Bayesian selection of best subsets via hybrid search

Abstract

Over the past decades, variable selection for high-dimensional data has drawn increasing attention. With a large number of predictors, there rises a big challenge for model fitting and prediction. In this paper, we develop a new Bayesian method of best subset selection using a hybrid search algorithm that combines a deterministic local search and a stochastic global search. To reduce the computational cost of evaluating multiple candidate subsets for each update, we propose a novel strategy that enables us to calculate exact marginal likelihoods of all neighbor models simultaneously in a single computation. In addition, we establish model selection consistency for the proposed method in the high-dimensional setting in which the number of possible predictors can increase faster than the sample size. Simulation study and real data analysis are conducted to investigate the performance of the proposed method.

Appendices

Appendix

Calculation of Equation (3)

For any $i \notin {\hat{{\varvec{\gamma }}}}$, from Eq. (2), we have

$$\begin{aligned} m(\mathbf{y}|{\hat{{\varvec{\gamma }}}}\cup \{i\})\propto & {} |\mathbf{X}_{{\hat{{\varvec{\gamma }}}}\cup \{i\} }^{{ \top }}\mathbf{X}_{{\hat{{\varvec{\gamma }}}}\cup \{i\} }+\tau ^{-1}\mathbf{I}_{k+1} |^{-1/2} \nonumber \\&\times \left( \mathbf{y}^{{ \top }}\mathbf{H}_{{\hat{{\varvec{\gamma }}}}\cup \{i\}}\mathbf{y}+b_{\sigma }\right) ^{-\frac{a_{\sigma }+n}{2} }. \end{aligned}$$

(13)

It follows from the Sherman–Morrison formula that

$$\begin{aligned} \mathbf{H}_{{\hat{{\varvec{\gamma }}}}\cup \{i\} }= \mathbf{H}_{{\hat{{\varvec{\gamma }}}}}-\frac{\mathbf{H}_{{\hat{{\varvec{\gamma }}}}}\mathbf{x}_i \mathbf{x}_i ^{{ \top }}\mathbf{H}_{{\hat{{\varvec{\gamma }}}}} }{\tau ^{-1}+\mathbf{x}_i^{{ \top }}\mathbf{H}_{{\hat{{\varvec{\gamma }}}}}\mathbf{x}_i}. \end{aligned}$$

(14)

Using the Sylvester’s determinant identity and the Sherman–Morrison formula, we obtain

$$\begin{aligned}&|\mathbf{X}_{{\hat{{\varvec{\gamma }}}}\cup \{i\} }^{{ \top }}\mathbf{X}_{{\hat{{\varvec{\gamma }}}}\cup \{i\} }+\tau ^{-1}\mathbf{I}_{k+1} | \nonumber \\&\quad = \tau ^{-(k+1)}|\tau \mathbf{X}_{{\hat{{\varvec{\gamma }}}}\cup \{i\} }\mathbf{X}_{{\hat{{\varvec{\gamma }}}}\cup \{i\} }^{{ \top }}+\mathbf{I}_n |\nonumber \\&\quad = \tau ^{-(k+1)}|\tau \mathbf{X}_{{\hat{{\varvec{\gamma }}}}}\mathbf{X}_{{\hat{{\varvec{\gamma }}}}}^{{ \top }}+\mathbf{I}_n+\tau \mathbf{x}_i \mathbf{x}_i^{{ \top }} |\nonumber \\&\quad = \tau ^{-(k+1)} |\tau \mathbf{X}_{{\hat{{\varvec{\gamma }}}}}\mathbf{X}_{{\hat{{\varvec{\gamma }}}}}^{{ \top }}+\mathbf{I}_n| \{1+ \tau \mathbf{x}_i^{{ \top }} (\tau \mathbf{X}_{{\hat{{\varvec{\gamma }}}}}\mathbf{X}_{{\hat{{\varvec{\gamma }}}}}^{{ \top }}+\mathbf{I}_n)^{-1}\mathbf{x}_i\}\nonumber \\&\quad =| \mathbf{X}_{{\hat{{\varvec{\gamma }}}}}^{{ \top }}\mathbf{X}_{{\hat{{\varvec{\gamma }}}}}+\tau ^{-1}\mathbf{I}_{k}| (\tau ^{-1}+\mathbf{x}_i^{{ \top }}\mathbf{H}_{{\hat{{\varvec{\gamma }}}}} \mathbf{x}_i). \end{aligned}$$

(15)

Applying (14) and (15) to (13), we thus have

$$\begin{aligned} m(\mathbf{y}|{\hat{{\varvec{\gamma }}}}\cup \{i\})\propto & {} \left\{ \mathbf{y}^{{ \top }}\mathbf{H}_{{\hat{{\varvec{\gamma }}}}}\mathbf{y}-\frac{( \mathbf{x}_i^{{ \top }}\mathbf{H}_{{\hat{{\varvec{\gamma }}}}}\mathbf{y})^2}{\tau ^{-1}+\mathbf{x}_i^{{ \top }}\mathbf{H}_{{\hat{{\varvec{\gamma }}}}} \mathbf{x}_i }+b_{\sigma }\right\} ^ {-\frac{a_{\sigma }+n}{2}}\\&\times (\tau ^{-1}+\mathbf{x}_i^{{ \top }}\mathbf{H}_{{\hat{{\varvec{\gamma }}}}} \mathbf{x}_i)^{-1/2} \end{aligned}$$

for any $i \notin {\hat{{\varvec{\gamma }}}}$.

