Title: Generator Identification for Linear SDEs with Additive and Multiplicative Noise
Venue: NeurIPS 2023
Accepted: yes
Keywords: Linear SDE

Identification

Causal inference
Abstract: In this paper, we present conditions for identifying the generator of a linear stochastic differential equation (SDE) from the distribution of its solution process with a given fixed initial state. These identifiability conditions are crucial in causal inference using linear SDEs as they enable the identification of the post-intervention distributions from its observational distribution. Specifically, we derive a sufficient and necessary condition for identifying the generator of linear SDEs with additive noise, as well as a sufficient condition for identifying the generator of linear SDEs with multiplicative noise. We show that the conditions derived for both types of SDEs are generic. Moreover, we offer geometric interpretations of the derived identifiability conditions to enhance their understanding. To validate our theoretical results, we perform a series of simulations, which support and substantiate the established findings.

Generator Identification for Linear SDEs with Additive and Multiplicative Noise

Abstract

In this paper, we present conditions for identifying the generator of a linear stochastic differential equation (SDE) from the distribution of its solution process with a given fixed initial state. These identifiability conditions are crucial in causal inference using linear SDEs as they enable the identification of the post-intervention distributions from its observational distribution. Specifically, we derive a sufficient and necessary condition for identifying the generator of linear SDEs with additive noise, as well as a sufficient condition for identifying the generator of linear SDEs with multiplicative noise. We show that the conditions derived for both types of SDEs are generic. Moreover, we offer geometric interpretations of the derived identifiability conditions to enhance their understanding. To validate our theoretical results, we perform a series of simulations, which support and substantiate the established findings.

1 Introduction

Stochastic differential equations (SDEs) are a powerful mathematical tool for modelling dynamic systems subject to random fluctuations. These equations are widely used in various scientific disciplines, including finance [11, 30, 40], physics [53, 55, 58], biology [2, 8, 61] and engineering [18, 44, 55]. In recent years, SDEs have garnered growing interest in the machine learning research community. Specifically, they have been used for tasks such as modelling time series data [19, 21, 33] and estimating causal effects [5, 36, 47].

To enhance understanding we first introduce the SDEs of our interest, which are multidimensional linear SDEs with additive and multiplicative noise, respectively. Consider an m -dimensional standard Brownian motion defined on a filtered probability space (Ω , F , P , {F t } ) , denoted by W := { W t = [ W 1 ,t , . . . , W m,t ] ⊤ : 0 ⩽ t < ∞} . Let X t ∈ R d be the state at time t and let x 0 ∈ R d be a constant vector denoting the initial state of the system, we present the forms of the aforementioned two linear SDEs.

Linear SDEs with additive noise.

2.

Linear SDEs are wildly used in financial modeling for tasks like asset pricing, risk assessment, and portfolio optimization [3, 10, 24]. Where they are used to model the evolution of financial variables, such as stock prices and interest rates. Furthermore, linear SDEs are also used in genomic research, for instance, they are used for modeling the gene expression in the yeast microorganism Saccharomyces Cerevisiae [17]. The identifiability analysis of linear SDEs is essential for reliable causal inference of dynamic systems governed by these equations. For example, in the case of Saccharomyces Cerevisiae, one aims to identify the system such that making reliable causal inference when interventions are introduced. Such interventions may involve deliberate knockout of specific genes to achieve optimal growth of an organism. In this regard, identifiability analysis plays a pivotal role in ensuring reliable predictions concerning the impact of interventions on the system.

Previous studies on identifiability analysis of linear SDEs have primarily focused on Gaussian diffusions, as described by the SDE (1) [6, 16, 23, 28, 35, 42]. These studies are typically based on observations located on one trajectory of the system and thus require restrictive identifiability conditions, such as the ergodicity of the diffusion or other restrictive requirements on the eigenvalues of matrix A . However, in practical applications, multiple trajectories of the dynamic system can often be accessed [15, 31, 45, 54]. In particular, these multiple trajectories may start from the same initial state, e.g., in experimental studies where repeated trials or experiments are conducted under the same conditions [9, 12, 26, 29] or when the same experiment is performed on multiple identical units [48]. To this end, this work presents an identifiability analysis for linear SDEs based on the distribution of the observational process with a given fixed initial state. Furthermore, our study is not restricted to Gaussian diffusions (1), but also encompasses linear SDEs with multiplicative noise (2). Importantly, the conditions derived for both types of SDEs are generic, meaning that the set of system parameters that violate the proposed conditions has Lebesgue measure zero.

Traditional identifiability analysis of dynamic systems focuses on deriving conditions under which a unique set of parameters can be obtained from error-free observational data. However, our analysis of dynamic systems that are described by SDEs aims to uncover conditions that would enable a unique generator to be obtained from its observational distribution. Our motivation for identifying generators of SDEs is twofold. Firstly, obtaining a unique set of parameters from the distribution of a stochastic process described by an SDE is generally unfeasible. For example, in the SDE (1), parameter G cannot be uniquely identified since one can only identify GG ⊤ based on the distribution of its solution process [17, 28]. Secondly, the identifiability of an SDE's generator suffices for reliable causal inferences for this system. Note that, in the context of SDEs, the main task of causal analysis is to identify the post-intervention distributions from the observational distribution. As proposed in [17], for an SDE satisfying specific criteria, the post-intervention distributions are identifiable from the generator of this SDE. Consequently, the intricate task of unraveling causality can be decomposed into two constituent components through the generator. This paper aims to uncover conditions under which the generator of a linear SDE attains identifiability from the observational distribution. By establishing these identifiability conditions, we can effectively address the causality task for linear SDEs.

In this paper, we present a sufficient and necessary identifiability condition for the generator of linear SDEs with additive noise (1), along with a sufficient identifiability condition for the generator of linear SDEs with multiplicative noise (2).

2 Background knowledge

In this section, we introduce some background knowledge of linear SDEs. In addition, we provide a concise overview of the causal interpretation of SDEs, which is a critical aspect of understanding the nature and dynamics of these equations. This interpretation also forms a strong basis for the motivation of this research.

2.1 Background knowledge of linear SDEs

The solution to the SDE (1) can be explicitly expressed as (cf. [50]):

Note that in the context of our study, the solution stands for the strong solution, refer to [22] for its detailed definition.

In general, obtaining an explicit expression for the solution to the SDE (2) is not feasible. In fact, an explicit solution can be obtained when the matrices A,G 1 , . . . , G k commute, that is when

holds for all k, l = 1 , . . . , m (cf. [25]). However, the conditions described in (4) are too restrictive and impractical. Therefore, this study will focus on the general case of the SDE (2).

We know that both the SDE (1) and the SDE (2) admit unique solutions that manifest as continuous stochastic processes [22]. A d -dimensional stochastic process is a collection of R d -valued random variables, denoted as X = { X t ; 0 ⩽ t < ∞} defined on some probability space. When comparing two stochastic processes, X and ˜ X , that are defined on the same probability space (Ω , F , P ) , various notions of equality may be considered. In this study, we adopt the notion of equality with respect to their distributions, which is a weaker requirement than strict equivalence, see [22] for relevant notions. We now present the definition of the distribution of a stochastic process.

Definition 2.1. Let X be a random variable on a probability space (Ω , F , P ) with values in a measurable space ( S, B ( S )) , i.e., the function X : Ω → S is F / B ( S ) -measurable. Then, the distribution of the random variable X is the probability measure P X on ( S, B ( S )) given by

When X := { X t ; 0 ⩽ t < ∞} is a continuous stochastic process on (Ω , F , P ) , and S = C [0 , ∞ ) , such an X can be regarded as a random variable on (Ω , F , P ) with values in ( C [0 , ∞ ) , B ( C [0 , ∞ ))) , and P X is called the distribution of X . Here C [0 , ∞ ) stands for the space of all continuous, realvalued functions on [0 , ∞ ] .

It is noteworthy that the distribution of a continuous process can be uniquely determined by its finite-dimensional distributions. Hence, if two stochastic processes, labelled as X and ˜ X , share identical finite-dimensional distributions, they are regarded as equivalent in distribution, denoted by X d = ˜ X . Relevant concepts and theories regarding this property can be found in [22].

The generator of a stochastic process is typically represented by a differential operator that acts on functions. It provides information about how a function evolves over time in the context of the underlying stochastic process. Mathematically, the generator of a stochastic process X t can be defined as

where f is a suitably regular function.

In the following, we present the generator of the SDEs under consideration. Obviously, both the SDE (1) and the SDE (2) conform to the general form:

where b and σ are locally Lipschitz continuous in the space variable x . The generator L of the SDE (5) can be explicitly computed by utilizing Itô's formula (cf. [50]).

Proposition 2.1. Let X be a stochastic process defined by the SDE (5) . The generator L of X on C 2 b ( R d ) is given by

for f ∈ C 2 b ( R d ) and x ∈ R d , where c ( x ) = σ ( x ) · σ ( x ) ⊤ is a d × d matrix, and C 2 b ( R d ) denotes the space of continuous functions on R d that have bounded derivatives up to order two.

2.2 Causal interpretation of SDEs

An important motivation for the identification of the generator of an SDE lies in the desire to infer reliable causality within dynamic models described by SDEs. In this subsection, we aim to provide some necessary background knowledge on the causal interpretation of SDEs. Consider the general SDE framework described as:

where Z is a p -dimensional semimartingale and a : R d → R d × p is a continuous mapping. By writing the SDE (7) in integral form

The authors of [17] proposed a mathematical definition of the SDE resulting from an intervention to the SDE (8). In the following, X ( -l ) denotes the ( d -1) -dimensional vector that results from the removal of the l -th coordinate of X ∈ R d .

