Structure learning of exponential family graphical model with false discovery rate control

Abstract

Probabilistic graphical models enjoy great popularity in a wide range of domains due to their ability to model the conditional dependency relationships among random variables. This paper explores the structure learning for the exponential family graphical model with false discovery rate (FDR) control. Most existing FDR-controlled structure learning procedures have been designed for the Gaussian graphical model (GGM). A systematic approach for more general exponential family graphical models is still lacking. In this paper, we introduce a unified procedure to learn the structure of the exponential family graphical model with FDR control utilizing the symmetrized data aggregation (SDA) technique via sample splitting, data screening, and information pooling. We show that our method controls FDR asymptotically under some mild conditions. Extensive simulation results and two real-data examples validate the effectiveness of our method.

Appendices

Appendix A: The proof of Theorem 1

Recall that

$$\begin{aligned} \textrm{FDP}&=\frac{\sum _{(r,t)\in E^c}\mathbb {I}\left( W_{rt}\ge L\right) }{1\vee \sum _{(r,t)}\mathbb {I}(W_{rt}\ge L)}= \frac{\sum _{(r,t)}\mathbb {I}\left( W_{rt}\le - L \right) }{1\vee \sum _{(r,t)}\mathbb {I}(W_{rt}\ge L)}\times \frac{\sum _{(r,t)\in E^c}\mathbb {I}\left( W_{rt}\ge L\right) }{\sum _{(r,t)}\mathbb {I}\left( W_{rt}\le -L\right) }\\&\le \alpha \times \frac{\sum _{(r,t)\in E^c}\mathbb {I}\left( W_{rt}\ge L\right) }{\sum _{(r,t)\in E^c}\mathbb {I}\left( W_{rt}\le -L\right) }. \end{aligned}$$

Let $q_{0n}=|\mathcal {S}\cap E^c|$, then $q_{0n}<p\bar{q}$. Let $\tilde{\Phi }(x)=1-\Phi (x)$, $G(s)=(q_{0n})^{-1}\sum _{(r,t)\in E^c}\Pr (W_{rt}\ge s\mid {\mathcal D}_{1})$, $G_{-}(s)=(q_{0n})^{-1}\sum _{(r,t)\in E^c}\Pr (W_{rt}\le -s\mid {\mathcal D}_{1})$ and $G^{-1}(y)=\inf \{s\ge 0: G(s)\le y\}$ for $0\le y\le 1$. In order to prove Theorem 1, we introduce an intermediate quantity $\tilde{W}_{rt}$, it is constructed as follows: define $\tilde{\varvec{\theta }}_r^{(2)}={\varvec{\theta }}_r^{*}-\left\{ \Upsilon \left( \varvec{\theta }_r^{*}\right) \right\} ^{-1} \nabla \mathcal {L}_{2n}\left( \varvec{\theta }_r^{*}\right)$. For $1\le r\le p,$ we define $T_{1rt}=\sqrt{n_1} \hat{\theta }_{rt}^{(1)} / {{\hat{\sigma }}}_{2rt}$ and $\tilde{T}_{2rt}=\sqrt{n_2} \tilde{\theta }_{rt}^{(2)} /{{\hat{\sigma }}}_{2rt}$, and $\tilde{W}^{1}_{rt}=T_{1rt}\tilde{T}_{2rt}$, then obtain $\tilde{W}_{rt}$ by the AND rule similar to formula (8). We first establish the symmetry property and uniform convergence of $\tilde{W}_{rt}$ under the null and then show the distance of $W_{rt}$ and $\tilde{W}_{rt}$ is negligible, thus proving Theorem 1. For $\tilde{W}_{rt},$ we accordingly define $\tilde{G}(s)=(q_{0n})^{-1}\sum _{(r,t)\in E^c}\Pr (\tilde{W}_{rt}\ge s\mid {\mathcal D}_{1})$, $\tilde{G}_{-}(s)=(q_{0n})^{-1}\sum _{(r,t)\in E^c}\Pr (\tilde{W}_{rt}\le -s\mid {\mathcal D}_{1})$ and $\tilde{G}^{-1}(y)=\inf \{s\ge 0: \tilde{G}(s)\le y\}$ for $0\le y\le 1$. We first provide a lemma that establishes the uniform bounds for $\tilde{\theta }_{rt}^{(2)}$, which is the counterpart of Lemma S.8 in Du et al. (2021).

