SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Prior on spline coefficients. The prior on  $A_{k}$  is chosen to serve multiple purposes. (i) It sorts the basis functions by decreasing smoothness and therefore helps to identify the spanning set of size  $K_{j}$  for the underlying smooth causally embedded space  $\mathcal{D}_j$ . (ii) Although not a strict requirement for modelling purpose, it forces  $\phi_{k}$ 's to be orthonormal, i.e.  $\int \phi_k(\omega)\phi_{k'}(\omega)d\omega = I(k = k')$ . As such, the orthogonality constraints help eliminate any information overlap between the basis functions, which keeps the total number of necessary basis functions that actually contribute to the causal structure learning to a minimum. (iii) It regularizes the roughness of  $\phi_{k}$ 's to prevent overfitting.
For (iii), more specifically, we restrict the roughness of the basis functions  $\phi_k(\cdot)$  by assigning a prior that penalizes its second derivatives [Gu, 1992, Wahba, 1978, Berry et al., 2002]:
$$
\boldsymbol {A} _ {k} \sim N (\boldsymbol {0}, \lambda_ {k} ^ {- 1} \boldsymbol {\Omega} ^ {-}),
$$
where  $\Omega^{-}$  is the pseudoinverse of  $\Omega = \int \pmb{b}^{\prime \prime}(t)[\pmb{b}^{\prime \prime}(t)]^{\top}dt$ . Let  $\Omega = UDU^{\top}$  be the singular value decomposition of  $\Omega$ . Following Wand and Ormerod, 2010, to facilitate computation, we reparameterize  $\phi_k(\cdot) = \sum_{r=1}^{R} \tilde{A}_{kr} \tilde{b}_k(\cdot)$  with  $\tilde{b}(\cdot) = (1, t, \pmb{b}^T(\cdot) \tilde{U} \tilde{D}^{-\frac{1}{2}})^{\top}$  where  $\tilde{D}$  is the  $(R-2) \times (R-2)$  submatrix of  $\pmb{D}$  corresponding to non-zero singular values (note that the rank of  $\Omega$  is  $R-2$  by definition) and  $\tilde{U}$  is the corresponding  $R \times (R-2)$  submatrix of  $\pmb{U}$ . This induces a prior on  $\tilde{A}_k$  given by
$$
\tilde {\boldsymbol {A}} _ {k} \sim N (\boldsymbol {0}, \boldsymbol {S} _ {k}) \text {w i t h} \boldsymbol {S} _ {k} = \operatorname {d i a g} \left(\infty , \infty , \lambda_ {k} ^ {- 1}, \dots , \lambda_ {k} ^ {- 1}\right).
$$
In other words, the intercept and the linear term are unpenalized but the non-linear terms are penalized, the degree of which is controlled by  $\lambda_{k}$ . In practice, we set the first two diagonal elements of  $S_{k}$  as  $10^{8}$ . We constrain the regularization parameters  $\lambda_1 > \dots >\lambda_S > 0$  by putting a uniform prior:
$$
\lambda_ {k} \sim \operatorname {U n i f o r m} \left(L _ {k}, U _ {k}\right), \forall k \in [ S ],
$$
$$
U _ {1} = 1 0 ^ {8}, L _ {k} = \lambda_ {k + 1} \forall k \in [ S - 1 ],
$$
$$
U _ {k} = \lambda_ {k - 1} \forall k \in \{2, \dots , S \}, L _ {S} = 1 0 ^ {- 8},
$$
which implies that the smoothness of  $\phi_k(\cdot)$  decreases as  $k$  gets larger.
Priors on the adjacency matrix. We propose to use an independent uniform-Bernoulli prior on each entry  $E_{j\ell}$  of  $\pmb{E}$ , i.e.,  $E_{j\ell}|\rho \stackrel{\mathrm{ind}}{\sim}\mathrm{Bernoulli}(\rho)$  and  $\rho \sim \mathrm{Uniform}(0,1)$ . The marginal distribution of  $\pmb{E}$  with  $\rho$  integrated out is given by
$$
p (\boldsymbol {E}) = \int p (\boldsymbol {E} | \rho) p (\rho) d \rho = \operatorname {B e t a} \left(\sum_ {j \neq \ell} E _ {j \ell} + 1, \sum_ {j \neq \ell} (1 - E _ {j \ell}) + 1\right).
$$
Now, for example, if  $E_0$  denotes the null adjacency matrix and  $E_1$  denotes the adjacency matrix with only one edge, then we can see that  $p(E_0) / p(E_1) = p^2 - p$ . Therefore, an empty graph is favored over a graph with one edge by a factor of  $p^2 - p$ , and, importantly, this penalty increases with  $p$ . Thus, the uniform-Bernoulli prior prevents false discoveries and leads to a sparse network by increasing the penalty against additional edges as the dimension  $p$  grows.
