual factors have the form \[ q(\beta_k,\gamma_k;\phi_k)=\alpha_kN(\beta_k|\mu_k.s_k^2)\mathbb I_{\gamma_k=1}+(1-\alpha_k)\delta_0(\beta_k)\mathbb I_{\gamma_k=0}. \]
This family assumes the factors are independent and each of them has a spike-and-slab prior. This is the case when \(X^TX\) is diagnal.
We set the partial derivatives to zero and solve for the best parameters. The formula goes as follows.
\[ Var(\beta_k|\gamma_k=1)\approx s^2_k=\frac{\sigma^2}{(\textbf X^T\textbf X)_{kk}+1/\sigma_{\beta}^2}. \]
\[ \mathbb E(\beta_k|\gamma_k=1)\approx \mu_k=\frac{s_k^2}{\sigma^2}\left((\textbf X^Ty)_k-\sum_{j\neq k}(\textbf X^T\textbf X)_{jk}\alpha_j\mu_j\right). \]
\[ \frac{\mathbb P(\gamma_k=1|\textbf X,y,\theta)}{\mathbb P(\gamma_k=0|\textbf X,y,\theta)}\approx \frac{\alpha_k}{1-\alpha_k}=\frac{\pi}{1-\pi}\cdot\frac{s_k}{\sigma_\beta\sigma}\cdot e^{\mu_k^2/(2s_k^2)}. \]
In our algorithm, we keep updating the values via the formulas above, until it converges. This is a coordinate-descent algorithm.
In the importance sampling scheme, the importance weight is \[ w(\theta)=\frac{\mathbb P(y|\textbf X,\theta)p(\theta)}{\tilde p(\theta)}. \]
Where \(p(\theta)\) is the prior of the parameters, and \(\tilde p(\theta)\) is the sampling distribution. These two are easy to get. But \(\mathbb P(y|\textbf X,\theta)\) is intractable for now. We use a lower bound to estimate it.