{'fraction_non_alphanumeric': 0.05466559226430298, 'fraction_numerical': 0.017082997582594683, 'mean_word_length': 4.892877492877493, 'pattern_counts': {'":': 0, '<': 13, '<?xml version=': 0, '>': 2, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 2, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Causal Inference for Complex Continuous-Time Longitudinal Studies

The existing causal inference frameworks for identifying causal effects for longitudinal studies typically assume that time advances in discrete time steps. However, medical studies nowadays with either irregular visit times or real-time monitoring have posed threats to the existing frameworks, rendering them invalid or to the very least, inefficient usage of the data. Therefore more general and advanced theory around causal inference for longitudinal data when confounders and treatments are measured continuously across time is needed. We develop a framework to identify causal effects under a user-specified treatment regime for continuous-time longitudinal studies. We provide sufficient identification assumptions including generalized consistency assumption, sequential randomization assumption, positivity assumption, and a novel ``achievable'' assumption designed for continuous time. Under these assumptions, we propose a g-computation process and an inverse probability weighting process, which suggest a g-computation formula and an inverse probability weighting formula for identification. For practical purposes, we also construct two classes of population estimating equations to identify these two processes, respectively, which further suggest a doubly robust formula that identifies causal effects under the user-specified treatment regime with extra robustness against process misspecification.

1. Introduction. Complex longitudinal studies, where treatment, confounders, and sometimes also the outcome, vary across time, pose extensive challenges to valid causal inference. This is because the past confounders affect the past treatment and the past treatment can further affect the confounders after it, which leads to so-called "treatment-confounder feedback" (Hernán and Robins, 2020). Treatment-confounder feedback invalidates traditional adjustment methods such as regression methods by including a history of treatment and confounders. Recently, Greenland and Robins (1986); Robins (1986Robins ( , 1987Robins ( , 1989Robins ( , 1998 developed extensive theories around causal inference for complex longitudinal studies. For nonparametrically inferring the marginal distribution of a potential outcome, a gcomputation formula (Greenland and Robins, 1986), an inverse probability weighting (IPW) estimator (Rosenbaum and Rubin, 1983;Hernán, Brumback and Robins, 2000, 2001, 2002 and a doubly robust estimator have been well established to correct for possible time-varying confounding. However, the aforementioned methods can only accommodate discrete-time longitudinal data where time advances in pre-specified and fixed discrete time steps. The literature on generalizing these methods to continuous-time longitudinal data is limited. For continuous-time longitudinal data, Lok (2001); Røysland (2011Røysland ( , 2012; Rytgaard, Gerds and van der Laan (2021) managed to formally develop identification frameworks where they allow time to advance in continuous time steps. However, all literature above assumed the treatment process and the confounder processes had step paths with finite jumps, as counting processes. Take pharmacoepidemiology studies considered in Rytgaard, Gerds and van der Laan (2021) as an example, they assumed that each patient had a random finite number of visits at random time points, and at each visit, they could change their treatment received and some confounders. This type of identification allows for longitudinal data with irregular visit times.