Definition 2.2. [17, Definition 2.4.] Consider some l ⩽ d and ζ : R d -1 → R . The SDE arising from (8) under the intervention X l t := ζ ( X ( -l ) t ) is the ( d -1) -dimensional equation

̸

̸

where b : R d -1 → R ( d -1) × p is defined by b ij ( y ) = a ij ( y 1 , . . . , ζ ( y ) , . . . , y d ) for i = l and j ⩽ p and the ζ ( y ) is on the l -th coordinate.

Definition 2.2 presents a natural approach to defining how interventions should affect dynamic systems governed by SDEs. We adopt the same notations as used in [17]. Assuming (8) and (9) have unique solutions for all interventions, we refer to (8) as the observational SDE, to its solution as the observational process, to the distribution of its solution as observational distribution, to (9) as the post-intervention SDE, to the solution of (9) as the post-intervention process, and to the distribution of the solution of (9) as the post-intervention distribution. The authors in [17] related Definition 2.2 to mainstream causal concepts by establishing a mathematical connection between SDEs and structural equation models (SEMs). Specifically, the authors showed that under regularity assumptions, the solution to the post-intervention SDE is equal to the limit of a sequence of interventions in SEMs based on the Euler scheme of the observational SDE. Despite the fact that the parameters of the SDEs are generally not identifiable from the observational distribution, the post-intervention distributions can be identified, thus enabling causal inference of the system. To this end, Sokol and Hansen [17] derived a condition under which the generator associated with the observational SDE allows for the identification of the post-intervention distributions. We present the corresponding theory as follows.

Lemma 2.1. [17, Theorem 5.3.] Consider the SDEs

where Z is a p -dimensional Lévy process and ˜ Z is a ˜ p -dimensional Lévy process. Assume that (10) and (11) have the same generator, that a : R d → R d × p and ζ : R d -1 → R are Lipschitz and that the initial values have the same distribution. Then the post-intervention distributions of doing X l := ζ ( X ( -l ) ) in (10) and doing ˜ X l := ζ ( ˜ X ( -l ) ) in (11) are equal for any choice of ζ and l .

A main task in the causality research community is to uncover the conditions under which the post-intervention distributions are identifiable from the observational distribution. In the context of dynamic systems modelled in SDEs, similar conditions need to be derived. Lemma 2.1 establishes that, for SDEs with a Lévy process as the driving noise, the post-intervention distributions can be identifiable from the generator. Nevertheless, a gap remains between the observational distribution and the SDE generator's identifiability. This work aims to address this gap by providing conditions under which the generator is identifiable from the observational distribution.

3 Main results

In this section, we present some prerequisites first, and then we present the main theoretical results of our study, which include the condition for the identifiability of generator that is associated with the SDE (1) / SDE (2) from the distribution of the corresponding solution process.

3.1 Prerequisites

We first show that both the SDE (1) and the SDE (2) satisfy the conditions stated in Lemma 2.1.

Lemma 3.1. Both the SDE (1) and the SDE (2) can be expressed as the form of (10) , with Z being a p -dimensional Lévy process, and a : R d → R d × p being Lipschitz.

The proof of Lemma 3.1 can be found in Appendix A.1. This lemma suggests that Lemma 2.1 applies to both the SDE (1) and the SDE (2), given that they meet the specified conditions. Therefore, for either SDE, deriving the conditions that allow for the generator to be identifiable from the observational distribution is sufficient. By applying Lemma 2.1, when the intervention function ζ is Lipschitz, the post-intervention distributions can be identified from the observational distribution under these conditions.

We then address the identifiability condition of the generator L defined by (6).

Proposition 3.1. Let L and ˜ L be generators of stochastic processes defined by the form of the SDE (5) on C 2 b ( R d ) , where L is given by (6) and ˜ L is given by the same expression, with ˜ b ( x ) and ˜ c ( x ) substituted for b ( x ) and c ( x ) . It then holds that the two generators L = ˜ L if and only if b ( x ) = ˜ b ( x ) and c ( x ) = ˜ c ( x ) for all x ∈ R d .

The proof of Proposition 3.1 can be found in Appendix A.2. This proposition states that for stochastic processes defined by the SDE (5), the generator is identifiable from functions associated with its coefficients: b ( x ) and c ( x ) = σ ( x ) · σ ( x ) ⊤ .

3.2 Conditions for identifying generators of linear SDEs with additive noise

Expressing the SDE (1) in the form given by (5) yields b ( x ) = Ax and c ( x ) = GG ⊤ . Therefore, based on Proposition 3.1, we define the identifiability of the generator of the SDE (1) as follows.

Definition 3.1 ( ( x 0 , A, G ) -identifiability) . For x 0 ∈ R d , A ∈ R d × d and G ∈ R d × m , the generator of the SDE (1) is said to be identifiable from x , if for all ˜ A ∈ R d × d and all ˜ G ∈ R d × m , with

̸

0 d .

̸

In the following, we begin by introducing two lemmas that serve as the foundation for deriving our main identifiability theorem.

Lemma 3.2. For x 0 ∈ R d , A, ˜ A ∈ R d × d and G, ˜ G ∈ R d × m , let X t := X ( t ; x 0 , A, G ) , ˜ X t := X ( t ; x 0 , ˜ A, ˜ G ) , then X ( · ; x 0 , A, G ) d = X ( · ; x 0 , ˜ A, ˜ G ) if and only if the mean E [ X t ] = E [ ˜ X t ] and the covariance E { ( X t + h -E [ X t + h ])( X t -E [ X t ]) ⊤ } = E { ( ˜ X t + h -E [ ˜ X t + h ])( ˜ X t -E [ ˜ X t ]) ⊤ } for all 0 ⩽ t < ∞ and 0 ⩽ h < ∞ .

The proof of Lemma 3.2 can be found in Appendix A.3. This lemma states that for stochastic processes modelled by the SDE (1), the equality of the distribution of two processes can be deconstructed as the equality of the mean and covariance of the state variables at all time points. Calculation shows

where V ( t ) := V ( t, t ) . Please refer to the proof A.5 of Theorem 3.4 for the detailed calculations. It can be easily checked that E [ X t ] follows the linear ordinary differential equation (ODE)

where ˙ m ( t ) denotes the first derivative of function m ( t ) with respect to time t . Similarly, each column of the covariance V ( t, t + h ) also follows the linear ODE (13) but with a different initial state: the corresponding column of V ( t ) . This observation allows us to leverage not only the characteristics of the SDE (1), but also the established theories [43, 57] on identifiability analysis for the ODE (13), to derive the identifiability conditions for the generator of the SDE (1).

We adopt the same setting as in [43], discussing the case where A has distinct eigenvalues. Because random matrix theory suggests that almost every A ∈ R d × d has d distinct eigenvalues with respect to the Lebesgue measure on R d × d [43]. And the Jordan decomposition of such a matrix A follows a straightforward form which is helpful for deriving the geometric interpretation of the proposed identifiability condition. The Jordan decomposition can be expressed as A = Q Λ Q -1 , where

where λ k is a real eigenvalue of A and v k is the corresponding eigenvector of λ k , for k = 1 , . . . , K 1 . For k = K 1 + 1 , . . . , K , [ v 2 k -K 1 -1 | v 2 k -K 1 ] are the corresponding 'eigenvectors' of complex eigenvalues a k ± b k i . Inspired by [43, Definition 2.3., Lemma 2.3.], we establish the following Lemma.

Lemma 3.3. Assuming A ∈ R d × d has d distinct eigenvalues, with Jordan decomposition A = Q Λ Q -1 . Let γ j ∈ R d and ˜ γ j := Q -1 γ j ∈ R d for all j = 1 , . . . , n with n ⩾ 2 . We define

where ˜ γ j,k denotes the k -th entry of ˜ γ j . rank ([ γ 1 | Aγ 1 | . . . | A d -1 γ 1 | . . . | γ n | Aγ n | . . . | A d -1 γ n ]) < d if and only if there exists k ∈ { 1 , . . . , K } , such that | w j,k | = 0 for all j = 1 , . . . , n , where | w j,k | is the absolute value of w j,k for k = 1 , . . . , K 1 , and the Euclidean norm of w j,k for k = K 1 +1 , . . . , K .

The proof of Lemma 3.3 can be found in Appendix A.4. From a geometric perspective, γ j can be decomposed into a linear combination of Q k 's

̸

Let L k := span ( Q k ) . According to [43, Theorem 2.2], each L k is an A -invariant subspace of R d . Recall that a space L is called A -invariant, if for all γ ∈ L , Aγ ∈ L . We say L is a proper subspace of R d if L ⊂ R d and L = R d . If | w j,k | = 0 (i.e., w j,k = 0 in R 1 or R 2 ), then γ j does not contain any information from L k . In this case, γ j is contained in an A -invariant proper subspace of R d that excludes L k , denoted as L -k . It is worth emphasizing that L -k ⊂ R d is indeed a proper subspace of R d . This further implies that the trajectory of the ODE (13) generated from initial state γ j is confined to L -k [57, Lemma 3.2]. Lemma 3.3 indicates that if rank ([ γ 1 | Aγ 1 | . . . | A d -1 γ 1 | . . . | γ n | Aγ n | . . . | A d -1 γ n ]) < d then all γ j for j = 1 , . . . , n are confined to an A -invariant proper subspace of R d , denoted as L . Therefore, all trajectories of the ODE (13) generated from initial states γ 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 ∈ 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

where H := GG T , and H · j stands for the j -th column vector of matrix H , for all j = 1 , · · · , 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 ∈ R d × d has d distinct eigenvalues concerning the Lebesgue measure on R d × d . Hence, this condition is both sufficient and necessary for almost every A in R d × 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 has Lebesgue measure zero in 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 · 1 , . . . , H · d are confined to an A -invariant proper subspace of R d . A key finding is that when all the vectors H · j , j = 1 , . . . , d are confined to an A -invariant proper subspace L of R d , each column of the covariance matrix V ( t ) in Equation (12) is also confined to L , for all 0 ⩽ t < ∞ . Thus, the identifiability of the generator of the SDE (1) can be fully determined by x 0 and the system parameters ( A,GG ⊤ ) . 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

Based on the identifiability condition (15), we derive the following corollary.