Lemma 1

Suppose Conditions 1–6 hold, then for $C>4$, as $n\rightarrow \infty$, we have,

$$\begin{aligned} \Pr \left( \max _{1\le r\le p}\max _{t\in \mathcal {S}_r}\sqrt{n_2} |\tilde{\theta }_{rt}^{(2)}-\theta _{rt}^{*}|/\sigma _{rt}>\sqrt{C\log (p\bar{q})}\right) =o(1/p\bar{q}) \end{aligned}$$

Proof

Let $\mathcal {B}=\left\{ \max _{1 \le r \le p}\max _{ 1\le i\le n_2}\left\| \left\{ {\Upsilon }\left( {\varvec{\theta }}_r^{*}\right) \right\} ^{-1} \nabla \mathcal {L}_{2n}\left( \varvec{\theta }^{*}_r;X\right) \right\| _{\infty } \le m_{n}\right\}$, where $m_{n}=(n_2p\bar{q})^{1 / \varpi +\gamma } K_{n_2}$ for some small $\gamma >0$. By Condition 6 and Markov inequality, $\Pr \left( \mathcal {B}^{c}\right) \le 1 / p\bar{q}$. Let $x=\sqrt{C \log p\bar{q} / n_{2}}$. Conditioned on the post-selection model $\mathcal {S}=\bigcup _{r=1}^{p}\mathcal {S}_r$ and $\mathcal {B}$, the Bernstein inequality in Lemma 9 yields that

$$\begin{aligned} \begin{aligned}&{\text {Pr}}\left( |\tilde{\theta }_{rt}^{(2)}-\theta _{rt}^{*}|/ \sigma _{rt}>x \text{ for } \text{ some } r,t \mid \mathcal {S}, \mathcal {B}\right) \\&\quad \le p\bar{q} \max _{r}\max _{t} {\text {Pr}}\left( |\textbf{e}_{t}^{\top }\left\{ \Upsilon \left( \varvec{\theta }^{*}_r;\textbf{X}_i\right) \right\} ^{-1}\sum _{i=1}^{n_{2}} \nabla \mathcal {L}\left( \varvec{\theta }^{*}_r; {\textbf{X}}_{i}\right) |>n_{2} \sigma _{rt} x \mid \mathcal {S}, \mathcal {B}\right) \\&\quad \le 2 p\bar{q} \max _{r}\max _{t} \exp \left\{ -\frac{n_{2}^{2} \sigma _{rt}^{2} x^{2}}{2 n_{2} \sigma _{rt}^{2} + 2 m_{n} n_{2} \sigma _{rt} x / 3}\right\} \\&\quad =o(1 / p\bar{q}) \end{aligned} \end{aligned}$$

holds uniformly for $1\le r\le p$ and $t\in \mathcal {S}_r$, where we use the fact that $m_{n} \sqrt{\log p\bar{q} / n_2} \rightarrow 0$. The lemma is proved. $\square$

The next lemma establishes the symmetry property of $\tilde{W}_{rt}$, and it is the counterpart of Lemma S.1 in Du et al. (2021).

Lemma 2

Suppose Conditions 1–6 hold. Define a sequence $a_{p}$ satisfying $a_{p} / \bar{q} \rightarrow \infty$ and $a_{p}=o(p \bar{q})$. Then for any $0 \leqslant s \leqslant \tilde{G}_{-}^{-1}(a_{p} /(p \bar{q}))$, we have

$$\begin{aligned} \frac{\tilde{G}(s)}{\tilde{G}_{-}(s)}-1 \rightarrow 0. \end{aligned}$$

The proof of Lemma 2 mainly utilizes Lemmas 1 and 10, and its proof is similar to the proof of Lemma S.1 in Du et al. (2021) and thus is omitted.

The next lemma characterizes the uniform convergence of $(p\bar{q})^{-1} \sum _{(r,t)\in E^c} \mathbb {I}(\tilde{W}_{rt} \ge s)$ and $(p \bar{q})^{-1} \sum _{(r,t)\in E^c} \mathbb {I}(\tilde{W}_{rt} \le -s)$.

Lemma 3

Suppose Conditions 1–8 hold. For any sequence $a_{p}$ satisfying $a_{p} / \bar{q} \rightarrow \infty$ and $a_{p}=o(p \bar{q})$, we have

$$\begin{aligned} \sup _{0 \le s \le \tilde{G}^{-1}\left( a_{p} /(p \bar{q})\right) }\left|\frac{\sum _{(r,t)\in E^c } \mathbb {I}\left( \tilde{W}_{rt} \ge s\right) }{p\bar{q} \tilde{G}(s)}-1\right|=o_{p}(1), \end{aligned}$$