Nowadays medical studies call for more general and advanced theory around causal inference for longitudinal data when confounders and treatments are also measured continuously over time. For example, in the intensive care and in-patient setting, vital status is continuously monitored as a matter of routine (Johnson et al., 2016(Johnson et al., , 2018. With the introduction of various wearable devices, real-time monitoring is also increasingly employed for long-term management of chronic diseases, for example, continuous glucose monitoring for diabetes (Mastrototaro, 2000;Klonoff, 2005;Rodbard, 2016). As healthcare providers increasingly consult real-time monitoring reports for decisions on disease management, real-time monitoring may emerge as the source of confounding in longitudinal studies. This raises a challenge to existing causal inference tools mostly defined on discrete-time frames. See Figure 1 for a comparison between possible simulated realizations of a covariate process among three scenarios: fixed visit times (Greenland and Robins, 1986;Robins, 1986Robins, , 1987Robins, , 1989Robins, , 1998, irregular visit times (Lok, 2001;Røysland, 2011Røysland, , 2012Rytgaard, Gerds and van der Laan, 2021), and real-time monitoring data. Applying the existing frameworks to real-time monitoring data can be computationally very intensive, invalidate causal analysis or at the very least, be an inefficient usage of the data. In this paper, we generalize the causal inference identification framework to longitudinal studies where confounders and treatments are measured continuously across time. Under our framework, confounder processes and treatment processes are left unrestricted, unlike existing literature (Greenland and Robins, 1986;Robins, 1986Robins, , 1987Robins, , 1989Robins, , 1998 or (Lok, 2004(Lok, , 2008Røysland, 2011;Rytgaard, Gerds and van der Laan, 2021). Therefore, our framework allows for studies with either irregular visit times or real-time monitoring data. We build our identification frameworks upon assumptions including a generalized consistency assumption, a generalized sequential randomization assumption, and a generalized positivity assumption, which are typically required in discrete-time longitudinal studies for identification. We additionally impose a novel "achievable" assumption on the target treatment regime and the observed data distribution, which is designed uniquely for the continuous-time case. We define a continuous-time g-computation formula via a g-computation process, and an inverse probability weighting (IPW) formula via an IPW process. For possible practical purposes, we also provide two classes of population estimating equations that identify these two processes, respectively. These population estimating equations further suggest a doubly robust formula that provides extra protection against process misspecification.
The paper is organized as follows. We devote Section 2 to a review of existing frameworks on causal inference including identification assumptions and identification formulas. The identification framework for continuous-time longitudinal studies is illustrated in Section 3, where we present the "achievable" assumption, a positivity assumption, a consistency assumption, and a sequential randomization assumption. In Section 4, we discuss the "achievable" assumption, summarize our contributions and discuss future directions.

A Review of Causal Identification in Discrete-Time Longitudinal Studies.
Suppose one is interested in evaluating treatment effect in a longitudinal study conducted on fixed time 0, · · · , K. Define Y as the final outcome measured at the end of the study. We define A(k) and L(k) as the treatment received and measured confounders at time k, k ∈ {0, 1, · · · , K}. We also writeĀ(k) = {A(j) : 0 ≤ j ≤ k} and likewise forL(k). We abbreviateĀ(K) andL(K) asĀ andL. We write A as the set of all possible paths ofĀ. We call Yā the counterfactual outcome (or potential outcome) if the treatment were to be set asā. Therefore the full data are {{Yā}ā ∈A ,Ā,L} and the observed data are {Y,Ā,L}. The temporal order in the observed data is {L(0), A(0), L(1), A(1), · · · , L(K), A(K), Y }. We use P(y,ā,l) as the observed data density or probability mass function. Then the observed data likelihood admits a unique Markov's factorization according to the temporal order as whereL(−1) =Ā(−1) = ∅ as convenience. We use the upper case for random variables and the lower case for their realized values. Suppose one is interested in evaluating the average treatment effect of a regime versus another one. One way that he/she could do so is to identify a weighted mean of some transformation of potential outcomes under both regimes and compare them. For instance, one might investigate where G is a user-specified treatment regime, that is, a priori defined rule which can be applied during the study period to decide if the treatment should be changed or continued. Essentially, the idea to identify (2) is to substitute P{Ā(i)|Ā(i − 1),L(i)} in (1) by G{Ā(i)|Ā(i − 1)} to obtain a target distribution One therefore can identify E * (Y ) (E * is the mathematical expectation with P * ) with observed data and the knowledge on G. Then one impose some assumptions to ensure causal interpretation, that is, The following positivity assumption is typically imposed to ensure that there are enough data observed to identify E * (Y ). It assumes that the target distribution P * defined by (3), is absolutely continuous against observed data distribution, that is where represents "absolutely continuous". In the discrete-time framework with a finite number of random variables when a density admits, (5) is equivalent to imposing P{a(j)|ā(j − 1),L(j)}, almost surely overL, or a stronger version, almost surely for any j andā. For instance, for a binary treatment process, when G(Ā) = 1(Ā =0), that is, "always under control", the positivity assumption states that for any patients with any characteristicsL, they have a non-vanishing probability to always remain under control.