Corollary 3.4.1. Let x 0 ∈ R d be fixed. If rank ([ G | AG | . . . | 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 ∈ R d when the pair [ A,G ] is controllable (rank ([ G | AG | . . . | 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 σ ( x ) = [ G 1 x | . . . | G m x ] ∈ R d × m , thus, c ( x ) = σ ( x ) σ ( x ) ⊤ = ∑ m k =1 G k xx ⊤ G ⊤ k . Let X ( t ; x 0 , A, { G k } m k =1 ) 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 } m k =1 ) -identifiability) . For x 0 ∈ R d , A, G k ∈ R d × d for all k = 1 , . . . , m , the generator of the SDE (2) is said to be identifiable from x 0 , if for all ˜ A, ˜ G k ∈ R d × d , there exists an x ∈ R d , such that ( A, ∑ m k =1 G k xx ⊤ G ⊤ k ) = ( ˜ A, ∑ m k =1 ˜ G k xx ⊤ ˜ G ⊤ k ) , it holds that X ( · ; x 0 , A, { G k } m ) d = X ( · ; x 0 , ˜ A, { ˜ G k } m ) .

̸

Based on Definition 3.2, we present the identifiability condition for the generator of the SDE (2).

Theorem 3.5. Let x 0 ∈ R d be fixed. The generator of the SDE (2) is identifiable from x 0 if the following conditions are satisfied:

where A = A ⊕ A + ∑ m k =1 G k ⊗ G k ∈ R d 2 × d 2 , ⊕ denotes Kronecker sum and ⊗ denotes Kronecker product, v is a d 2 -dimensional vector defined by v := vec ( x 0 x ⊤ 0 ) , where 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 ∑ m k =1 G k xx ⊤ G ⊤ k holds for all x ∈ R d .

Remark. The identifiability condition stated in Theorem 3.5 is generic, that is, let

S has Lebesgue measure zero in 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 ) := E [ X t ] and P ( t ) := E [ X t X ⊤ t ] , 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]):

An important trick to deal with the ODE (16) is to vectorize P ( t ) , then it can be expressed as:

where 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 R d [57, Lemma 3.1.]. And condition A2 implies that the vectorization of x 0 x ⊤ 0 is not confined to an A -invariant proper subspace of W , with W ⊂ 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:

where E ij denotes a d × d matrix whose ij -th and ji -th elements are 1 , and all other elements are 0 , for all i, j = 1 , . . . , d and i ⩾ 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 ⊤ 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 ⊤ 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 := ∑ m k =1 G k xx ⊤ G ⊤ k , where x is a randomly generated vector from R 2 (in these simulations, x = [1 . 33 , 0 . 72] ⊤ ). 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.

5 Related work

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 equallyspaced observations sampled from one trajectory, and then attempting to determine conditions under which ( A,GG ⊤ ) 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, Boogers 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].

6 Conclusion and discussion

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 ζ .

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.

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Appendix for "Generator Identification for Linear SDEs with Additive and Multiplicative Noise"

A Detailed proofs

A.1 Proof of Lemma 3.1

Proof. We start by presenting the mathematical definition of a Lévy process. (cf. [51])

Definition A.1. A stochastic process X := { X t : 0 ⩽ t < ∞} is said to be a Lévy process if it satisfies the following properties:

X 0 = 0 almost surely;

Independence of increments: For any 0 ⩽ t 1 < t 2 < . . . < t n < ∞ , X t 2 -X t 1 , X t 3 -X t 2 , . . . , X t n -X t n -1 are independent;

Stationary increments: For any s < t , X t -X s is equal in distribution to X t -s ;

Continuity in probability: For any ε > 0 and 0 ⩽ t < ∞ it holds that lim h → 0 P ( | X t + h -X t | > ε ) = 0 .

In the following, we first show that the SDE (1) can be expressed as the form of (10), with Z being a p -dimensional Lévy process and a : R d → R d × p being Lipschitz. The first equation in the SDE (1) can be rearranged as

with and

where 0 i × j denotes an i × j zero matrix, let ˜ W := { ˜ W t : 0 ⩽ t < ∞} with ˜ W t = [ W 0 ,t , W 1 ,t , . . . , W m,t ] ⊤ denote a ( m +1) -dimensional standard Brownian motion, then one can find a process Z := { Z t : 0 ⩽ t < ∞} with

satisfying dZ t described in Equation (19). Then we will show that the process Z described in (20) is a Lévy process, that is, it satisfies the four properties stated in Definition A.1.

Property 1: The first property is readily checked since Z 0 = 0 .

Property 2: For any 0 ⩽ t 1 < t 2 < t 3 < ∞ ,

Similarly,

Since W t 2 -W t 1 and W t 3 -W t 2 are independent, Z t 3 -Z t 2 and Z t 2 -Z t 1 are independent.

Property 3: when s < t ,

And

Therefore, property 3 is checked.

Property 4: Obviously, process Z described in (20) is continuous with probability one at t for all 0 ⩽ t < ∞ , therefore, Z has continuity in probability.

Now that we have shown that process Z is a p -dimensional Lévy process with p = m +1 . Then we will show that a ( X t ) = [ AX t G ] is Lipschitz.

where ∥ M ∥ F denotes the Frobenius norm of matrix M and ∥ v ∥ 2 denotes the Euclidean norm of vector v . Now it is readily checked that function a : R d → R d × p is Lipschitz.

Similarly, we will show that the SDE (2) can also be expressed as the form of (10), with Z being a p -dimensional Lévy process and a : R d → R d × p being Lipschitz. Let us rearrange the first equation in the SDE (2):

Since the dZ t here has the same form as that of the SDE (1), we use the same process Z described in Equation (20), which has been shown to be a Lévy process.

As for the function a ( X t ) ,

it is readily checked that function a : R d → R d × p is Lipschitz.

A.2 Proof of Proposition 3.1

Proof. For the backward direction, when b ( x ) = ˜ b ( x ) and c ( x ) = ˜ c ( x ) for all x ∈ R d , it is obviously that ( L f )( x ) = ( ˜ L f )( x ) for all f ∈ C 2 b ( R d ) and x ∈ R d , that is L = ˜ L .

For the forward direction, since ( L f )( x 1 ) = ( ˜ L f )( x 1 ) for all f ∈ C 2 b ( R d ) and x 1 ∈ R d .

We first set

Then we set one has

Thus, in particular, one has

For the backward direction, we know that the solution of the SDE (1) is a Gaussian process. The distribution of a Gaussian process can be fully determined by its mean and covariance functions. Therefore, the two processes have the same distribution when the mean and covariance are the same for both processes for all 0 ⩽ t, h < ∞ .

A.4 Proof of Lemma 3.3

Proof. For the forward direction, since

then for all l = [ l 1 , . . . , l n ] ⊤ ∈ R n ,

where β := l 1 γ 1 + . . . + l n γ n . Consequently, the corresponding ODE system

where x p denotes the p -th component of variable x . It is readily checked that

for all x 1 ∈ R d and p = 1 , . . . , d . As a result,

where x 1 p denotes the p -th component of x 1 . It is readily checked that

for all x 1 ∈ R d and p, q = 1 , . . . , d . Consequently,

A.3 Proof of Lemma 3.2

Proof. For the forward direction, since

is not identifiable from β by [57, Theorem 2.5.], where ˙ x ( t ) denotes the first derivative of x ( t ) with respect to time t .

Let and

Simple calculation shows that therefore, one has

By [43, Theorem 2.4], we know that for any l ∈ R n , there always exists k ∈ { 1 , . . . , K } such that w k = 0 ( ∈ R 1 or R 2 ) since the ODE (21) is not identifiable from initial state β . Next, we will show that this result is satisfied only when there exists a k such that w j,k = 0 ( ∈ R 1 or R 2 ) for all j = 1 , . . . , n . Let us rearrange the Equation (22) as

̸

assume for any k ∈ { 1 , . . . , K } , [ w 1 ,k , . . . , w n,k ] ⊤ = 0 , then there always exists a l ∈ R n such that w k = 0 for all k = { 1 , . . . , K } . The reason is that under this circumstance, for any k ∈ { 1 , . . . , K } , the set of l 's such that w k = 0 has Lebesgue measure zero in R n . Therefore, the set of l 's such that there exists a k such that w k = 0 has Lebesgue measure zero in R n . This result creates a contradiction. Thus, there must exist a k , such that [ w 1 ,k , . . . , w n,k ] ⊤ = 0 , that is | w j,k | = 0 for all j = 1 , . . . , n .

̸

For the backward direction, there exists k such that | w j,k | = 0 for all j = 1 , . . . , n , that is w j,k = 0 ( ∈ R 1 or R 2 ) for all j = 1 , . . . , n . Simple calculation shows that

and recall that

Then matrix where

and matrix C denotes:

We know that

When k ∈ { 1 , . . . , K 1 } , Q -k ∈ R d × ( d -1) , and rank ( Q -k ) = d -1 , while when k ∈ { K 1 + 1 , . . . , K } , Q -k ∈ R d × ( d -2) , and rank ( Q -k ) = d -2 . In both cases, rank ( Q -k ) < d , thus rank ( M ) < d .

A.5 Proof of Theorem 3.4

Proof. Let ˜ A ∈ R d × d and ˜ G ∈ R d × m , such that X ( · ; x 0 , A, G ) d = X ( · ; x 0 , ˜ A, ˜ G ) , we denote as X d = ˜ X . For simplicity of notation, in the following, we denote A 1 := A , A 2 := ˜ A , G 1 := G and G 2 := ˜ G , and denote X d = ˜ X as X 1 d = X 2 .

Sufficiency . We will show that under the identifiability condition (14), one has ( A 1 , G 1 G ⊤ 1 ) = ( A 2 , G 2 G ⊤ 2 ) .