(A1)

and

$$\begin{aligned} \sup _{0 \le s \le \tilde{G}_{-}^{-1}\left( a_{p} /(p \bar{q})\right) }\left|\frac{\sum _{(r,t)\in E^c} \mathbb {I}\left( \tilde{W}_{rt}\le -s\right) }{p \bar{q} \tilde{G}_{-}(s)}-1\right|=o_{p}(1). \end{aligned}$$

(A2)

Under the Condition 8, Lemma 3 is easily proved using the technique in Du et al. (2021) and thus its proof is omitted. Lemmas 2 and 3 respectively establish the symmetry property and uniform consistency for $\tilde{W}_{rt}$’s. In the following, we show that the distance between $W_{rt}$ and $\tilde{W}_{rt}$ is small. To show the distance between $W_{rt}$ and $\tilde{W}_{rt}$ is negligible, we need the following lemmas. For the rth node ($1\le r\le p$), define the event

$$\begin{aligned} {\small \xi _r:=\left\{ \frac{1}{n}\sum _{i=1}^{n} M_r\left( \textbf{X}_{i}\right) \le 2 L_r,\left\| \nabla ^{2} \mathcal {L}_{2n}\left( \varvec{\theta }_r^{*}\right) -\Upsilon \left( \varvec{\theta }_r^{*}\right) \right\| \le \frac{\rho _r l_{-}}{2}, \left\| \nabla \mathcal {L}_{2n}\left( \varvec{\theta }_r^{*}\right) \right\| < \frac{(1-\rho _r) l_{-} \delta _{\rho _r}}{2}\right\} ,} \end{aligned}$$

where $\delta _{\rho _r}=\min \left\{ \rho _r,\rho _rl_{-} /(4 L_r)\right\}$. And we further define the event $\xi =\bigcap _{r=1}^{p}\xi _r$.

Lemma 4

Suppose Conditions 3–6 hold. Under the event $\xi _r$, we have,

$$\begin{aligned} \left\| \hat{\varvec{\theta }}^{(2)}_r-\varvec{\theta }_r^{*}\right\| \le \frac{2\left\| \nabla \mathcal {L}_{2n}\left( \varvec{\theta }_r^{*}\right) \right\| }{(1-\rho _r) l_{-}}. \end{aligned}$$

Proof

This lemma follows exactly the same as Lemma 6 in Zhang et al. (2013), and thus its proof is omitted. $\square$

Lemma 5

Suppose Conditions 3–6 hold. Under the event $\xi _r$, we have

$$\begin{aligned}&\left\| \hat{\varvec{\theta }}_r^{(2)}-\varvec{\theta }_r^{*}+\left\{ {\varvec{\Upsilon }}\left( \varvec{\theta }_r^{*}\right) \right\} ^{-1}\nabla \mathcal {L}_{2n}\left( \varvec{\theta }_r^{*}\right) \right\| _{\infty }\\&\quad \le \frac{2}{(1-\rho _r) l_{-}^{2}}\left\| \nabla ^{2} \mathcal {L}_{2n}\left( \varvec{\theta }_r^{*}\right) -{\Upsilon }\left( \varvec{\theta }_r^{*}\right) \right\| \left\| \nabla \mathcal {L}_{2n}\left( \varvec{\theta }_r^{*}\right) \right\| +\frac{8 L_r}{(1-\rho _r)^{2} l_{-}^{3}}\left\| \nabla \mathcal {L}_{2n}\left( \varvec{\theta }_r^{*}\right) \right\| ^{2}. \end{aligned}$$

Proof

By the integral form of Taylor’s expansion, we have

$$\begin{aligned} 0=\nabla \mathcal {L}_{2n}(\hat{\varvec{\theta }}_r^{(2)} )=\nabla \mathcal {L}_{2n}\left( \varvec{\theta }_r^{*}\right) +\textbf{H}_{n}\left( \hat{\varvec{\theta }}_r^{(2)}-\varvec{\theta }_r^{*}\right) , \end{aligned}$$

where $\textbf{H}_{n}=\int _{0}^{1} \nabla ^{2} \mathcal {L}_{2n}\left( \varvec{\theta }_r^{*}+u\left( \hat{\varvec{\theta }}_r^{(2)}-\varvec{\theta }_r^{*}\right) \right) du$. Then simple linear algebra yields

$$\begin{aligned} \hat{\varvec{\theta }}_r^{(2)}-\varvec{\theta }^{*}_r=-\left\{ \varvec{\Upsilon }\left( \varvec{\theta }^{*}_r\right) \right\} ^{-1} \nabla \mathcal {L}_{2n}\left( \varvec{\theta }^{*}_r\right) -\left\{ \Upsilon \left( \varvec{\theta }^{*}_r\right) \right\} ^{-1} \textbf{U}_{n}\left( \hat{\varvec{\theta }}_r^{(2)}-\varvec{\theta }^{*}_r\right) -\left\{ \varvec{\Upsilon }\left( \varvec{\theta }^{*}_r\right) \right\} ^{-1} \textbf{V}_{n}\left( \hat{\varvec{\theta }}_r^{(2)}-\varvec{\theta }^{*}_r\right) , \end{aligned}$$