The following two assumptions are imposed for causal interpretation (4). The first one is the consistency assumption: almost surely. This assumption links the observed outcome and the potential outcome via the treatment regime actually received. It says that if an individual receives the treatment regimē A =ā, then his/her observed outcome Y matches Yā.
The second assumption is typically referred to as the "sequential randomization" assumption (SRA), "sequentially exchangeability", or "no unmeasured confounders" assumption, which assumes that the treatment A(k) received at time k is independent of the potential outcome Yā given the history of treatmentĀ(k − 1) and confoundersL(k), that is, for any k = 0, · · · , K. It requires that the history of treatment receivedĀ(k − 1) and covariates collectedL(k) before treatment decision A(k) at time k are sufficient to render subjects exchangeable across observed treatment values within measured covariates strata. We remark that there is also literature not relying on this assumption (Tchetgen Tchetgen, Michael and Cui, 2018;Tchetgen et al., 2020;Ying et al., 2021), which is beyond the scope of this review.
From SRA, intuitively, Yā and A(k) shall be symmetric in identification. In fact, this conjecture is true. There are two commonly adopted dual approaches identifying E(Yā)d P(ā) based on observed data distribution. The first one is the g-computation formula (Greenland and Robins, 1986;Robins, 2000;Bang and Robins, 2005), which identifies E(Yā)d P(ā) via adjustment over an iterative steps of conditional expectations of Yā overL. Indeed, by the consistency assumption, the sequential randomization assumption, and the positivity assumption, the parameter of interest (2) is identified by the g-computation formula (Greenland and Robins, 1986), Note that the consistency assumption and the sequential randomization assumption imply a causal interpretation of the formula. The positivity assumption ensures that this function is well defined.
Another formula named inverse probability weighting (IPW) (Rosenbaum and Rubin, 1983;Hernán, Brumback and Robins, 2000, 2001, 2002 borrows the idea of reweighting, that is, one can leverage the Radon-Nikodym derivative between the target distribution P * and the underlying distribution P as pseudoweights to mimic the distribution as if the treatment were randomly allocated under G, which thus leads to an inverse probability weighting formula As one can observe, g-computation formula relies on correct knowledge of outcome regression model E k≤j≤K, * {Y |Ā(j),L(j)} whereas IPW formula hinges on knowing the propensity model, 0≤j≤K P{A(j)|Ā(j − 1),L(j)}, both of which are subject to model misspecification. Define the g-computation process, for any 0 ≤ k ≤ K, H out, * (K + 1) = Y, and the inverse probability weighting process, for any 0 ≤ k ≤ K, and Q trt, * (−1) = 1.
Typically one learns H out, * (k) recursively via regression from K + 1 to 0 and Q trt, * (k) recursively via multiplying G(Ā(k)|Ā(k−1)) P{Ā(k)|Ā(k−1),L(k)} from −1 to K. There is also a third doubly robust (DR) formula, which incorporates both the gcomputation formula and IPW formula, The doubly robust formula is also known as augmented inverse probability weighting (Van der Laan and Robins, 2003;Bang and Robins, 2005;Tsiatis, 2006), which was also heavily investigated in missing data literature Zhao, 1994, 1995;Rotnitzky, Robins and Scharfstein, 1998;Kang and Schafer, 2007). A unique property of the doubly robust formula is that it provides extra robustness and protection against model misspecification. That is, it remains unbiased for (2) either the g-computation process (6) or the IPW process (7) is correct, but not necessarily both. We recommend interested readers Robins (1986Robins ( , 1987Robins ( , 1989; Hernán and Robins (2020) for a thorough review of causal inference for discrete-time longitudinal studies. Note that discrete-time longitudinal studies can be seen as a special type of continuous-time longitudinal studies since it is assuming thatĀ andL take values in the space of stepwise functions with jumps only allowed at time k ∈ {0, 1, · · · , K}, which essentially restricts the path space to be those stepwise functions.