We first show that H 1 = H 2 ( H i := G i G T i ). Indeed, since X 1 , X 2 have the same distribution, one has

for all 0 ⩽ t < ∞ and f ∈ C ∞ ( R d ) . By differentiating (23) at t = 0 , one finds that

where L i is the generator of X i ( i = 1 , 2 ). Based on the Proposition 2.1,

where ( M ) kl denotes the kl -entry of matrix M , and x 0 l is the l -th component of x 0 . Since (24) is true for all f , by taking

for all p, q = 1 , . . . , d . As a result, H 1 = H 2 . Let us call this matrix H .

Next, we show that A 1 = A 2 . We first show the relationship between A i and x 0 , and then show the relationship between A i and H . To this end, one first recalls that

Set m i ( t ) := E [ X i t ] , we know that m i ( t ) satisfies the ODE

where ˙ f ( t ) denotes the first derivative of function f ( t ) with respect to time t .

Simple calculation shows that it is readily checked that

Since X 1 d = X 2 , one has for all 0 ⩽ t < ∞ . That is

Taking k -th derivative of e A i t x 0 with respect to t , one finds that

Let us denote this vector A k x 0 . Obviously, one gets

In the following, we show the relationship between A i and H . Let us denote

for all k = 1 , 2 , . . . . Consequently, and

Simple calculation shows that

where V i ( t ) := V i ( t, t ) .

Since X 1 d = X 2 , by Lemma 3.2, one has

To obtain information about A i , let us fix t for now and take k -th derivative of (27) with respect to h . One finds that for all k = 1 , 2 , . . . .

On the other hand, the function V i ( t ) satisfies the ODE [56]

By differentiating (28) at t = 0 , it follows that

for all k = 1 , 2 , . . . . Consequently,

In particular,

Let us denote this matrix A k H . Obviously, by rearranging this matrix, one gets

Recall our identifiability condition is that rank ( M ) = d with

If we denote the j -th column in M as M · j , one gets

for all j = 1 , . . . , d + d 2 by Equations (26) and (29).

This means one can find a full-rank matrix B ∈ R d × d by horizontally stacking d linearly independent columns of matrix M , such that A 1 B = A 2 B . Since B is invertible, one thus concludes that A 1 = A 2 . Hence, the sufficiency of the condition is proved.

̸

Necessity . In the following, we will show that when A has distinct eigenvalues. The condition (14) stated in Theorem 3.4 is also necessary. Specifically, we will show that when the identifiability condition (14) is not satisfied, one can always find a ˜ A with ( A,GG ⊤ ) = ( ˜ A,GG ⊤ ) such that X d = ˜ X . Recall that for simplicity of notation, we denote A 1 := A , A 2 := ˜ A , and denote X d = ˜ X as X 1 d = X 2 , where process X i = { X i t : 0 ⩽ t < ∞} , and X i t = X ( t ; x 0 , A i , G ) following the form described in the solution process (3). In the following, we may use both A and A 1 interchangeably according to the context.

By Lemma 3.2, to guarantee X 1 d = X 2 one only needs to show that

where V ( t ) := V 1 ( t ) = V 2 ( t ) .

Recall that H = GG ⊤ . For simplicity of notation, abusing notation a bit, we denote H · 0 := x 0 . Let

and

When the identifiability condition (14) is not satisfied, that is

by Lemma 3.3, there exists k such that | w j,k | = 0 , i.e., w j,k = 0 ( ∈ R 1 or R 2 ) , for all j = 0 , . . . , d . Recall that

where and some calculation shows that

That is,

recall that

Since w j,k = 0 ( ∈ R 1 or R 2 ) , for all j = 0 , 1 , . . . , d , then if k ∈ { 1 , . . . , K 1 } , the k -th row of W

and if k ∈ { K 1 +1 , . . . , K } , then the (2 k -K 1 -1) -th and the (2 k -K 1 ) -th rows

where W k · denotes the k -th row vector of matrix W .

If we denote

and

Then by multiplying Q -1 in both sides of Equation (31), one obtains that

for all j = 1 , . . . , d and all 0 ⩽ t < ∞ . This indicates that when all the vectors H · j for j = 1 , . . . , d are confined to an A -invariant proper subspace of R d , denotes as L , then each column of the covariance matrix V ( t ) in Equation (30) is also confined to L , for all 0 ⩽ t < ∞ . Therefore, under condition (14), x 0 , H · j (for all j = 1 , . . . , d ) and each column of the covariance matrix V ( t ) (for all 0 ⩽ t < ∞ ) are confined to an A -invariant proper subspace of R d . Thus, a matrix A 2 exists, with A 2 = A 1 such that the first two equations in Equation (30) are satisfied.

̸

̸

In particular, by [43, Theorem 2.5], when k ∈ { 1 , . . . , K 1 } , there exists matrix D ∈ R d × d , with the kk -th element D kk = c = 0 and all the other elements of D are zeros. Let

̸

then A 1 and A 2 satisfy the first two equations in Equation (30). Then we will show that such a A 2 also satisfy the third equation in Equation (30).

Some calculation shows that

where P 1 := Q -1 H ( Q T ) -1 . And

where P 2 := e D ( t -s ) Q -1 H ( Q T ) -1 e D ( t -s ) . If one can show that P 1 = P 2 , then it is readily checked that V 1 ( t ) = V 2 ( t ) for all 0 ⩽ t < ∞ . Recall that

where ˜ H = [ ˜ H · 1 | . . . | ˜ H · d ] . And when condition (14) is not satisfied, the k -th row of ˜ H :

Simple calculation shows that matrix P 1 is symmetric, thus, the k -th column of P 1 :

It is easy to obtain that e D ( t -s ) is a diagonal matrix expressed as

where e c ( t -s ) is the kk -th entry. Then, simple calculation shows that

Therefore, one obtains that

Since therefore, the k -th row of P 1 :

then

̸

Hence, when k ∈ { 1 , . . . , K 1 } , we find a A 2 , with A 2 = A 1 such that Equation (30) is satisfied.

When k ∈ { K 1 +1 , . . . , K } , there exists matrix D ′ ∈ R d × d , with

̸

where M i,j denotes the ij -th entry of matrix M , c = [ c 1 , c 2 , c 3 , c 4 ] ⊤ = 0 , and all the other elements of D ′ are zeros. Let

̸

then A 1 and A 2 satisfy the first two equations in Equation (30). Similar to the case where k ∈ { 1 , . . . , K 1 } , one can also show that such a A 2 also satisfies the third equation in Equation (30).

̸

Therefore, assuming A has distinct eigenvalues, then when the identifiability condition (14) is not satisfied, one can always find a A 2 with ( A 1 , GG ⊤ ) = ( A 2 , GG ⊤ ) such that Equation (30) is satisfied, i.e., X 1 d = X 2 . Hence, the necessity of the condition is proved.

A.6 Proof of Corollary 3.4.1

Proof. There are two ways to prove this corollary, we will present both of them in the following.

where span ( M ) denotes the linear span of the columns of the matrix M . therefore, when

thus,

Therefore,

since the rank of a matrix is the dimension of its span. Then by Theorem 3.4, the generator of the SDE (1) is identifiable from x 0 .

Way2. Let ˜ A ∈ R d × d and ˜ G ∈ R d × m , such that X ( · ; x 0 , A, G ) d = X ( · ; x 0 , ˜ A, ˜ G ) , we denote as X d = ˜ X , we will show that under our identifiability condition ( A,GG ⊤ ) = ( ˜ A, ˜ G ˜ G ⊤ ) . By applying the same notations used in the proof of Theorem 3.4, in the following, we denote A 1 := A , A 2 := ˜ A , G 1 := G and G 2 := ˜ G .

In the proof of Theorem 3.4, we have shown that G 1 G ⊤ 1 = G 2 G ⊤ 2 , thus, we only need to show that under the condition stated in this corollary, A 1 = A 2 . According to the proof of Theorem 3.4, for all 0 ⩽ t < ∞ , we have

Let V ( t ) := V i ( t )( i = 1 , 2) , one gets

Therefore, if there exists a 0 ⩽ t < ∞ , such that V ( t ) is nonsingular, then one can conclude that A 1 = A 2 .

By [20, Theorem 3.2], the covariance V ( t ) is nonsingular for all t > 0 , if and only if

that is the pair [ A,G ] is controllable. Therefore, under the condition stated in this corollary, A 1 = A 2 , thus the generator of the SDE (1) is identifiable from x 0 .

A.7 Proof of Theorem 3.5

Proof. Let ˜ A, ˜ G k ∈ R d × d for all k = 1 , . . . , m , such that X ( · ; x 0 , A, { G k } m k =1 ) d = X ( · ; x 0 , ˜ A, { ˜ G k } m k =1 ) , we denote as X d = ˜ X , we will show that under our identifiability condition, for all x ∈ R d , ( A, ∑ m k =1 G k xx ⊤ G ⊤ k ) = ( ˜ A, ∑ m k =1 ˜ G k xx ⊤ ˜ G ⊤ k ) . For simplicity of notation, in the following, we denote A 1 := A , A 2 := ˜ A , G 1 ,k := G k and G 2 ,k := ˜ G k , and denote X d = ˜ X as X 1 d = X 2 .

We first show that A 1 = A 2 . Set m i ( t ) := E [ X i t ] , we know that m i ( t ) satisfies the ODE

where ˙ f ( t ) denotes the first derivative of function f ( t ) with respect to time t .

Simple calculation shows that

Since X 1 d = X 2 , one has for all 0 ⩽ t < ∞ . That is

Taking j -th derivative of e A i t x 0 with respect to t , one finds that

for all j = 1 , 2 , . . . . Consequently, with an explicit solution

Let us denote this vector A j x 0 . Obviously, one gets

By condition A1, it is readily checked that A 1 = A 2 from Equation (35).