Prior on the causal effect matrix. Now given  $\pmb{E}$ , we assume an independent spike and slab prior on the entries of  $\pmb{B} = (B_{j\ell})_{j,\ell=1}^{p}$ :
$$
\boldsymbol {B} _ {j \ell} | E _ {j \ell} \sim (1 - E _ {j \ell}) M V N (\boldsymbol {B} _ {j \ell}; \mathbf {0}, s \gamma \boldsymbol {I} _ {K _ {j}}, \boldsymbol {I} _ {K _ {\ell}}) + E _ {j \ell} M V N (\boldsymbol {B} _ {j \ell}; \mathbf {0}, \gamma \boldsymbol {I} _ {K _ {j}}, \boldsymbol {I} _ {K _ {\ell}}),
$$
where  $MVN(\pmb{B}_{j\ell}; \mathbf{0}, \gamma \pmb{I}_{K_j}, \pmb{I}_{K_\ell})$  is a matrix-variate normal distribution with row and column covariance matrices as  $\gamma \pmb{I}_{K_j}$  and  $\pmb{I}_{K_\ell}$ , respectively. We assume a conjugate inverse-gamma prior on the causal effect size,  $\gamma \sim \text{InverseGamma}(a_\gamma, b_\gamma)$ . We choose  $a_\gamma = b_\gamma = 1$ . We fix  $s = 0.02$  so that when  $E_{j\ell} = 0$ ,  $\pmb{B}_{j\ell}$  is negligibly small.
Priors on the parameters of the Gaussian mixture distribution. We choose conjugate priors for the parameters of the Gaussian mixture distribution:
$$
\begin{array}{l} \left(\pi_ {j k 1}, \dots , \pi_ {j k M _ {j k}}\right) \sim \operatorname {D i r i c h l e t} (\beta , \dots , \beta), \quad \forall j \in [ p ], k \in [ S ] \\ \mu_ {j k m} \sim N \left(a _ {\mu}, b _ {\mu}\right), \tau_ {j k m} \sim \text {I n v e r s e G a m m a} \left(a _ {\tau}, b _ {\tau}\right), \forall j \in [ p ], k \in [ S ], m \in [ M _ {j k} ] \\ \end{array}
$$
We have fixed values for the hyperparameters,  $\beta = 1, a_{\mu} = 0, b_{\mu} = 100, a_{\tau} = b_{\tau} = 1$
Prior on the noise variances. We assume a conjugate prior for  $\sigma_{j} \sim \text{InverseGamma}(a_{\sigma}, b_{\sigma})$ ,  $\forall j \in [p]$ . We choose  $a_{\sigma} = b_{\sigma} = 0.01$ .
We simulate posterior samples through Markov chain Monte Carlo (MCMC). Details are given in Section B of the Supplementary Materials. Sensitivity analyses will be conducted to test the hyperparameters including  $(a_{\gamma}, b_{\gamma})$ ,  $(a_{\tau}, b_{\tau})$ ,  $(a_{\sigma}, b_{\sigma})$ ,  $s, R, S, M$  and  $\beta$ .
4 Simulation study
Data generation The data were simulated according to various combinations of sample size  $(n)$ , number of nodes  $(p)$ , and grid size  $(m_j = d\forall j\in [p])$  where  $n\in \{75,150,300\}$ ,  $p\in \{20,40,60\}$ , and  $d\in \{125,250\}$ . The grid evenly spans the unit interval  $[0,1]$ ; the results with unevenly spaced grids are presented in Section C of the Supplementary Materials. The true causal graph  $\mathcal{G}$  was generated randomly with edge formation probability  $2 / p$ . Given  $\mathcal{G}$ , each non-zero block  $B_{j\ell}$  of the causal effect matrix was generated from the standard matrix-variate normal distribution. We set the true number of basis functions to be  $K = 4$ . In order to generate  $K = 4$  orthonormal basis functions, we first simulated unnormalized basis functions by expanding them further with 6 cubic B-spline basis functions where the coefficients were drawn from the standard normal distribution and then empirically orthonormalized them. The basis coefficients  $\tilde{\alpha}$  were generated following (8) with the exogenous variables  $\tilde{\epsilon}_j$  drawn independently from Laplace distribution with location parameter  $\mu = 0$  and scale parameter  $b = 0.2$ . We have also considered other non-Gaussian distributions for the exogenous variables; the corresponding results are provided in Section C of the Supplementary Materials. Finally, noisy observations were simulated following (6) with the signal-to-noise ratio, i.e., the mean value of  $|Y_j^{(i)}(t)| / \sigma_j$  across all  $i$  and  $t$ , set to 5. Here, superscript  $(i)$  denotes the  $i$ th sample, where  $i\in [n]$ .