A Review of Existing Literature on Applications Involving Irregular Visit Times.
Causal inference framework for continuous-time longitudinal studies developed by Lok (2004Lok ( , 2008; Røysland (2011); Rytgaard, Gerds and van der Laan (2021) hinges heavily on assuming that the treatment process and the confounder process are counting processes, that is, they can only change finite times at random time points within a finite period of time, like shown in Figure 1. This restricts the path space to be the set of all stepwise functions. The identification results in Rytgaard, Gerds and van der Laan (2021) is most relevant to ours in that they also focused on identifying intervention-specific parameters. This pathwise assumption on restricting the path space to be the set of all stepwise functions greatly simplifies the question in Rytgaard, Gerds and van der Laan (2021) because counting processes can be equivalently characterized by a finite number of random variables. Therefore the stochastic processes remain to admit a joint density like in (1) and the substitution as in (3) is still achievable. The existing frameworks for discrete-time longitudinal studies therefore can be applied with some efforts. This pathwise assumption is practical enough for their data applications. However, for a typical stochastic process on [0, ∞) involving uncountably infinite numbers of random variables like real-time monitoring data, a joint density does not exist, which motivate us to extend identification frameworks to those cases. To that end, we develop a causal inference framework for continuous-time longitudinal studies without restrictions on the path space.

Identification in Continuous Time.
Throughout the rest of the paper, suppose that one has conducted a continuous-time longitudinal study during 0 to ∞. Define (Ω, F, P) as the underlying sample space, σ-algebra, and probability measure. The full data and the ob- • Also, write the counterfactual outcome Yā, for anyā ∈ A, as the outcome if the treatment were to set asā. • We also define counterfactual confounders • When there is no confusion, we may write P(dydādl) to represent the distribution on the path space induced by the stochastic processes and the probability measure P on the sample space Ω. Note that this is not a density. We assume the event space is Polish, so that conditional probability can be chosen to be regular. We understand conditional distribution as a function over a sigma-algebra multiplied with a path set. For instance, P(dā|l) can be seen as a function over Borel space generated by {ā} and the path set {l}. • We use the upper case for random variables and lower case for their realized values.
is a finite sequence of n + 1 numbers of the form Suppose one is interested in evaluating the average treatment effect of a treatment regime versus another one. One way that he/she could do is to identify the mean of some transformation of the potential outcome under both treatment regimes and compare them. As an illustration, throughout the draft, suppose we are interested in learning a marginal mean of a potential outcome under a user-specified treatment regime, where G is a priori defined measure onā and we assume E(Yā) is integrable against G.
Our framework can be easily generalized to learn the marginal mean of outcomes under a dynamic treatment regime, either deterministic (Rytgaard, Gerds and van der Laan, 2021) or random (Cain et al., 2010;Young et al., 2011), that has been considered for a discrete-time longitudinal study.
3.1. Identification Assumptions. As we mentioned before, since a Markov's factorization like (1) and a substitution like (3) fail in the continuous-time case, the existing frameworks cannot apply here. However, the intuition shall remain correct, that is, one needs to substitute conditional distributions of the treatment process at uncountably infinite time points by G. Our idea is to approximate such target distribution by discrete-time distributions and to ensure such approximation is well-defined, that is, it is independent of choices of distribution times. Our idea is illustrated as follows. For any sequences of partitions {∆ n [s, t]} with |∆ n [s, t]| → 0, denote DEFINITION 1. We call a treatment regime G coupled with a probability distribution P achievable if, P * ∆n[0,∞] {dādl|ā(0),l(0)}G{dā(0)} P{dl(0)} converges to the same probability measure in the total variation norm on the path space, regardless of the choices of partitions. We write the limit as P * (dādl).