In the following, we show that under condition A2, for all x ∈ R d ,

We know the function P i ( t ) := E [ X i t ( X i t ) ⊤ ] satisfies the ODE

Since X 1 d = X 2 ,

let us call it P ( t ) . By differentiating P i ( t ) one also gets that

Since we have shown that A 1 = A 2 under condition A1, from Equation (36) one observes that

By vectorizing P ( t ) , some calculation shows that the ODE (36) can be expressed as

where A = A ⊕ A + ∑ m k =1 G k ⊗ G k ∈ R d 2 × d 2 , and vec ( M ) denotes the vector by stacking the columns of matrix M vertically.

By definition, P ( t ) ∈ R d × d is symmetric, thus vec ( P ( t )) for all 0 ⩽ t < ∞ is confined to a proper subspace of R d 2 , let us denote this proper subspace W , simple calculation shows that

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 find a basis of W denoting as

where E ij stands for a d × d matrix, with the ij -th and ji -th elements are 1 and all other elements are 0 , for all i, j = 1 , . . . , d and i ⩾ j .

Suppose there exists t i 's, for i = 1 , . . . , ( d 2 + d ) / 2 , such that vec ( P ( t 1 )) , . . . , vec ( P ( t ( d 2 + d ) / 2 )) are linearly independent, then for all x ∈ R d ,

that is

where l := { l 1 , . . . , l ( d 2 + d ) / 2 } ∈ R ( d 2 + d ) / 2 . According to Equation (37), it is readily checked that for all x ∈ R d ,

By [57, Lemma 6.1], there exists ( d 2 + d ) / 2 t i 's such that vec ( P ( t 1 )) , . . . , vec ( P ( t ( d 2 + d ) / 2 )) are linearly independent, if and only if the orbit of vec ( P ( t )) (i.e., the trajectory of ODE (38) started from initial state v ), denoting as γ ( A , v ) with v = vec ( x 0 x ⊤ 0 ) , is not confined to a proper subspace of W .

Next, we show that under condition A2, orbit γ ( A , v ) is not confined to a proper subspace of W .

̸

Assume orbit γ ( A , v ) is confined to a proper subspace of W . Then there exists w = 0 ∈ W such that

By taking j -th derivative with respect to t , we have

In particular, for t = 0 ,

Therefore,

Since w ∈ W , w ∈ span { vec ( E 11 ) , vec ( E 21 ) , . . . , vec ( E dd ) } , set vec ( w ) := w , then w is a d × d symmetric matrix. Since P ( t ) is symmetric for all 0 ⩽ t < ∞ , according to Equation (36), simple calculation shows that the j -th derivative of P ( t ) is also symmetric for all 0 ⩽ t < ∞ , for j = 0 , 1 , . . . . Recall that

where P ( j ) ( t ) denotes the j -th derivative of P ( t ) with respect to t . In particular, when t = 0 , one has

then if we denote matrix A j v by setting vec ( A j v ) := A j v , matrices A j v are symmetric for all j = 0 , 1 , . . . .

Therefore, we can say that there are only ( d 2 + d ) / 2 distinct elements in each of the vectors: w,v, A v, . . . , A ( d 2 + d -2) / 2 v in Equation (39). Moreover, since these vectors all correspond to d × d symmetric matrices, the repetitive elements in each vector appear in the same positions in each vector. Hence, we can focus on checking those distinct elements in each vector, that is Equation (39) can be expressed as

where w ∈ R ( d 2 + d ) / 2 denotes as

̸

where w ij denotes the ij -th element of w , with i, j = 1 , . . . , d and i ⩾ j . When i = j , the element is multiplied by a √ 2 , that is, w only keeps the distinctive elements in w , and for each of the repetitive element, we multiply √ 2 . We define v, A v, . . . using the same way.

Under condition A2, matrix

is easily to be checked to be invertible, then w = 0 , thus w = 0 . This contradicts to w = 0 , therefore, under condition A2, orbit γ ( A , v ) is not confined to a proper subspace of W . Hence, the theorem is proved.

̸

B Genericity of the derived identifiability conditions

B.1 The identifiability condition stated in Theorem 3.5 is generic

We will show that the identifiability condition stated in Theorem 3.5 is generic. Specifically, we will show that for the set of ( x 0 , A, { G k } m k =1 ) ∈ R d +( m +1) d 2 such that either condition A1 or A2 stated in Theorem 3.5 is violated, has Lebesgue measure zero in R d +( m +1) d 2 . In the following, we first present a lemma we will use to prove our main proposition.

Lemma B.1. Let p : R n → R be a non-zero polynomial function. Let Z := { x ∈ R n : p ( x ) = 0 } . Then Z has Lebesgue measure zero in R n .

Proof. When n = 1 , suppose the degree of x is k ⩾ 1 , then by the fundamental theorem of algebra, there are at most k x 's such that x ∈ Z . Therefore, Z has Lebesgue measure zero, since a finite set has measure zero in R .

Suppose the lemma is established for polynomials in n -1 variables. Let p be a non-zero polynomial in n variables, say of degree k ⩾ 1 in x n , then we can write

where x = { x 1 , . . . , x n -1 } and p 0 , . . . , p k are polynomials in the n -1 variables { x 1 , . . . , x n -1 } , and there exists j ∈ { 0 , . . . , k } such that p j is a non-zero polynomial since p is a non-zero polynomial. Then we can denote Z as

Suppose ( x , x n ) ∈ Z , then there are two possibilities:

̸

case 2 there exists i ∈ { 0 , . . . , k } such that p i ( x ) = 0 .

Let then Z = A ∪ B .

For case 1, recall that there exists j ∈ { 0 , . . . , k } such that p j is a non-zero polynomial, let

then by the induction hypothesis, A j has Lebesgue measure zero in R n -1 . Therefore, A j × R has Lebesgue measure zero in R n . Since A ⊆ A j × R , A has Lebesgue measure zero in R n .

For case 2, let λ n be Lebesgue measure on R n , then where

✶ The inner integral in Equation (41) is equal to zero, since for a fixed x , there are finitely many (indeed, at most k ) x n 's such that p ( x , x n ) = 0 under the condition of case 2. Thus, λ n ( B ) = 0 , that is, B has Lebesgue measure zero in R n . Then it is readily checked that Z has Lebesgue measure zero.

Now we are ready to present the main proposition.

Proposition B.1. Let

S := { ( x 0 , A, { G k } m k =1 ) ∈ R d +( m +1) d 2 : either condition A1 or A2 in Theorem 3.5 is violated } , then S has Lebesgue measure zero in R d +( m +1) d 2 .

Proof. Let

we first show that S A has Lebesgue measure zero in R d + d 2 . Suppose ( x 0 , A ) ∈ S A , then ( x 0 , A ) satisfies

that is the set of vectors { x 0 , Ax 0 , . . . , A d -1 x 0 } are linearly dependent, this means that

It is a simple matter of algebra that the left side of (42) can be expressed as some universal polynomial of the entries of x 0 and entries of A , denotes p ( x 0 , A ) = p ( x 01 , . . . , x 0 d , a 11 , a 12 , . . . , a dd ) , where x 0 j denotes the j -th entry of x 0 and a ij denotes the ij -th entry of A . Therefore, one concludes that

Thus, S A can be expressed as

Some calculation shows that

where ( M ) · j denotes the j -th column vector of matrix M . Obviously, p ( x 0 , A ) is a non-zero polynomial function of entries of x 0 and entries of A , therefore, by Lemma B.1, S A has Lebesgue measure zero in R d + d 2 . Let

then it is readily checked that S 1 has Lebesgue measure zero in R d +( m +1) d 2 .

Let

we then show that S 2 has Lebesgue measure zero in R d +( m +1) d 2 . Suppose ( x 0 , A, { G k } m k =1 ) ∈ S 2 , then ( x 0 , A, { G k } m k =1 ) satisfies

recall that A = A ⊕ A + ∑ m k =1 G k ⊗ G k ∈ R d 2 × d 2 and v = vec ( x 0 x ⊤ 0 ) ∈ R d 2 . According to the proof A.7 of Theorem 3.5, we obtain that the set of vectors { v, A v, . . . , A ( d 2 + d -2) / 2 } are linearly dependent. Because all of these vectors are transferred from vectorizing d × d symmetric matrices, thus each of these vectors has only ( d 2 + d ) / 2 distinct elements and the repetitive elements appear in the same positions in all vectors. Hence, abuse notation a little bit, we can focus on checking those distinct elements in each vector, that is

where v ∈ R ( d 2 + d ) / 2 denotes the vector of deleting the repetitive elements of v . Since the set of vectors { v, A v, . . . , A ( d 2 + d -2) / 2 } are linearly dependent, the set of vectors { v |A v | . . . |A ( d 2 + d -2) / 2 v } are linearly dependent, that is

Each entry of v can be written as a non-zero polynomial function of entries of x 0 since v = vec ( x 0 x ⊤ 0 ) . Each entry of A can be written as a non-zero polynomial function of entries of A and G k with k = 1 , . . . , m , since A = A ⊕ A + ∑ m k =1 G k ⊗ G k ∈ R d 2 × d 2 . Hence, the left side of Equation (45) can be expressed as some universal polynomial of the entries of x 0 , A and G k 's, denotes

where G k,ij denotes the ij -th entry of matrix G k . Therefore, one concludes that

Thus, S 2 can be expressed as

Similar to the calculation of p ( x 0 , A ) in Equation (43), p ( x 0 , A, { G k } m k =1 ) can be expressed as a non-zero polynomial function of entries of v and A , thus it can also be expressed as a non-zero polynomial function of entries of x 0 , A and G k 's. Therefore, by Lemma B.1, S 2 has Lebesgue measure zero in R d +( m +1) d 2 .

We know that S ⊆ S 1 ∪ S 2 , let λ be Lebesgue measure on R d +( m +1) d 2 , then one has

Thus S has Lebesgue measure zero in R d +( m +1) d 2 .

