SlowGuess/NeurIPS_2023_OCR huggingface dataset:
The proof of Lemma 3.3 can be found in Appendix A.4. From a geometric perspective,  $\gamma_{j}$  can be decomposed into a linear combination of  $Q_{k}$ 's
$$
\gamma_ {j} = Q \tilde {\gamma} _ {j} = \sum_ {k = 1} ^ {K} Q _ {k} w _ {j, k}.
$$
Let  $L_{k} \coloneqq \operatorname{span}(Q_{k})$ . According to [43, Theorem 2.2], each  $L_{k}$  is an  $A$ -invariant subspace of  $\mathbb{R}^d$ . Recall that a space  $L$  is called  $A$ -invariant, if for all  $\gamma \in L$ ,  $A\gamma \in L$ . We say  $L$  is a proper subspace of  $\mathbb{R}^d$  if  $L \subset \mathbb{R}^d$  and  $L \neq \mathbb{R}^d$ . If  $|w_{j,k}| = 0$  (i.e.,  $w_{j,k} = 0$  in  $\mathbb{R}^1$  or  $\mathbb{R}^2$ ), then  $\gamma_{j}$  does not contain any information from  $L_{k}$ . In this case,  $\gamma_{j}$  is contained in an  $A$ -invariant proper subspace of  $\mathbb{R}^d$  that excludes  $L_{k}$ , denoted as  $L_{-k}$ . It is worth emphasizing that  $L_{-k} \subset \mathbb{R}^d$  is indeed a proper subspace of  $\mathbb{R}^d$ . This further implies that the trajectory of the ODE (13) generated from initial state  $\gamma_{j}$  is confined to  $L_{-k}$  [57, Lemma 3.2]. Lemma 3.3 indicates that if  $\mathrm{rank}([ \gamma_1 | A\gamma_1 | \ldots | A^{d - 1}\gamma_1 | \ldots | \gamma_n | A\gamma_n | \ldots | A^{d - 1}\gamma_n ]) < d$  then all  $\gamma_{j}$  for  $j = 1, \ldots, n$  are confined to an  $A$ -invariant proper subspace of  $\mathbb{R}^d$ , denoted as  $L$ . Therefore, all trajectories of the ODE (13) generated from initial states  $\gamma_{j}$  are also confined to  $L$ . Furthermore, based on the identifiability conditions proposed in [57], the ODE (13) is not identifiable from observational data collected in these trajectories. This lemma provides an approach to interpreting our identifiability conditions from a geometric perspective.
Now we are ready to present our main theorem.
Theorem 3.4. Let  $x_0 \in \mathbb{R}^d$  be fixed. Assuming that the matrix  $A$  in the SDE (1) has  $d$  distinct eigenvalues. The generator of the SDE (1) is identifiable from  $x_0$  if and only if
$$
\operatorname {r a n k} \left( \right.\left[ \right. x _ {0} \mid A x _ {0} \left. \right| \dots \mid A ^ {d - 1} x _ {0} \mid H _ {. 1} \mid A H _ {. 1} \mid \dots \mid A ^ {d - 1} H _ {. 1} \mid \dots \mid H _ {. d} \mid A H _ {. d} \mid \dots \mid A ^ {d - 1} H _ {. d} \left. \right]\left. \right) = d, \tag {14}
$$
where  $H\coloneqq GG^T$  , and  $H_{j}$  stands for the  $j$  -th column vector of matrix  $H,$  for all  $j = 1,\dots ,d$
The proof of Theorem 3.4 can be found in Appendix A.5. The condition in Theorem 3.4 is both sufficient and necessary when the matrix  $A$  has distinct eigenvalues. It is worth noting that almost every  $A \in \mathbb{R}^{d \times d}$  has  $d$  distinct eigenvalues concerning the Lebesgue measure on  $\mathbb{R}^{d \times d}$ . Hence, this condition is both sufficient and necessary for almost every  $A$  in  $\mathbb{R}^{d \times d}$ . However, in cases where  $A$  has repetitive eigenvalues, this condition is solely sufficient and not necessary.
Remark. The identifiability condition stated in Theorem 3.4 is generic, that is, let
$$
S := \left\{\left(x _ {0}, A, G\right) \in \mathbb {R} ^ {d + d ^ {2} + d m}: \text {c o n d i t i o n (1 4) i s v i o l a t e d} \right\},
$$
$S$  has Lebesgue measure zero in  $\mathbb{R}^{d + d^2 + dm}$ . Refer to Appendix B.2 for the detailed proof.