The "achievable" definition requires that such discretization approximates to the same limit regardless of the choice of partitions so that P * is well-defined. Note that when P * is well-defined, we may also refer P * (dydādl) = P(dy|ā,l) P * (dādl).
ASSUMPTION 1 (Achievable). The target treatment regime G coupled with the probability distribution P is achievable.
We note that Assumption 1 is a novel assumption imposed uniquely for the continuoustime framework yet holds trivially for discrete-time cases. Intuitively, Assumption 1 claims any sequence of discrete-time studies represented by a sequence of partitions that approximates our continuous-time study, the substitution in probability measures approximates to the same limit, regardless of the choices of partitions on time. Assumption 1 might be abstract for readers to comprehend and hence we discuss about it, which includes a plausible sufficient assumption on the ancillary treatment process (Robins, 1998) in Section 4. We take Assumption 1 as the primitive assumption throughout the paper.
Next, we impose the positivity assumption.
ASSUMPTION 2 (Continuous-Time Positivity). The target probability distribution P * induced by the target regime G is absolutely continuous against P, that is, as the corresponding Radon-Nikodym derivative.
This assumption implies that there are enough data to infer the expectation E * (Y ) of the outcome Y under the target distribution, where E * is the mathematical expectation under P * . The following generalized consistency assumption and sequential randomization assumption ensure us to show that E * (Y ) has a causal interpretation, that is, it equals the target parameter (8). Like in the discrete case, the consistency assumption links the observed outcome and the potential outcome via the treatment actually received. It says that if an individual receives the treatmentĀ =ā, then his/her observed outcome Y matches Yā. Before stating the last assumption, we define · p , · p, * as the L p distance and we denote L p (P) as the space of p-th order integrable functions over P and P * .
THEOREM 1 (G-computation Formula). Under Assumptions 1 -4, the marginal mean of potential outcomes under a user-specified regime G (8) is identified via a continuous-time g-computation formula as PROOF. Since P * (dydādl) is a limit of measures in total variation norm of for any indicator functions f (y,ā,l). Therefore, with linearity of expectations and possibly monotone convergence theorem, we have and hence the conclusion.

Inverse Probability Weighting Formula.
DEFINITION 3 (Inverse Probability Weighting Process). Under Assumption 1, 2, define as a Radon-Nikodym derivative at any time t and apparently a P-martingale. We call Q trt, * (t) the inverse probability weighting process. In particular we set Q trt, * (0−) = 1.
THEOREM 2 (Inverse Probability Weighting Formula). Under Assumptions 1 -4, the marginal mean of potential outcomes under a user-specified regime G (8) is identified via a continuous-time inverse probability weighting formula as PROOF. The proof is immediate by noting that by Theorem 1.

Identification of Processes via Population Estimating Equations.
In the previous two subsections, we have defined two processes H out, * (t), Q trt, * (t) in Definitions 2, 3, which provided two identification formulas for (8) in Theorems 1, 2. However, such definitions of H out, * (t), Q trt, * (t) via projections and Radon-Nikodym derivatives are not useful for identifying themselves. Recursive identification approaches across time for the discrete-time version g-computation process (6) and IPW process (7) outlined in Section 2.1 cannot be utilized because of the continuous-time nature. In this subsection, we establish identification (also can be seen as alternative definitions) of both processes via two classes of population estimating equations which can be more useful for practice and may inspire Z-estimators (Ghassami et al., 2021) or M-estimators (Kompa et al., 2022) in the future. Identification for H out, * (t) and Q trt, * (t) also turns out to be useful for constructing the doubly robust formula later.