B.2 The identifiability condition stated in Theorem 3.4 is generic

We will show that the identifiability condition stated in Theorem 3.4 is generic.

Proposition B.2. Let

then S has Lebesgue measure zero in R d + d 2 + dm .

Proof. Suppose ( x 0 , A, G ) ∈ S , then ( x 0 , A, G ) satisfies

recall that H := GG T , and H · j stands for the j -th column vector of matrix H , for all j = 1 , · · · , d . Let

one observes that S ⊆ S ′ . According to the proof of Proposition B.1, it is readily checked that S ′ has Lebesgue measure zero in R d + d 2 + dm . Thus, S has Lebesgue measure zero in R d + d 2 + dm .

,

C Simulation settings

We present the true underlying system parameters along with the initial states of the SDEs employed in the simulation experiments. We randomly generate the true system parameters that satisfy or violate the corresponding identifiability conditions.

For the SDE (1):

identifiable case: satisfy condition (14) stated in Theorem 3.4:

unidentifiable case: violate condition (14):

For the SDE (2):

identifiable case: satisfy both A1 and A2 stated in Theorem 3.5:

unidentifiable case1: violate A1 satisfy A2:

unidentifiable case2: satisfy A1 violate A2:

We have discussed in Section 4 that we use MLE method to estimate the system parameters from discrete observations sampled from the corresponding SDEs. Specifically, the negative log-likelihood function was minimized using the 'scipy.optimize.minimize' library in Python.

For all of our experiments, we initialized the parameter values as the true parameters plus 2. In the case of the SDE (1), we utilized the 'trust-constr' method with the hyper-parameter 'gtol' = 1 e3 and 'xtol' = 1 e-3. On the other hand, for the SDE (2), we applied the 'BFGS' method and set the hyper-parameter 'gtol' = 1 e2 . The selection of the optimization method and the corresponding hyper-parameters was determined through a series of experiments aimed at identifying the most suitable configuration.

D Examples for reliable causal inference for linear SDEs

D.1 Example for reliable causal inference for the SDE (1)

This example is inspired by [17, Example 5.4]. Recall that the SDE (1) is defined as

where 0 ⩽ t < ∞ , A ∈ R d × d and G ∈ R d × m are constant matrices, W is an m -dimensional standard Brownian motion. Let X ( t ; x 0 , A, G ) denote the solution to the SDE (1). Let ˜ A ∈ R d × d and ˜ G ∈ R d × m define the following SDE:

Then under our proposed identifiability condition stated in Theorem 3.4, we have shown that the generator of the SDE (1) is identifiable, i.e., ( A,GG ⊤ ) = ( ˜ A, ˜ G ˜ G ⊤ ) . Till now, we have shown that under our proposed identifiability conditions, the observational distribution identity - - - - → the generator of the observational SDE. Then we will show that the post-intervention distribution is also identifiable. For notational simplicity, we consider intervention on the first coordinate, making the intervention X 1 t = ξ and ˜ X 1 t = ξ for 0 ⩽ t < ∞ . It will suffice to show equality of the distributions of the non-intervened coordinates (i.e., X ( -1) t and ˜ X ( -1) t , note the superscripts do not denote reciprocals, but denote the ( d -1) -coordinates without the first coordinate). Express the matrices of A and G in blocks

where A 11 ∈ R 1 × 1 , A 22 ∈ R ( d -1) × ( d -1) , G 1 ∈ R 1 × m and G 2 ∈ R ( d -1) × m . Also, consider corresponding expressions of matrices ˜ A and ˜ G . By making intervention X 1 t = ξ , one obtains the post-intervention process of the first SDE satisfies:

which is a multivariate Ornstein-Uhlenbeck process, according to [59, Corollary 1], this process is a Gaussian process, assuming A 22 is invertible, then the mean vector can be described as

and based on [59, Theorem 2], the cross-covariance can be described as

Similarly, one can obtain that the mean vector and cross-covariance of the distribution of the postintervention process of the second SDE by making intervention ˜ X 1 t = ξ satisfy:

such that and

Then we will show that E [ X ( -1) t ] = E [ ˜ X ( -1) t ] , and V ( X ( -1) t + h , X ( -1) t ) = V ( ˜ X ( -1) t + h , ˜ X ( -1) t ) for all 0 ⩽ t, h < ∞ . Recall that we have shown ( A,GG ⊤ ) = ( ˜ A, ˜ G ˜ G ⊤ ) , thus, A 22 = ˜ A 22 and A 21 = ˜ A 21 , then it is readily checked that E [ X ( -1) t ] = E [ ˜ X ( -1) t ] for all 0 ⩽ t < ∞ .

Since

thus, G 2 G ⊤ 2 = ˜ G 2 ˜ G ⊤ 2 , then it is readily checked that V ( X ( -1) t + h , X ( -1) t ) = V ( ˜ X ( -1) t + h , ˜ X ( -1) t ) for all 0 ⩽ t, h < ∞ . Since both of these two post-intervention processes are Gaussian processes, according to Lemma 3.2, the distributions of these two post-intervention processes are the same. That is, the post-intervention distribution is identifiable.

D.2 Example for reliable causal inference for the SDE (2)

Recall that the SDE (2) is defined as

where 0 ⩽ t < ∞ , A,G k ∈ R d × d for k = 1 , . . . , m are some constant matrices, W := { W t = [ W 1 ,t , . . . , W m,t ] ⊤ : 0 ⩽ t < ∞} is an m -dimensional standard Brownian motion. Let X ( t ; x 0 , A, { G k } m k =1 ) denote the solution to the SDE (2). Let ˜ A, ˜ G k ∈ R d × d for k = 1 , . . . , m define the following SDE:

Then under our proposed identifiability condition stated in Theorem 3.5, we have shown that the generator of the SDE (2) is identifiable, i.e., ( A, ∑ m k =1 G k xx ⊤ G ⊤ k ) = ( ˜ A, ∑ m k =1 ˜ G k xx ⊤ ˜ G ⊤ k ) for all x ∈ R d . Till now, we have shown that under our proposed identifiability conditions, the observational distribution identity - - - - → the generator of the observational SDE. Then we aim to show that the postintervention distribution is also identifiable. For notational simplicity, we consider intervention on the first coordinate, making the intervention X 1 t = ξ and ˜ X 1 t = ξ for 0 ⩽ t < ∞ . It will suffice to show equality of the distributions of the non-intervened coordinates (i.e., X ( -1) t and ˜ X ( -1) t ). Express the matrices of A and G k for k = 1 , . . . , m in blocks

where A 11 , G k, 11 ∈ R 1 × 1 , A 22 , G k, 22 ∈ R ( d -1) × ( d -1) . Also consider corresponding expressions of matrices ˜ A and ˜ G k for k = 1 , . . . , m . By making intervention X 1 t = ξ , one obtains the postintervention process of the first SDE satisfies:

Since this post-intervention process is not a Gaussian process, one cannot explicitly show that the post-intervention distribution is identifiable. Instead, we check the surrogate of the post-intervention distribution, that is the first- and second-order moments of the post-intervention process X ( -1) t . Which denote as m ( t ) ( -1) = E [ X ( -1) t ] and P ( t ) ( -1) = E [ X ( -1) t ( X ( -1) t ) ⊤ ] respectively. Then m ( t ) ( -1) and P ( t ) ( -1) satisfy the following ODE systems:

and

Similarly, one can obtain the ODE systems describing the ˜ m ( t ) ( -1) and ˜ P ( t ) ( -1) . Then we will show that m ( t ) ( -1) = ˜ m ( t ) ( -1) and P ( t ) ( -1) = ˜ P ( t ) ( -1) for all 0 ⩽ t < ∞ . Recall that we have such that

shown A = ˜ A , thus A 21 = ˜ A 21 and A 22 = ˜ A 22 , then it is readily checked that m ( t ) ( -1) = ˜ m ( t ) ( -1) for all 0 ⩽ t < ∞ .

In the proof of Theorem 3.5, we have shown that

for all 0 ⩽ t < ∞ , where P ( t ) = E [ X t X ⊤ t ] . Simple calculation shows that

Then one can get that the (2 , 2) -th block entry of the matrix ∑ m k =1 G k P ( t ) G ⊤ k is the same as the ∑ m k =1 ( . . . ) part in the ODE corresponds to P ( t ) ( -1) (i.e., Equation (46)), since ∑ m k =1 G k P ( t ) G ⊤ k = ∑ m k =1 ˜ G k P ( t ) ˜ G ⊤ k , then the ∑ m k =1 ( . . . ) part in the ODEs correspond to both P ( t ) ( -1) and ˜ P ( t ) ( -1) are the same. Thus, it is readily checked that P ( t ) ( -1) = ˜ P ( t ) ( -1) for all 0 ⩽ t < ∞ .

Though we cannot explicitly show that the post-intervention distribution is identifiable, showing that the first- and second-order moments of the post-intervention process is identifiable can indicate the identification of the post-intervention distribution to a considerable extent.

E Conditions for identifying the generator of a linear SDE with multiplicative noise when its explicit solution is available

Proposition E.1. Let x 0 ∈ R d be fixed. The generator of the SDE (2) is identifiable from x 0 if the following conditions are satisfied:

where H := ∑ m k =1 G k x 0 x ⊤ 0 G ⊤ k , and H · j stands for the j -th column vector of matrix H , for all j = 1 , · · · , d . And A = A ⊕ A + ∑ m k =1 G k ⊗ G k ∈ R d 2 × d 2 , ⊕ denotes Kronecker sum and ⊗ denotes Kronecker product, v is a d 2 -dimensional vector defined by v := vec ( x 0 x ⊤ 0 ) , where vec ( M ) denotes the vectorization of matrix M .