where $\textbf{U}_{n}=\textbf{H}_{n}-\nabla ^{2} \mathcal {L}_{2n}\left( \varvec{\theta }^{*}_r\right)$ and $\textbf{V}_{n}=\nabla ^{2} \mathcal {L}_{2n}\left( \varvec{\theta }^{*}_r\right) -\Upsilon \left( \varvec{\theta }^{*}_r\right)$. Then, the claimed expansion is a direct consequence of Lemma 4 and Conditions 5–6. $\square$

Lemma 6

Suppose Conditions 1–6 hold. Under the event $\xi =\bigcap _{r=1}^{p}\xi _r$, we have,

$$\begin{aligned} \left|\frac{\sqrt{n}\left( \hat{\theta }_{ rt}^{(2)}-\theta _{rt}^{*}\right) }{\hat{\sigma }_{2 rt}}-\frac{\sqrt{n}\left( \tilde{\theta }^{(2)}_{rt}- \theta _{rt}^{*}\right) }{\sigma _{rt}}\right|= o_{p}\left( a_{n}\right) , \end{aligned}$$

holds uniformly for $1\le r\le p$ and $t\in \mathcal {S}_r$, where, $a_{n} /\left( \bar{q}^{3 / 2} \log p\bar{q} / \sqrt{n}\right) \rightarrow \infty .$

Proof

By the Bernstein inequality and Conditions 5–6, we can show that $\mathop {\max }\nolimits _{1\le r \le p}\mathop {\max }\nolimits _{t\in \mathcal {S}_r}|\sigma _{rt}^{-1}\hat{\sigma }_{2 rt}-1|=O_{p}(\sqrt{\log p\bar{q} / n})$. Similar to Lemma 1, we can show that $\left\| \max _{1\le r\le p}\nabla \mathcal {L}_{2 n}\left( \varvec{\theta }^{*}_r\right) \right\| _{\infty }=O_{p}(\sqrt{\log p\bar{q} / n})$, and accordingly $\left\| \max _{1\le r\le p}\nabla \mathcal {L}_{2n}\left( \varvec{\theta }^{*}_r\right) \right\| \le \sqrt{\bar{q}}\left\| \max _{1\le r\le p}\nabla \mathcal {L}_{2 n}\left( \varvec{\theta }^{*}_r\right) \right\| _{\infty }=O_{p}(\sqrt{\bar{q} \log p\bar{q} / n})$, uniformly in $1\le r\le p$ and $t\in \mathcal {S}_r$. By using similar arguments in the proof of Lemma 1 and Conditions 5–6, we can also show that $\left\| \max _{1\le r\le p}\{\nabla ^{2} \mathcal {L}_{2 n}\left( \varvec{\theta }^{*}_r\right) -{\Upsilon }\left( \varvec{\theta }^{*}_r\right) \}\right\| =O_{p}(\bar{q} \sqrt{\log p\bar{q} / n})$. The assertion for $\hat{\theta }^{(2)}_{rt}$ follows immediately from Lemma 5. $\square$

Lemma 7

Under Conditions 1–6, we have $\Pr (\xi ) \lesssim o(1 / p\bar{q})$.

Proof

Since $\Pr (\xi _r)\lesssim o(1/\bar{q})$, we have

$$\begin{aligned} \Pr (\xi )=\Pr \left( \bigcap _{r=1}^{p}\xi _r\right) \lesssim o(1/p\bar{q}) \end{aligned}$$

. $\square$

The following Lemma establishes the distance between $W_{rt}$ and $\tilde{W}_{rt}$.