For the implementation of the proposed FENCE, we fixed the number of mixture components to be 10 and ran MCMC for 5,000 iterations (discarding the first 2,000 iterations as burn-in and retaining every 5th iteration after burn-in). The causal graph  $G$  was then estimated by using the median probability model [Barbieri and Berger, 2004], i.e., by thresholding the posterior probability of inclusion at 0.5.
Methods for comparison. We compared our method with fLiNG [Zhou et al., 2022b], a recently proposed directed acyclic graph (DAG) for multivariate functional data. Codes for Lee and Li [2022], Yang and Suzuki [2022] are not publicly available. Hence for more comparison, we considered two ad hoc two-step approaches. In the first step of both approaches, we obtained the basis coefficients by carrying out functional principal component analysis (fPCA) using the fdapace [Zhou et al., 2022c] package in R. Then in the second step, given the basis coefficients, we estimated causal graphs using existing causal discovery methods, (i) LiNGAM [Shimizu et al., 2006] (ii) PC [Spirtes and Glymour, 1991] and (iii) CCD [Richardson, 1996]; we call these three approaches fPCA-LiNGAM, fPCA-PC and fPCA-CCD respectively. Note that we did not use SEM-based cyclic discovery algorithm, LiNG-D [Lacerda et al., 2008] in the second step due to the unavailability of the code.
LiNGAM estimates a causal DAG based on the linear non-Gaussian assumption whereas PC generally returns only an equivalence class of DAGs based on conditional independence tests. CCD algorithm is a constraint-based causal discovery method, which yields an equivalence class of cyclic causal graphs. LiNGAM and PC are implemented in the pcalg package [Kalisch et al., 2018] in R. CCD algorithm is implemented in the py-tetrad [Ramsey et al., 2018] package in python.
Performance metrics. To assess the graph recovery performance, we calculated the true positive rate (TPR), false discovery rate (FDR), and Matthew's correlation coefficient (MCC). For TPR and MCC, higher is better, whereas lower FDR is better.
Results. Table 1 summarizes the results of 50 repeat simulations, demonstrating that the proposed FENCE model outperforms all competitors (fLiNG, fPCA-LiNGAM, and fPCA-CCD) across all combinations of  $n$ ,  $p$ , and  $d$ . We provide the results of fPCA-PC in Section D of the Supplementary Materials, which are similar to those of fPCA-LiNGAM. We favored fPCA-PC and fPCA-CCD by counting a non-invariant edge between two nodes as a true positive as long as the two nodes are adjacent in the true graph. The superiority of FENCE is not unexpected for three reasons. First, fLiNG, fPCA-LiNGAM, and fPCA-PC are not specifically designed for learning cyclic graphs. Second, two-step approaches like fPCA-LiNGAM, fPCA-PC, and fPCA-CCD do not necessarily capture the causally embedded space through the functional principal components. Third, although fPCA-CCD can handle cyclic graphs, it, being a two-step approach, fails to capture true functional dependencies. Overall, these findings provide strong evidence of the effectiveness of FENCE compared to existing methods.
Additional simulations. We considered additional simulation scenarios with unevenly spaced grids, general exogenous variable distributions, true acyclic graphs and data generated using non-linear SEM, and also conducted sensitivity analyses of FENCE with respect to several hyperparameters; the results are presented in Section C of the Supplementary Materials. The performance of the proposed method is consistently better than competing methods and is relatively robust with respect to hyperparameter choices.