Proof. Let ˜ A, ˜ G k ∈ R d × d and ˜ A ˜ G k = ˜ G k ˜ A , ˜ G k ˜ G l = ˜ G l ˜ G k for all k, l = 1 , . . . , m , such that X ( · ; x 0 , A, { G k } m k =1 ) d = X ( · ; x 0 , ˜ A, { ˜ G k } m k =1 ) , we denote as X d = ˜ X , we will show that under our identifiability condition, for all x ∈ R d , ( A, ∑ m k =1 G k xx ⊤ G ⊤ k ) = ( ˜ A, ∑ m k =1 ˜ G k xx ⊤ ˜ G ⊤ k ) . By applying the same notations used in the proof of Theorem 3.5, in the following, we denote A 1 := A , A 2 := ˜ A , G 1 ,k := G k and G 2 ,k := ˜ G k , and denote X d = ˜ X as X 1 d = X 2 .

We first show that H 1 = H 2 ( H i := ∑ m k =1 G i,k x 0 x ⊤ 0 G ⊤ i,k ). Indeed, since X 1 , X 2 have the same distribution, one has

for all 0 ⩽ t < ∞ and f ∈ C ∞ ( R d ) . By differentiating (47) at t = 0 , one finds that

where L i is the generator of X i ( i = 1 , 2 ). Based on the Proposition 2.1,

where ( M ) kl denotes the kl -entry of matrix M , and x 0 l is the l -th component of x 0 . Since (48) is true for all f , by taking

for all p, q = 1 , . . . , d . As a result, H 1 = H 2 . Let us call this matrix H . That is

In the proof of Theorem 3.5, we have shown that

next, we will derive the relationship between A i and H . Under condition C3, the SDE system (2) has an explicit solution (cf. [25]):

it is readily checked that

then the covariance of X t , P ( t, t + h ) = E [ X t X ⊤ t + h ] can be calculated as

=

e

where

because the Brownian motion W k,t has independent increments.

It is known that, for Z ∼ N (0 , 1) , we have that the j th moment is

Since W k,t ∼ N (0 , t ) , we have

At

e

-

1

2

∑

m

k

=1

G

2

k

t

[

e

∑

m

k

=1

G

k

W

k,t

x

x

T

0

e

∑

m

k

=1

G

⊤

k

W

k,t

+

h

]

e

-

1

2

∑

m

k

=1

(

G

2

k

)

⊤

(

t

+

h

)

A

e

⊤

(

t

+

h

)

,

Similarly, we have

Simple calculation shows that

By combining Equations (51), (52) and (53), one readily obtains that

, it follows that we denote P ( t ) := P ( t, t ) . Set P i ( t, t + h ) = E [ X i t ( X i t + h ) ⊤ ] , since X 1 d = X 2 P 1 ( t, t + h ) = E [ X 1 t ( X 1 t + h ) ⊤ ] = E [ X 2 t ( X 2 t + h ) ⊤ ] = P 2 ( t, t + h ) ∀ t, h ⩾ 0 .

To obtain information about A , let us fix t for now and take j -th derivative of (54) with respect to h . One finds that

E

0

for all j = 1 , 2 , . . . . It is readily checked that

We know the function P i ( t ) satisfies the ODE

In particular,

By differentiating (56) with respect to t at t = 0 , it follows that

Since we have known that A j 1 x 0 = A j 2 x 0 for all j = 1 , 2 , . . . , it is readily checked that

that is A j 1 H = A j 2 H for all j = 1 , 2 , . . . . Let us denote this matrix A j H . Obviously, by rearranging this matrix, one gets

Therefore, under condition C1, that is rank ( M ) = d with

if we denote the j -th column in M as M · j , one gets A 1 M · j = A 2 M · j for all j = 1 , . . . , d + d 2 by equations (49) and (58).

This means one can find a full-rank matrix B ∈ R d × d by horizontally stacking d linearly independent columns from matrix M , such that A 1 B = A 2 B . Since B is invertible, one thus concludes that A 1 = A 2 .

In the proof of Theorem 3.5, we have shown that when A 1 = A 2 , under condition C2, for all x ∈ R d ,

Thus the proposition is proved.

It is noteworthy that Proposition E.1 is established on the explicit solution assumption of the SDE (2), which requires both sets of vectors { A, { G k } m k =1 } and { ˜ A, { ˜ G k } m k =1 } to satisfy condition C3. As aforementioned, condition C3 is very restrictive and impractical, rendering the identifiability condition derived in this proposition unsatisfactory. Nonetheless, this condition is presented to illustrate that condition C1 is more relaxed compared to condition A1 stated in Theorem 3.5 when identifying A with the incorporation of G k 's information.

Peer reviews:
comments: SUMMARY:
This paper presents conditions for identifying the generator of a linear stochastic differential equation (SDE) with additive and multiplicative noise. The authors derive sufficient conditions for identifying the generator of both types of SDEs and offer geometric interpretations.

STRENGTHS:
I think this paper is theoretically sound.

WEAKNESSES:
1. It would be great if there are practical examples for motivating readers. It’s hard to evaluate the papers’ practical importance because there are no practical scenarios and simulations. 
2. In the introduction, the link between causal inference and the identiability of the SDE is weak. Is the goal to identify the “generator” of the post-interventional stochastic process when only observational process is available?

QUESTIONS:
1. Could you please give some examples or practical scenarios of the proposed method 
2. Could you please provide DAGs or any graphs that describing the data generating process? I want to see graphical settings that enables identification of post-interventional processes.

LIMITATIONS:
1. I believe that the paper's writing lacks motivation because it lacks examples or practical scenarios to describe practical importance.
2. Without a graphical description of the data generating processes, it's difficult to assess how this result is related to causality.
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2023-07-03

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper addresses the interventional identifiability of two classes of stochastic differential equations. Theoretical results are illustrated with simulation of identifiable and non-identifiable cases.\n\nSTRENGTHS:\nThe paper is carefully written and the theoretical results appear sound, although I could not check the proofs in detail.\n\nWEAKNESSES:\nMy main concern is whether the target audience for this paper is really NeurIPS. Considering myself one of relatively few in the NeurIPS community with some familiarity with both causality and (to a limited extent) with SDE, I have a hard time making sense of the contribution. Although the presentation of the framework of reference [16] for causality with SDE is interesting, it is quite challenging to get an intuition of the concrete cases for which this framework is relevant: the defined interventions are static (not time dependent) and I have a hard time to imagine in which application we need the heavy toolbox of SDEs to model static interventions, which moreover are constrained to always start from the exact same point in state space. The paper does not really help, neither with justifying this framework nor with providing some intuitions. \n\nMoreover, the paper is written in a quite confusing way, at least for a non expert in SDEs: after introducing \"standard\" SDE based on Brownian motion in Eqs. (1) and (3), we are introduced in Lemma 2.1 and above with SDEs based on integrating Levy processes (and not being explained what is this). Then we move on to talking about Ito diffusion processes (Propostion 2.1) with an equation of the generator exhibited, although we have not being told at any time, even roughly, what a generator is for a stochastic process. \n\nI can imagine the results may be relevant for a mathematical journal, but if the authors what to target the ML/causality community, there is a lot of work to do on the manuscript to keep only the essential mathematical concepts, explain them well with reduced formalism, and convey intuition of why this is relevant to the NeurIPS readership interested in causality and time series.\n\nAdditionally, if I risk myself on evaluating the importance/novelty of the key results of this work, they resemble standard results on characterizing the set of states reachable by some classes of Markov chains in general state space. I can imagine there are also fairly standard results for this question in the context of SDEs, and the rank constraint of Theorem 3.3 is not very surprising when considering the dimension of the subspace reachable from a fixed initial point. Given the whole state space is explored, again I can imagine identifying the generator boils down to exploiting standard results. I am left wondering whether this result really deserves a new paper that addresses it exclusively (meaning: without exhibiting any concrete problem where it is relevant and helpful).\n\nQUESTIONS:\nHow the continuous time identifiability results relate to identifiability of equivalent discrete time models (Markov chains with additive/multiplicative noise in general state space)?\n\nLIMITATIONS:\nLimitations are addressed in the discussion. However, how restrictive is the interventional framework for concrete problems is not addressed (see \"weaknesses\" section).", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 2, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-11"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis article derives relationships for identifying generators of linear SDEs in both the case of additive and multiplicative noise. It is written in a very classically statistical manor, with a great number of citations from the statistical research community.\n\nSTRENGTHS:\nThis article identifies two apparently unresolved issues from stochastic differential equations and solves them.\n\nI thank the authors for their thoughtful discussion in the rebuttal process.\n\nWEAKNESSES:\nThis article assumes a high understanding of statistics, and unfortunately this is not my field. I do not have the expertise to review the proofs in this article. I think the article could do a better job at explaining the relevance of this topic to the broader machine learning community. Excluding the unpublished (arxiv) citations, it looks to me like there are only two published articles cited in the ML community, and both of those articles seem only tangentially related to the work in this article. The presentation of this article appears to be quite good for a pure-statistical research community.\n\n- All of the proofs claimed in this article are in the appendices of their article. While I know this is standard practice, technically NeurIPS appendices are un-reviewed (and in practice they are criminally under-reviewed). In my own field, this has led to an increased number of incorrect results being published, cited, and propagated through the research community. Because reviewers like me cannot assess the validity of the results claimed in this article, I worry that this may also happen in statistics. For this reason, I am giving a low-confidence rejection. If I see that other reviewers make good points attesting to the validity of the theory, I will change my recommendation.\n- Throughout the article there are many forward-references (e.g., on line 25, the reader has no idea what the matrix \"A\" means, in lines 31 and 32, the equations (1) and (3) are referenced without being introduced first, etc.) which make for a hard experience for the reader.\n- Clerically, one could order the citations so that they appear in increasing order (e.g., in line 19, [4, 34, 45])\n- Line 39 - 41: The argument for uniqueness is not clear from this statement.\n- Line 69: I think you mean \"are commutative\"\n- The sectioning in this article does not quite make sense to me. The pagraph \"We know that both the SDE (1) and SDE (3) ...\" which appears in Section 2.2, actually talks about both the content in Sections 2.1 and 2.2.\n- Inconsistent citations throughout. For instance, some articles are cited using the lowercase \"volume\" while others do not use the abbreviation for a volume at all (e.g., 61(11): 1038-1044) -- this citation style should be consistent.\n- The citation also contain many mis-capitalizations (odes vs ODEs, SIAM vs Siam), typos, duplicated authors [21], and clerical errors.\n- There are a lot of citations of unpublished material; a small number of recent arxiv publications is fine in this community, but some are missing any references alltogether, e.g.,  [13] and [24].\n\nQUESTIONS:\nWhy is this article relevant to the machine learning community?\n\nLIMITATIONS:\nThis article is written for pure-statisticans and it is not really legible for many in the ML community.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 1, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-17"}

comments: SUMMARY:
This paper derives some sufficient conditions for identifying linear stochastic differential equations (SDEs). The validity of the identifiability conditions is clarified on synthetic simulation data.