From the geometric perspective, suppose matrix  $A$  has distinct eigenvalues, the generator of the SDE (1) is identifiable from  $x_0$  when not all of the vectors:  $x_0, H_{\cdot 1}, \ldots, H_{\cdot d}$  are confined to an  $A$ -invariant proper subspace of  $\mathbb{R}^d$ . A key finding is that when all the vectors  $H_{\cdot j}, j = 1, \ldots, d$  are confined to an  $A$ -invariant proper subspace  $L$  of  $\mathbb{R}^d$ , each column of the covariance matrix  $V(t)$  in Equation (12) is also confined to  $L$ , for all  $0 \leqslant t < \infty$ . Thus, the identifiability of the generator of the SDE (1) can be fully determined by  $x_0$  and the system parameters  $(A, GG^{\top})$ . Further details can be found in the proof A.5 of Theorem 3.4.
By rearranging the matrix in (14), the identifiability condition can also be expressed as
$$
\operatorname {r a n k} \left(\left[ x _ {0} \right| A x _ {0} | \dots | A ^ {d - 1} x _ {0} | G G ^ {\top} | A G G ^ {\top} | \dots | A ^ {d - 1} G G ^ {\top} ]\right) = d. \tag {15}
$$
Based on the identifiability condition (15), we derive the following corollary.
Corollary 3.4.1. Let  $x_0 \in \mathbb{R}^d$  be fixed. If  $\text{rank}([G|AG] \ldots |A^{d-1}G]) = d$ , then the generator of the SDE (1) is identifiable from  $x_0$ .
The proof of Corollary 3.4.1 can be found in Appendix A.6. This corollary indicates that the generator of the SDE (1) is identifiable from any initial state  $x_0 \in \mathbb{R}^d$  when the pair  $[A, G]$  is controllable (rank  $([G|AG| \ldots |A^{d-1}G]) = d$ ). Notably, this identifiability condition is stricter than that proposed in Theorem 3.4, as it does not use the information of  $x_0$ .
3.3 Conditions for identifying generators of linear SDEs with multiplicative noise
Expressing the SDE (2) in the form given by (5) yields  $b(x) = Ax$  and  $\sigma(x) = [G_1x|\ldots|G_mx] \in \mathbb{R}^{d \times m}$ , thus,  $c(x) = \sigma(x)\sigma(x)^\top = \sum_{k=1}^m G_kxx^\top G_k^\top$ . Let  $X(t;x_0,A,\{G_k\}_{k=1}^m)$  denote the solution to the SDE (2), then based on Proposition 3.1, we define the identifiability of the generator of the SDE (2) as follows.
Definition 3.2  $((x_0, A, \{G_k\}_{k=1}^m)$ -identifiability). For  $x_0 \in \mathbb{R}^d$ ,  $A, G_k \in \mathbb{R}^{d \times d}$  for all  $k = 1, \ldots, m$ , the generator of the SDE (2) is said to be identifiable from  $x_0$ , if for all  $\tilde{A}, \tilde{G}_k \in \mathbb{R}^{d \times d}$ , there exists an  $x \in \mathbb{R}^d$ , such that  $(A, \sum_{k=1}^m G_k xx^\top G_k^\top) \neq (\tilde{A}, \sum_{k=1}^m \tilde{G}_k xx^\top \tilde{G}_k^\top)$ , it holds that  $X(\cdot; x_0, A, \{G_k\}_{k=1}^m) \stackrel{\mathrm{d}}{\neq} X(\cdot; x_0, \tilde{A}, \{\tilde{G}_k\}_{k=1}^m)$ .
Based on Definition 3.2, we present the identifiability condition for the generator of the SDE (2).
Theorem 3.5. Let  $x_0 \in \mathbb{R}^d$  be fixed. The generator of the SDE (2) is identifiable from  $x_0$  if the following conditions are satisfied:
A1 rank([  $x_0|Ax_0|\ldots |A^{d - 1}x_0]$ ) = d,
A2 rank([v|Av|...|A $(d^{2} + d - 2) / 2$  v]）=  $(d^2 +d) / 2$
where  $\mathcal{A} = A\oplus A + \sum_{k = 1}^{m}G_{k}\otimes G_{k}\in \mathbb{R}^{d^{2}\times d^{2}}$ $\oplus$  denotes Kronecker sum and  $\otimes$  denotes Kronecker product,  $v$  is a  $d^2$ -dimensional vector defined by  $v\coloneqq \operatorname {vec}(x_0x_0^\top)$ , where  $\operatorname {vec}(M)$  denotes the vectorization of matrix  $M$ .