Lemma 8

Suppose conditions 1–6 hold and $\bar{q}^{3/2}\left( \log p\bar{q}\right) ^{2+c}c_{np} \rightarrow 0$ for a small $c>0$. Then, for any $M>0$, we have,

$$\begin{aligned} \sup _{M \le s \le G^{-1}\left( \alpha \eta _{n} /p\bar{q}\right) }\left|\frac{\sum _{(r,t)\in E^c} \mathbb {I}\left( \tilde{W}_{rt} \ge s\right) }{\sum _{(r,t)\in E^c} \mathbb {I}\left( W_{rt} \ge s\right) }-1\right|= & {} o_{p}(1), \\ \sup _{M \le s \le G_{-}^{-1}\left( \alpha \eta _{n} /p\bar{q}\right) }\left|\frac{\sum _{(r,t)\in E^c} \mathbb {I}\left( \tilde{W}_{rt} \le -s\right) }{\sum _{(r,t)\in E^c} \mathbb {I}\left( W_{rt}\le -s\right) }-1\right|= & {} o_{p}(1). \end{aligned}$$

Proof

By similar arguments in the proof of Lemma S.4 and S.5 in Du et al. (2021), Lemma 8 can be proved, thus its proof is omitted. $\square$

Proof of Theorem 1

In order to prove Theorem 1, we consider another EFC procedure with the statistics $\tilde{W}_{rt}$, and choose a threshold $\tilde{L}>0$ by setting $\tilde{L}=\inf \left\{ s>0:\frac{\#\{(r,t): \tilde{W}_{rt}\le -s\}}{\#\{(r,t): \tilde{W}_{rt}\ge s\}\vee 1}\le \alpha \right\}$. Define $\mathcal {G}=\left\{ (r,t): \theta _{rt}=o\left( c_{n p}\right) \right\}$. Similar to the statements of the proof of Theorem 4 in Du et al. (2021), for any $(r,t)\in \mathcal {G}$, under the condition that $\bar{q}^{3/2}\left( \log p\bar{q}\right) ^{2+c}c_{np} \rightarrow 0$ for a small $c>0$, the absolute difference between $W_{rt}$ and $\tilde{W}_{rt}$ is negligible from Lemma 6, and for $(r,t)\in \mathcal {G}^{c}$, we have, $\tilde{W}_{rt}=W_{rt}\{1+o_p(1)\}$. From Lemma 8, we conclude that,

$$\begin{aligned} {\textrm{FDP}}_{\tilde{W}}(\tilde{L}):=\frac{\#\left\{ (r,t): \tilde{W}_{rt} \ge \tilde{L}, (r,t)\in E^{c}\right\} }{\#\left\{ (r,t): \tilde{W}_{rt} \ge \tilde{L}\right\} \vee 1}={\textrm{FDP}}_{W}(L)\left\{ 1+o_{p}(1)\right\} . \end{aligned}$$

Under Conditions 1–8, similar to the proof of Theorem 2 of Du et al. (2021), we can show that $\textrm{FDP}_{\tilde{W}}(\tilde{L})$ is controlled at the nominal level asymptotically. Thus the desired result follows. $\square$

1.1 Additional lemmas

Lemma 9

(Bernstein’s inequality) Let $X_{1}, \ldots , X_{n}$ be independent centered random variables a.s. bounded by $A<\infty$ in absolute value. Let $\sigma ^{2}=n^{-1}\sum _{i=1}^{n} \mathbb {E}\left( X_{i}^{2}\right)$. Then for all $x>0$,

$$\begin{aligned} \Pr \left( \sum _{i=1}^{n} X_{i} \ge x\right) \le \exp \left( -\frac{x^{2}}{2 n \sigma ^{2}+2 A x / 3}\right) . \end{aligned}$$

Lemma 10

(Moderate deviation for the independent sum) Suppose that $X_{1}, \ldots , X_{n}$ are independent random variables with mean zero, satisfying $\mathbb {E}\left( |X_{i}|^{2+\delta }\right) <\infty \ (i=1,2,\ldots )$. Let $B_{n}=\sum _{i=1}^{n} \mathbb {E}\left( X_{i}^{2}\right)$ and assume that $\liminf _{n} B_{n} / n>0$. Then,

$$\begin{aligned} \frac{\Pr \left( \sum _{i=1}^{n} X_{i}>x \sqrt{B_{n}}\right) }{1-\Phi (x)} \rightarrow 1 \end{aligned}$$

as $n \rightarrow \infty$ uniformly in x in the domain $0 \le x \le C\left\{ 2 \log \left( 1 / L_{n}\right) \right\} ^{1 / 2}$, where $L_{n}=B_{n}^{-1-\delta / 2} \sum _{i=1}^{n} \mathbb {E}|X_{i}|^{2+\delta }$ and C is a positive constant satisfying the condition $C<1$.

Appendix B: Additional simulation results

See Tables 7 and 8.

Table 7 The empirical FDR(%)s of the EFC method with its test statistic matrix W combined by the OR rule, the significance level $\alpha =5\%,10\%,20\%$ when $(n,\delta )=(500,0.3)$ under Case (Ia)

Table 8 The empirical FDR and TDR results of the EFC, GFC_L, and the EBIC methods under the PGM setup when $n=1000,p=50$