STRENGTHS:
- The identification problem of ordinary/stochastic differential equations has been a hot topic in machine learning community, but the condition for identification seems to be relatively overlooked. The paper could raise a good question about it. 
- The paper is well-motivated and makes a technically solid contribution.

WEAKNESSES:
- The paper focuses on linear SDEs. They are very simple models and, to the best of my knowledge, are rarely applied to modern problems of system identification. Thus, the paper’s contribution is very limited and not impactful. 
- The paper emphases its contribution to reliable causal inference, but the experiments don’t include causal inference task. How the derived sufficient conditions benefits the causal inference task remains to be clear.

QUESTIONS:
What are the situations where the derived conditions for identifying linear SDEs so useful in practice?

LIMITATIONS:
N/A
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 2
DATE: 2023-07-21

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe present work considers the problem of identifying the generators of autonomous linear\nstochastic differential equations (SDEs) with an additive or multiplicative\nnoise term from the law of its solution process (e.g., in case of additive noise\nthis is a Gaussian process) with a given fixed initial state. The authors derive\ngeneric sufficient conditions for this and use the knowledge gained to expand\nthe model interpretability from a geometric point of view. The applicability of\nthe theoretical results is shown in the context of causal inference using linear\nSDEs and empirical evidence is provided by a (small) set of simulations.\n\nSTRENGTHS:\nIn recent years, modeling random dynamics had great impact into deep learning by\nproposing so called Neural SDE (NSDE) models. Since SDEs are ubiquitously used\nfor modelling real world data, research interest in NSDEs rapidly grew.\nHowever, we still know very little about the often hidden systematics of these\nmodels and in particular the design of problem-specific function classes is a\nchallenge without a deeper understanding of it. Therefore, I appreciate this\nwork's attempt to make a systematic contribution to solving the mystery behind\nrandom dynamic modeling.\n\nFurther, the content is well organized, with very few exceptions, has no\nspelling or grammatical flaws and is original to the best of my knowledge. Empirical evidence is accessed through\nexperiments on toy data. Code examples are included in the supplementary\nmaterial.\n\nWEAKNESSES:\nIt is not the value of the approach I am concerning here, but the shortcomings\nrelated to the clarity of the submission and the technical soundness which I\nwill address with examples in the following:\n\n- W1: Although the use of SDEs has rapidly gained prominence in the NeurIPS\n  contributor landscape in recent years, I would argue it still represents a\n  rather subtopic, not easily accessible to every contributor. On the other hand, the generator theory for SDEs is more of\n  an advanced topic. Hence, in the context of its central role in this paper, it is\n  desirable to add a (gentle) introduction to the topic for the wider NeurIPS\n  audience. To illustrate this deficiency with an example: In Sec. In Section\n  2.3 the authors talk about SDEs generators without giving a definition, and\n  later in Section 2.1 the reader discovers that it is an operator on function\n  spaces.\n\n- W2: Starting from l. 184 including Lemma 3.2 and the interpretation of\n  identifiability conditions from a geometric perspective, i.e., the generator\n  of the SDE (1) is identifiable from $x_0$ when not all the given vectors\n  are confined to an $A$-invariant proper subspace of $\\mathbb{R}^d$, I found\n  difficult to access; similar in the case of multiplicative noise (l. 269ff.).\n  Can you phrase the idea in an easier manner? I think what I'm saying\n  is that the amount of technical detail presented diminishes the value of the\n  result.\n\n- W3: In my opinion, the presentation of the causal interpretation of SDEs in\n  Section. 2.3 as a subsection to \"Model and Problem Formulation\" is confusing\n  since this topic is more an example of the application of generator\n  identification of SDEs than a contribution to the problem definition of the\n  latter in general.\n\n- W4: Identifying the solution process of autonomous linear SDEs with additive\n  noise term as in Eq. (1) by Gaussian processes is necessary for the results in\n  the present work, e.g., cf. Lemma 3.1. This statement is justified by the\n  identification of generators by properties of the law of such Gaussian\n  solution processes with a given fixed initial state. I am wondering if the\n  authors also assume the solution process to be Gaussian in case of linear SDEs with multiplicative noise as in Eq.\n  (3) (because the method of solving for m(t) and P(t) is involved in the proof\n  of Theorem 3.4 in Sec. A.7 in the appendix) because this proves wrong in the general case; e.g., in case of a geometric\n  Brownian motion which is a solution to a linear SDE with multiplicative noise\n  term.\n\n- W5: Simulations are only presented in a very limited scenario on (very) low\n  dimensional toy datasets. In real world applications, e.g., modeling (multivariate) time\n  series data, the state variable dimension $d$ is often significantly larger\n  than 10 (cf. I. Silva et al. [52]). In order to create added value for such models, e.g., NSDEs, the\n  approach has to be tested in this extended framework.\n\nHowever, I would like to reiterate my appreciation for the authors' interest in\nexpanding the field of modelling random dynamics. I'm sure that by working on\nthe deficiencies mentioned, the work will gain in value\nand will deliver an enriching contribution to the community.\n\nQUESTIONS:\n- l. 34f. Check wording.\n- l. 63 What type of solution?\n- l. 73 Can you provide a reference here?\n- l. 75 Can you provide a definition for the notion of a stochastic process\n  here?\n- l. 82 What do you mean by: \"..., we recover the notion of stochastic\n  process,...\" ?\n- l. 85 Do you mean: \"..., $S$-valued functions on $[0,\\infty)$?\n- l. 124 \"... can be identifiable from their ...\", check wording.\n- l. 129 What is that gap you are mentioning here?\n- l. 142 Eq. 10: how are $b$ and $\\sigma$ defined?\n- l. 147 Prop. 2.1: \"... denotes the space of **continuous** functions ...\"\n- l. 185 Can you provide reference for this statement?\n- l. 195 Lemma 3.2: How is $Q$ defined?\n\nLIMITATIONS:\nLimitations are addressed (see e.g. Chapter 6) and mainly concern the practical\nverification of the identifiable conditions. From my perspective there are important aspects that need to be\ndiscussed additionally, e.g., computational limitations in the case of system\ndimension $d$, and Brownian motion dimension, $m$, greater 2. Further, the\nauthors consider the class of linear SDEs with either additive or multiplicative\nnoise in the autonomous setting (i.e., drift and diffusion do not explicitly\ndepend on time); can the presented results be transferred to the non-autonomous case?\n\nNo evidence was found that the authors addressed potential negative societal impact. Please make the effort to include this mandatory information in your work.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2023-07-26"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nAs I am not an expert in causal inference nor in SDEs, I will start with summarising the goal of the paper how I understood by reading it and some references therein. This summary will certainly reveal my ignorance regarding this research area, but I prefer to give this transparency and hope my review will be weighted accordingly. \nOrdinary and stochastic differential equations (SDEs) can be interpreted as modelling the influence among elements of a dynamical system. Therefore, they offer a tool for causal inference. For example, when making an intervention to a dynamical system, this intervention can be defined by a change to the corresponding SDE, resulting in “post-intervention” SDE. Previously, it was shown that if we can identify the generator of the original SDE, then we can use this result to obtain the distribution of the post-intervention SDE. \n\nThis paper derives sufficient conditions for the identifiability of the generator of linear SDEs with both additive and multiplicative noise. The authors conduct simulation experiments to assess the validity of their theoretical results.\n\nSTRENGTHS:\nThe paper is generally well-written and clearly structured and, although technical, well readable. \n\nThe theoretical results which seem to be the major contribution of the paper are tested in an intuitive simulation study. The simulation experiments and results are described clearly and concisely.\n\nWEAKNESSES:\nThe beginning of the introduction is good, but in the following paragraphs, the motivation for the importance of identifiability in SDE from a causal inference point of view is not well-explained. \nI had to refer to [16] to find a clear explanation of how SDE can be used for causal inference. \n\nSDE (1) is referenced in the introduction but appears only later in section 2.1.\n\nThe simulation experiment nicely illustrate the derived identifiability conditions. However, an example of how this can used for causal inference is not shown. Given that performing causal inference with SDE is one of the central motivations of this work, I think it would be essential to provide a concrete example of how this can be achieved using the contributions of this paper.\n\nQUESTIONS:\nYou are using MLE to estimate the system parameters (let’s call them $\\theta$) given discrete observational data $x$, i.e., you are obtaining a point for $\\theta$. Given there is uncertainty in the system and due to limited amount of observational data, there should be uncertainty w.r.t. to the estimate of $\\theta$ as well. Have you considered performing fully Bayesian inference over $\\theta$ instead, i.e., obtaining $p(\\theta | x)$ instead of performing $\\hat{\\theta}$ = MLE? \nFor example, you could use simulation-based inference (SBI) to obtain a approximate posterior from simulated data. \n\nThis would enable you add the uncertainty estimates to your analysis of the different “Identifiable” and “unidentifiable” cases reported in section 4.\n\nLIMITATIONS:\nThe authors discuss the limitations of their work.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-26"}