The proof of Theorem 3.5 can be found in Appendix A.7. This condition is only sufficient but not necessary. Specifically, condition A1 guarantees that matrix  $A$  is identifiable, and once  $A$  is identifiable, condition A2 ensures that the identifiability of  $\sum_{k=1}^{m} G_k xx^\top G_k^\top$  holds for all  $x \in \mathbb{R}^d$ .
Remark. The identifiability condition stated in Theorem 3.5 is generic, that is, let
$S \coloneqq \{(x_0, A, \{G_k\}_{k=1}^m) \in \mathbb{R}^{d+(m+1)d^2} : \text{either condition A1 or A2 in Theorem 3.5 is violated}\}$ ,
$S$  has Lebesgue measure zero in  $\mathbb{R}^{d + (m + 1)d^2}$ . This signifies that the conditions are satisfied for most of the combinations of  $x_0$ ,  $A$  and  $G_{k}$ 's, except for those that lie in a set of Lebesgue measure zero. The corresponding proposition and detailed proof can be found in Appendix B.1.
Since obtaining an explicit solution for the SDE (2) is generally infeasible, we resort to utilizing the first- and second-order moments of this SDE to derive the identifiability conditions. Let  $m(t) \coloneqq \mathbb{E}[X_t]$  and  $P(t) \coloneqq \mathbb{E}[X_t X_t^\top]$ , it is known that these moments satisfy ODE systems. Specifically,  $m(t)$  satisfies the ODE (13), while  $P(t)$  satisfies the following ODE (cf. [56]):
$$
\dot {P} (t) = A P (t) + P (t) A ^ {\top} + \sum_ {k = 1} ^ {m} G _ {k} P (t) G _ {k} ^ {\top}, \quad P (0) = x _ {0} x _ {0} ^ {\top}. \tag {16}
$$
An important trick to deal with the ODE (16) is to vectorize  $P(t)$ , then it can be expressed as:
$$
\operatorname {v e c} (\dot {P} (t)) = \mathcal {A} \operatorname {v e c} (P (t)), \quad \operatorname {v e c} (P (0)) = v, \tag {17}
$$
where  $\mathcal{A}$  and  $v$  are defined in Theorem 3.5. In fact, the ODE (17) follows the same mathematical structure as that of the ODE (13), which is known as homogeneous linear ODEs. Thus, in addition to the inherent properties of the SDE (2), we also employ some existing identifiability theories for homogeneous linear ODEs to establish the identifiability condition for the generator of the SDE (2).
From the geometric perspective, condition A1 indicates that the initial state  $x_0$  is not confined to an  $A$ -invariant proper subspace of  $\mathbb{R}^d$  [57, Lemma 3.1]. And condition A2 implies that the vectorization of  $x_0x_0^\top$  is not confined to an  $A$ -invariant proper subspace of  $W$ , with  $W \subset \mathbb{R}^{d^2}$ , and  $\dim(W) = (d^2 + d)/2$ , where  $\dim(W)$  denotes the dimension of the subspace  $W$ , that is the number of vectors in any basis for  $W$ . In particular, one can construct a basis for  $W$  as follows:
$$
\left\{\operatorname {v e c} \left(E _ {1 1}\right), \operatorname {v e c} \left(E _ {2 1}\right), \operatorname {v e c} \left(E _ {2 2}\right), \dots , \operatorname {v e c} \left(E _ {d d}\right) \right\},
$$
where  $E_{ij}$  denotes a  $d \times d$  matrix whose  $ij$ -th and  $ji$ -th elements are 1, and all other elements are 0, for all  $i,j = 1,\ldots,d$  and  $i \geq j$ . Refer to the proof A.7 of Theorem 3.5 for more details.
4 Simulations and examples
In order to assess the validity of the identifiability conditions established in Section 3, we present the results of simulations. Specifically, we consider SDEs with system parameters that either satisfy or violate the proposed identifiability conditions. We then apply the maximum likelihood estimation (MLE) method to estimate the system parameters from discrete observations sampled from the corresponding SDE. The accuracy of the resulting parameter estimates serves as an indicator of the validity of the proposed identifiability conditions.