Population Estimating Equations for the G-Computation
Process. For any two processes H(t) = h{t;Ā(t),L(t)} and Q(t) = q{t;Ā(t),L(t)} with H(∞+) = Y , and any partitions ∆ n [0, ∞] = {0 = t 0 < · · · < t n = ∞}, we write t −1 = 0− and t n+1 = ∞+ for convenience and define We also define Ξ out (H, Q) as the limit of Ξ out,∆n[0,∞] (H, Q) in probability whenever it exists. We define With some regularity conditions, for any locally bounded Q. This can happen, for example, defining such limit Ξ out is similar to defining stochastic integral in L 2 limit or Riemann-Stieltjes integral. The discussion on this details is beyond the scope of this paper but definitely of probabilistic interest. To prove the next theorem, we need a further assumption on the rate of how fast P * ∆n[0,∞] approaches P * in a tiny interval.
REMARK 4. One may understand this approximating rate assumption via Riemann integral as an example. This type of assumptions characterizes how fast the Riemann sum approaches its limit, the Riemann integral. Typically, for a continuous function, α = 2.
With the help of this assumption, one may obtain the following theorem, which state that a process is the g-computation process if and only if it is the solution to some class of estimating equations. Therefore, if one imposes a parametric or semiparametric form on H out, * (t) as H out, * (t; θ), one might estimate θ via some Z-estimation. Or, this theorem may inspire nonparametric estimation of H out, * (t) via minimax estimation (Ghassami et al., 2021) or maximum moment restriction (Kompa et al., 2022).
Then the process H(t) equals the g-computation process H out, * (t) in Definition 2 almost surely for any t ∈ [0, ∞].
PROOF. We first prove the first part of the theorem that H out, * is a solution to the class of estimating equations (9) given in the theorem. To that end, we need to show that for any process Q(t) with (H out, * , Q) ∈ M out and sup t H out, * (t)Q(t) 1 < ∞, (9) holds. Indeed, We now prove the second part of the theorem. Now suppose H(t) is a solution to the class of estimating equations (10). To prove H(t) = H out, * (t), it suffices to show that H(t) is a P * martingale with initial condition H(∞+) = Y , that is, by Doob's theorem, also equivalent to showing for a sufficiently rich set of Q(t). We plug in a process Q (t) = Q trt, * (t)Q(t) for some bounded process Q(t). We have sup t H(t)Q (t) 1 = sup t H(t)Q(t) 1, * < ∞. It hence follows that (H, Q) ∈ M out and E Ξ out (H, Q ) = 0.
THEOREM 4 (Identification of the Inverse Probability Weighting Process). Under Assumptions 1 -5, for any process H(t) with (H, Q trt, * ) ∈ M trt and sup t H(t)Q trt, * (t) 1 < ∞, Ξ trt (H, Q trt, * ) is unbiased for zero, that is, E {Ξ trt (H, Q trt, * )} = 0. Moreover, suppose there exists a process Q(t) = q{t;Ā(t),L(t)} satisfying sup 0≤s≤t Q(s) 1 < ∞ with initial value Q(0−) = 1, so that for any process H(t) with sup t H(t)Q(t) 1 < ∞, PROOF. We prove the first part of the theorem that Q trt, * is a solution to the class of estimating equations (11) given in the theorem. To that end, we need to show that for any P *integrable process H(t) with (H, Q trt, * ) ∈ M trt and sup t H(t)Q trt, * (t) 1 < ∞, (11) holds. Indeed, We now prove the second part of the theorem. Now suppose Q(t) is a solution to the class of estimating equations (12). To prove Q(t) = Q trt, * (t), it suffices to show that Q(t) satisfies the property of a Radon-Nikodym derivative process, that is, for any u, for a sufficiently rich set of H(t). To that end, we plug in any bounded process H(t) with H(t) = 0 for any t > u and H(t) = E * {H(u)|F(t)} for any t ≤ u, for some u > 0, treating Therefore M (u) satisfies the definition of Radon-Nikodym derivative and by the uniqueness of the Radon-Nikodym derivative we conclude that Q(t) = Q trt, * (t) almost surely for any t ∈ [0, ∞].