Calculation of Eq. (4)

For any $j \in {\tilde{{\varvec{\gamma }}}}$, Eq. (2) leads to

$$\begin{aligned} m(\mathbf{y}|{\tilde{{\varvec{\gamma }}}}{\setminus }\{j\})\propto & {} |\mathbf{X}_{{\tilde{{\varvec{\gamma }}}}{\setminus }\{j\} }^{{ \top }}\mathbf{X}_{{\tilde{{\varvec{\gamma }}}}{\setminus }\{j\}}+\tau ^{-1}\mathbf{I}_{k} |^{-1/2} \nonumber \\&\times \left( \mathbf{y}^{{ \top }}\mathbf{H}_{{\tilde{{\varvec{\gamma }}}}{\setminus }\{j\}}\mathbf{y}+b_{\sigma }\right) ^{-\frac{a_{\sigma }+n}{2} }. \end{aligned}$$

(16)

From the Sherman-Morrison formula, we have

$$\begin{aligned} \mathbf{H}_{{\tilde{{\varvec{\gamma }}}}{\setminus }\{j\} }= \mathbf{H}_{{\tilde{{\varvec{\gamma }}}}}+\frac{\mathbf{H}_{{\tilde{{\varvec{\gamma }}}}}\mathbf{x}_j \mathbf{x}_j ^{{ \top }}\mathbf{H}_{{\tilde{{\varvec{\gamma }}}}} }{\tau ^{-1}-\mathbf{x}_j^{{ \top }}\mathbf{H}_{{\tilde{{\varvec{\gamma }}}}}\mathbf{x}_j}. \end{aligned}$$

(17)

From the Sylvester’s determinant identity and the Sherman-Morrison formula, we obtain

$$\begin{aligned}&|\mathbf{X}_{{\tilde{{\varvec{\gamma }}}}{\setminus }\{j\}}^{{ \top }}\mathbf{X}_{{\tilde{{\varvec{\gamma }}}}{\setminus }\{j\} }+\tau ^{-1}\mathbf{I}_{k} | \nonumber \\&\quad = \tau ^{-k}|\tau \mathbf{X}_{{\tilde{{\varvec{\gamma }}}}{\setminus }\{j\} }\mathbf{X}_{{\tilde{{\varvec{\gamma }}}}{\setminus }\{j\} }^{{ \top }}+\mathbf{I}_n |\nonumber \\&\quad = \tau ^{-k}|\tau \mathbf{X}_{{\tilde{{\varvec{\gamma }}}}}\mathbf{X}_{{\tilde{{\varvec{\gamma }}}}}^{{ \top }}+\mathbf{I}_n-\tau \mathbf{x}_j \mathbf{x}_j^{{ \top }} |\nonumber \\&\quad = \tau ^{-k} |\tau \mathbf{X}_{{\tilde{{\varvec{\gamma }}}}}\mathbf{X}_{{\tilde{{\varvec{\gamma }}}}}^{{ \top }}+\mathbf{I}_n| \{1- \tau \mathbf{x}_j^{{ \top }} (\tau \mathbf{X}_{{\tilde{{\varvec{\gamma }}}}}\mathbf{X}_{{\tilde{{\varvec{\gamma }}}}}^{{ \top }}+\mathbf{I}_n)^{-1}\mathbf{x}_j\}\nonumber \\&\quad =\tau ^2 | \mathbf{X}_{{\tilde{{\varvec{\gamma }}}}}^{{ \top }}\mathbf{X}_{{\tilde{{\varvec{\gamma }}}}}+\tau ^{-1}\mathbf{I}_{k+1}| (\tau ^{-1}-\mathbf{x}_j^{{ \top }}\mathbf{H}_{{\tilde{{\varvec{\gamma }}}}} \mathbf{x}_j). \end{aligned}$$

(18)

Hence, applying (17) and (18) to (16), we have

$$\begin{aligned}&m(\mathbf{y}|{\tilde{{\varvec{\gamma }}}}{\setminus } \{j\}) \propto \left\{ \mathbf{y}^{{ \top }}\mathbf{H}_{{\tilde{{\varvec{\gamma }}}}}\mathbf{y}+\frac{( \mathbf{x}_j^{{ \top }}\mathbf{H}_{{\tilde{{\varvec{\gamma }}}}}\mathbf{y})^2}{\tau ^{-1}-\mathbf{x}_j^{{ \top }}\mathbf{H}_{{\tilde{{\varvec{\gamma }}}}} \mathbf{x}_j }+b_{\sigma }\right\} ^ {-\frac{a_{\sigma }+n}{2}}\\&\quad \times (\tau ^{-1}-\mathbf{x}_j^{{ \top }}\mathbf{H}_{{\tilde{{\varvec{\gamma }}}}} \mathbf{x}_j)^{-1/2} \end{aligned}$$

for any $j \in {\tilde{{\varvec{\gamma }}}}$.