Simulations. We conduct five sets of simulations, which include one identifiable case and one unidentifiable case for the SDE (1), and one identifiable case and two unidentifiable cases with either condition A1 or A2 in Theorem 3.5 unsatisfied for the SDE (2). We set both the system dimension,  $d$ , and the Brownian motion dimension,  $m$ , to 2. Details on the true underlying system parameters for the SDEs can be found in Appendix C. We simulate observations from the true SDEs for each of the five cases under investigation. Specifically, the simulations are carried out for different numbers of trajectories  $(N)$ , with 50 equally-spaced observations sampled on each trajectory from the time interval  $[0, 1]$ . We employ the Euler-Maruyama (EM) method [34], a widely used numerical scheme for simulating SDEs, to generate the observations.
Estimation. We use MLE [38, 50] to estimate the system parameters. The MLE method requires knowledge of the transition probability density function (pdf) that governs the evolution of the system. For the specific case of the SDE (1), the transition density follows a Gaussian distribution, which can be computed analytically based on the system's drift and diffusion coefficients (cf. [50]). To compute the covariance, we employ the commonly used matrix fraction decomposition method [4, 49, 50]. However, in general, the transition pdf of the SDE (2) cannot be obtained analytically due to the lack of a closed-form solution. To address this issue, we implement the Euler-Maruyama approach [32, 34], which has been shown to be effective in approximating the transition pdf of SDEs.
Metric. We adopt the commonly used metric, mean squared error (MSE), to assess the accuracy of the parameter estimates. To ensure reliable estimation outcomes, we perform 100 independent random replications for each configuration and report the mean and variance of their MSEs.
Results analysis. Table 1 and Table 2 present the simulation results for the SDE (1) and the SDE (2), respectively. In Table 1, the simulation results demonstrate that in the identifiable case, as the number of trajectories  $N$  increases, the MSE for both  $A$  and  $GG^{\top}$  decreases and approaches zero. However, in the unidentifiable case, where the identifiable condition (14) stated in Theorem 3.4 is not
satisfied, the MSE for both  $A$  and  $GG^{\top}$  remains high regardless of the number of trajectories. These findings provide strong empirical evidence supporting the validity of the identifiability condition proposed in Theorem 3.4. The simulation results presented in Table 2 show that in the identifiable case, the MSE for both  $A$  and  $Gsx$  decreases and approaches zero with the increase of the number of trajectories  $N$ . Here,  $Gsx := \sum_{k=1}^{m} G_k xx^{\top} G_k^{\top}$ , where  $x$  is a randomly generated vector from  $\mathbb{R}^2$  (in these simulations,  $x = [1.33, 0.72]^{\top}$ ). Interestingly, even in unidentifiable case 1, the MSE for both  $A$  and  $Gsx$  decreases with an increasing number of trajectories  $N$ , indicating that the generator of the SDE utilized in this particular case is still identifiable, although a larger number of trajectories is required compared to the identifiable case to achieve the same level of accuracy. This result is reasonable, because it aligns with our understanding that condition A1 is only sufficient but not necessary for identifying  $A$ , as the lack of an explicit solution for the SDE (2) results in condition A1 not incorporating any information from  $G_k$ 's. The identifiability condition derived for the SDE (1) in Theorem 3.4 leverages the information of  $G$ , similarly, if information regarding  $G_k$ 's is available, a weaker condition for identifying  $A$  could be obtained. For illustration, in Appendix E, we present such a condition assuming the SDE (2) has a closed-form solution. In the case of unidentifiable case 2, the MSE for  $A$  decreases with an increasing number of trajectories  $N$ ; however, the MSE for  $Gsx$  remains high, indicating that  $A$  is identifiable, while  $Gsx$  is not, albeit requiring more trajectories compared to the identifiable case to achieve the same level of accuracy of  $A$  (since the  $Gsx$  is far away from its true underlying value). This finding is consistent with the derived identifiability condition, as condition A1 is sufficient to identify  $A$ , whereas condition A2 governs the identifiability of  $Gsx$ . Worth noting that in cases where neither condition A1 nor condition A2 is satisfied, the estimated parameters barely deviate from their initial values, implying poor estimation of both  $A$  and  $Gsx$ . These results indicate the validity of the identifiability condition stated in Theorem 3.5.