Table 1: Comparison of performance of various methods under 50 replicates. Since LiNGAM is not applicable to cases where  $q > n$  with  $q = Kp$  being the total number of extracted basis coefficients across all functions, the results from those cases are not available and indicated by "-". The metrics reported are based on 50 repetitions are reported; standard deviations are given within the parentheses.
n	p	d	FENCE	fLiNG	fPCA-LINGAM	fPCA-CCD
TPR	FDR	MCC	TPR	FDR	MCC	TPR	FDR	MCC	TPR	FDR	MCC
75	20	125	0.85(0.09)	0.19(0.07)	0.88(0.05)	0.41(0.09)	0.79(0.05)	0.36(0.04)	0.35(0.19)	0.84(0.04)	0.11(0.08)	0.69(0.03)	0.41(0.04)	0.23(0.03)
75	40	125	0.79(0.08)	0.23(0.06)	0.86(0.04)	0.37(0.08)	0.82(0.06)	0.33(0.05)	-	-	-	0.73(0.02)	0.47(0.04)	0.21(0.05)
75	60	125	0.75(0.07)	0.27(0.05)	0.83(0.04)	0.34(0.07)	0.83(0.06)	0.32(0.04)	-	-	-	0.68(0.03)	0.61(0.05)	0.19(0.03)
150	20	125	0.88(0.07)	0.14(0.06)	0.89(0.05)	0.45(0.07)	0.75(0.06)	0.39(0.05)	0.28(0.22)	0.86(0.05)	0.08(0.09)	0.71(0.03)	0.42(0.03)	0.25(0.04)
150	40	125	0.81(0.07)	0.21(0.06)	0.87(0.05)	0.39(0.06)	0.79(0.05)	0.37(0.04)	0.35(0.22)	0.91(0.02)	0.08(0.06)	0.73(0.04)	0.47(0.05)	0.23(0.03)
150	60	125	0.79(0.06)	0.24(0.05)	0.86(0.04)	0.36(0.04)	0.80(0.06)	0.36(0.05)	-	-	-	0.72(0.05)	0.54(0.04)	0.22(0.02)
300	20	125	0.91(0.03)	0.09(0.04)	0.90(0.04)	0.51(0.04)	0.73(0.06)	0.41(0.04)	0.30(0.19)	0.84(0.05)	0.11(0.09)	0.81(0.03)	0.39(0.04)	0.26(0.03)
300	40	125	0.87(0.04)	0.15(0.05)	0.87(0.05)	0.47(0.05)	0.75(0.06)	0.38(0.05)	0.27(0.20)	0.91(0.02)	0.08(0.06)	0.77(0.03)	0.45(0.02)	0.24(0.03)
300	60	125	0.85(0.05)	0.17(0.03)	0.86(0.03)	0.45(0.05)	0.76(0.04)	0.38(0.03)	0.28(0.17)	0.91(0.05)	0.05(0.03)	0.72(0.03)	0.49(0.02)	0.22(0.03)
75	20	250	0.81(0.04)	0.23(0.02)	0.85(0.05)	0.39(0.07)	0.80(0.05)	0.39(0.04)	0.32(0.14)	0.82(0.03)	0.09(0.04)	0.67(0.03)	0.46(0.03)	0.22(0.04)
75	40	250	0.73(0.04)	0.28(0.05)	0.82(0.04)	0.35(0.04)	0.85(0.06)	0.33(0.05)	-	-	-	0.68(0.02)	0.51(0.04)	0.21(0.03)
75	60	250	0.67(0.03)	0.34(0.05)	0.79(0.04)	0.34(0.04)	0.85(0.03)	0.31(0.04)	-	-	-	0.63(0.04)	0.56(0.04)	0.19(0.04)
150	20	250	0.83(0.06)	0.17(0.05)	0.86(0.05)	0.46(0.07)	0.73(0.07)	0.42(0.05)	0.32(0.19)	0.79(0.05)	0.13(0.05)	0.73(0.04)	0.43(0.02)	0.24(0.03)
150	40	250	0.79(0.02)	0.26(0.06)	0.82(0.03)	0.41(0.05)	0.71(0.05)	0.40(0.03)	0.31(0.14)	0.81(0.02)	0.13(0.06)	0.71(0.03)	0.47(0.04)	0.23(0.03)
150	60	250	0.69(0.05)	0.31(0.05)	0.79(0.04)	0.43(0.03)	0.79(0.06)	0.43(0.05)	-	-	-	0.69(0.02)	0.52(0.03)	0.21(0.02)
300	20	250	0.86(0.02)	0.16(0.04)	0.85(0.04)	0.68(0.02)	0.77(0.07)	0.47(0.04)	0.45(0.13)	0.86(0.05)	0.17(0.09)	0.78(0.02)	0.44(0.06)	0.27(0.03)
300	40	250	0.79(0.08)	0.16(0.05)	0.84(0.06)	0.73(0.05)	0.71(0.06)	0.43(0.05)	0.39(0.16)	0.87(0.05)	0.16(0.07)	0.76(0.05)	0.49(0.06)	0.23(0.05)
300	60	250	0.76(0.05)	0.21(0.03)	0.80(0.03)	0.77(0.05)	0.74(0.03)	0.42(0.03)	0.28(0.17)	0.90(0.04)	0.13(0.04)	0.72(0.06)	0.53(0.03)	0.22(0.04)
5 Real data application
Brain EEG data. We demonstrate the proposed FENCE model on a brain EEG dataset from an alcoholism study [Zhang et al., 1995]. This dataset was earlier used to demonstrate functional undirected graphical models [Zhu et al., 2016, Qiao et al., 2019] and functional Bayesian network [Zhou et al., 2022b]. Data were initially obtained from 64 electrodes placed on subjects' scalp, which captured EEG signals at  $256\mathrm{Hz}$  (3.9 ms epoch) during a one-second period. The study consists of 122 subjects, out of which 77 are in the alcoholic group and 45 are in the control group. Each subject completed 120 trials. During each trial, the subject was exposed to either a single stimulus (a single picture) or two stimuli (a pair of pictures) shown on a computer monitor. We particularly focus on the EEG signals filtered at  $\alpha$  frequency bands between 8 and  $12.5\mathrm{Hz}$  using the eegfi1t function of the eeglab toolbox of Matlab as  $\alpha$  band signals are associated with inhibitory control [Knyazev, 2007]. Given that the EEG measurements were recorded from each subject over multiple trials, these