3.5. Doubly Robust Formula. Finally, we establish a doubly robust formula identifying (8). For any two processes H(t) = h{t;Ā(t),L(t)} and Q(t) = q{t;Ā(t),L(t)} with H(∞+) = Y and Q(0−) = 1, and any partitions ∆ n [0, ∞] = {0 = t 0 < · · · < t n = ∞}, we write t −1 = 0− and t n+1 = ∞+ for convenience and define The following theorem is immediate from the previous two theorems and fact that sup t H out, * Q trt, * 1 = E * |Y | < ∞. Furthermore, Ξ(H out, * , Q trt, * ) is a doubly robust for E(Yā)G(dā), the marginal mean of potential outcomes under a user-specified regime G, in the sense that it remains unbiased when either H out, * or Q trt, * is correct but not necessarily both. That is, for any processes Q(t) and H(t) with (H out, * , Q) ∈ M out , (H, Q trt, * ) ∈ M trt and sup t H out, * Q(t) 1 < ∞, sup t H(t)Q trt, * 1 < ∞, we have As one can see, the doubly robust formula provides extra protection against possible misspecification on either the g-computation process or the IPW process. However, unlike the discrete-time cases, one has to pay an additional price: Assumption 5 and regularity conditions like boundedness, to obtain a doubly robust formula. We find this natural for a continuous-time case, to ensure interchange of integrals and expectations.

Discussion.
4.1. Discussion on Assumption 1. As a specially designed assumption for continuoustime causal inference, here we discuss Assumption 1. There are some remarks. First of all, Assumption 1 holds trivially when a joint density exists, which includes those cases considered in (Greenland and Robins, 1986;Robins, 1986Robins, , 1987Robins, , 1989Robins, , 1998 or (Lok, 2004(Lok, , 2008Røysland, 2011;Rytgaard, Gerds and van der Laan, 2021). Secondly, it is a sufficient condition for identification but not a necessary one. For instance, one may weaken the requirements of a unique limit but allow for the dependence of choices of partitions. However, then one must have knowledge of such a choice of partitions over time. Thirdly, the definition of "achievable" is greatly inspired by the definition of the Riemann integral: with the mesh of partitions going to zero, the Riemann sum approaches its limit, the Riemann integral, independent of the choices of partitions. We further give a sufficient condition for Assumption 1.

4.2.
A Summarization and Future Directions. In this work, we establish a framework for valid causal inference in longitudinal studies where time advances continuously and data are allowed to be collected continuously as well. We provide sufficient assumptions for causal identification, which fully generalizes that in the existing literature. Indeed, this can be confirmed by noting that all additional assumptions are designed for continuous-time longitudinal studies, which trivially hold in discrete-time longitudinal studies. We also provide population estimating equation type of identification for the g-computation process and the IPW process, which might inspire future estimation works.
There are tons of open questions remaining. First, more understanding of Assumption 1 including sufficient conditions, hypothesis testing, and sensitivity analysis is of interest. Secondly, longitudinal studies are usually subject to loss to follow-up, so the extension of this framework considering right censoring is of interest. Thirdly, semiparametric models, including, for instance, marginal structural models (Robins, 1998;Røysland, 2011) and structural nested models (Robins, 1999;Lok, 2008) developed in the discrete case can also be developed in a fairly straightforward way. Generalization of the framework when Assumption 4 fails, including time-dependent IV (Tchetgen Tchetgen, Michael and Cui, 2018), timedependent proxies (Tchetgen et al., 2020;Ying et al., 2021) can also be considered similarly. Though focusing on extending the identification result of causal inference from discrete-time into continuous-time longitudinal studies, possible approaches for estimation when one has n i.i.d. samples are apparently interesting, the extension to which is currently being explored by the author. The author believes that with appropriate modeling of the g-computation process, IPW process and the help of the doubly robust formula (13), one can achieve model doublerobustness and rate double-robustness (Smucler, Rotnitzky and Robins, 2019) for estimation. Last but not least, a formal characterization of assumptions and theory under which limit in probability of pairs (H, Q) in M out, * and M trt, * is of probabilistic interest.