Illustrative instances of causal inference for linear SDEs (with interventions). To illustrate how our proposed identifiability conditions can guarantee reliable causal inference for linear SDEs, we present examples corresponding to both the SDE (1) and the SDE (2). In these examples, we show that under our proposed identifiability conditions, the post-intervention distributions are identifiable from their corresponding observational distributions. Please refer to Appendix D.1 and D.2 for the details of the examples.
Most current studies on the identifiability analysis of SDEs are based on the Gaussian diffusion processes that conform to the form described in the SDE (1). In particular, the authors of [27, 28, 42]
have conducted research on the identifiability or asymptotic properties of parameter estimators of Gaussian diffusions in view of continuous observations of one trajectory, and have highlighted the need for the diffusion to be ergodic. A considerable amount of effort has also been directed towards the identifiability analysis of Gaussian diffusions, relying on the exact discrete models of the SDEs [6, 16, 23, 35, 41]. Typically, these studies involve transferring the continuous-time system described in the SDE (1) to a discrete-time model such as a vector autoregressive model, based on equally-spaced observations sampled from one trajectory, and then attempting to determine conditions under which  $(A, GG^{\top})$  is identifiable from the parameters of the corresponding exact discrete models. These conditions often have requirements on eigenvalues of  $A$  among other conditions, such as requiring the eigenvalues to have only negative real parts, or the eigenvalues to be strictly real. Due to the limitation of the available observations (continuous or discrete observations located on one trajectory of the SDE system), the identifiability conditions proposed in these works are restrictive.
Causal modelling theories have been well-developed based on directed acyclic graphs (DAGs), which do not explicitly incorporate a time component [39]. In recent years, similar concepts of causality have been developed for dynamic systems operating in both discrete and continuous time. Discrete-time models, such as autoregressive processes, can be readily accommodated within the DAG-based framework [13, 14]. On the other hand, differential equations offer a natural framework for understanding causality in dynamic systems within the context of continuous-time processes [1, 52]. Consequently, considerable effort has been devoted to establishing a theoretical connection between causality and differential equations. In the deterministic case, Mooij et al. [37] and Rubenstein et al. [46] have established a mathematical link between ODEs and structural causal models (SCMs). Wang et al. [60] have proposed a method to infer the causal structure of linear ODEs. Turning to the stochastic case, Boegers and Mooij have built a bridge from random differential equations (RDEs) to SCMs [7], while Hansen and Sokol have proposed a causal interpretation of SDEs by establishing a connection between SDEs and SEMs [17].
In this paper, we present an investigation into the identifiability of the generators of linear SDEs under additive and multiplicative noise. Specifically, we derive the conditions that are fully built on system parameters and the initial state  $x_0$ , which enables the identification of a linear SDE's generator from the distribution of its solution process with a given fixed initial state. We establish that, under the proposed conditions, the post-intervention distribution is identifiable from the corresponding observational distribution for any Lipschitz intervention  $\zeta$ .
The main limitation of our work is that the practical verification of these identifiability conditions poses a challenge, as the true underlying system parameters are typically unavailable in real-world applications. Nevertheless, our study contributes to the understanding of the intrinsic structure of linear SDEs. By offering valuable insights into the identifiability aspects, our findings empower researchers and practitioners to employ models that satisfy the proposed conditions (e.g., through constrained parameter estimation) to learn real-world data while ensuring identifiability. We believe the paramount significance of this work lies in providing a systematic and rigorous causal interpretation of linear SDEs, which facilitates reliable causal inference for dynamic systems governed by such equations. It is worth noting that in our simulations, we employed the MLE method to estimate the system parameters. This necessitates the calculation of the transition pdf from one state to the successive state at each discrete temporal increment. Consequently, as the state dimension and Brownian motion dimension increase, the computational time is inevitably significantly increased, rendering the process quite time-consuming. To expedite parameter estimation for scenarios involving high dimensions, alternative estimation approaches are required. The development of a more efficient parameter estimation approach remains an important task in the realm of SDEs, representing a promising direction for our future research. We claim that this work does not present any foreseeable negative social impact.
YW was supported by the Australian Government Research Training Program (RTP) Scholarship from the University of Melbourne. XG was supported by ARC DE210101352. MG was supported by ARC DE210101624.
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