measurements are not independent of each other due to the time dependency of the trials. Moreover, since the measurements were obtained under various stimuli, the signals may have been affected by different stimulus effects. To mitigate these issues, we calculated the average of the band-filtered EEG signals for each subject across all trials under a single stimulus, resulting in a single event-related potential curve per electrode per subject. By doing so, we eliminated the potential dependence between the measurements and the influence of different stimulus types. We performed separate analyses of the two groups to identify both the similarities and dissimilarities in their brain effective connectivity.
We conducted a Shapiro-Wilk normality test on the observed functions for each of the  $p = 64$  scalp positions at each of the  $m_j = 256 \forall j \in [p]$  time points to evaluate their Gaussianity. The results showed that for numerous combinations of scalp position and time point, the null hypothesis (which assumes that the observations are marginally Gaussian) was rejected. Thus, we conclude that the non-Gaussianity of the proposed model is appropriate. Next, for posterior inference, we ran MCMC for 20,000 iterations, discarded the first half as burn-in, and retained every 10th iteration after burn-in. The estimated causal graph by thresholding the posterior inclusion probability to 0.9 is given below in Figure 2.
Figure 2: Estimated causal brain connectivity from EEG records by FENCE with posterior probability of inclusion  $\geq 0.9$ , separately for the alcoholic (left) and control (right) group. The bi-directed edges are just directed cycles, i.e.,  $i \leftrightarrow j$  means  $i \rightarrow j$  and  $i \gets j$ .
Results. There are some interesting findings. First, for both groups (alcoholic and control), brain regions that are located in adjacent positions tend to be more connected than the brain regions that are far apart. Second, dense connectivity is observed in the frontal region of the brain in both groups, with multiple cycles being formed. Third, compared to the control group, the alcoholic group has more connectivity across the left parietal and occipital lobes. Fourth, the same cycle of Iz, Cz, and RPA is observed in both groups.
Validity. We now discuss the validity of our real data results. In Hayden et al., 2007, it was observed that alcohol-dependent subjects exhibited frontal asymmetry, distinguishing them from the control group. Our own investigation aligns well with these results, as we have identified denser connectivity across various brain regions in the middle and left areas of the frontal lobe among alcoholic subjects, when compared to controls. Furthermore, Winterer et al., 2003 documented coherent differences between alcoholics and controls in the posterior hemispheres, specifically in the temporal, parietal, and occipital lobes. In accordance with their findings, our study provides additional support for this claim, as we have observed heightened activity with several cycles formed in those same regions within the alcoholic group when compared to the control group.
We briefly highlight here several potential avenues for future development of our current work. First, an intriguing and important direction would be to explore the relaxation of the causal sufficiency assumption in the model identifiability. Second, our current model is based on a linear non-Gaussian assumption over the exogenous variables, but a nonlinear model could be considered as an alternative. Lastly, an alternative approach to determining the effective number of basis functions that span the lower-dimensional causal embedded space would be to utilize the ordered shrinkage priors [Bhattacharya and Dunson, 2011, Legramanti et al., 2020] in order to adaptively eliminate redundant components, resulting in a more flexible methodology.
Ni's research was partially supported by NSF DMS-2112943 and NIH 1R01GM148974-01.
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