Causal Identification for Complex Functional Longitudinal Studies

## Abstract

Abstract Real-time monitoring in modern medical research introduces functional longitudinal data, characterized by continuous-time measurements of outcomes, treatments, and confounders. This complexity leads to uncountably infinite treatment-confounder feedbacks, which traditional causal inference methodologies cannot handle. Inspired by the coarsened data framework, we adopt stochastic process theory, measure theory, and net convergence to propose a nonparametric causal identification framework. This framework generalizes classical g-computation, inverse probability weighting, and doubly robust formulas, accommodating time-varying outcomes subject to mortality and censoring for functional longitudinal data. We examine our framework through Monte Carlo simulations. Our approach addresses significant gaps in current methodologies, providing a solution for functional longitudinal data and paving the way for future estimation work in this domain.

## 1 Introduction

The advent of real-time monitoring technologies in healthcare has led to the continuous-time measurement of outcomes, treatments, and confounders, which we term “functional longitudinal data.” For example, the Medical Information Mart for Intensive Care IV (MIMIC-IV) (Johnson et al., ) is a freely accessible electronic health record (EHR) database that records ICU care data, including physiological measurements, laboratory values, medication administration, and clinical events. Another example is Continuous Glucose Monitoring (CGM) (Rodbard, ; Klonoff et al., ), an increasingly adopted technology for insulin-requiring patients that provides insights into glycemic fluctuations. CGM offers a real-time, high-resolution stream of data, capturing the intricate fluctuations in interstitial fluid glucose levels every few minutes.

These examples illustrate the recent prevalence of functional longitudinal data, highlighting the necessity of a causal framework, as understanding treatment effects is of paramount interest in these settings. However, there is a great lack of investigation of causal inference at the intersection of longitudinal data and functional data. Even identifying causal parameters of interest through observed data becomes highly nontrivial in this setting, due to the issue of uncountably infinite treatment-confounder feedbacks (Hernán and Robins, ) within functional longitudinal data.

To bridge this gap, we aim to propose a novel identification framework for functional longitudinal data with time-varying outcomes subject to mortality and censoring, who enjoys the nonparametric property, making it more flexible and adaptable to various datasets.

We first define a causal quantity representing the mean of counterfactual outcomes under an idealized randomized world. To connect the observed data distribution to this idealized world, inspired by the coarsened data framework (Heitjan and Rubin, ) and through the application of continuous-time stochastic process theory and measure theory, we upgrade classical causal assumptions to accommodate functional longitudinal data nonparametrically. These together resolve the issue of uncountably infinite treatment-confounder feedbacks (Hernán and Robins, ) for functional longitudinal data. We generalize the well-known g-computation formula, inverse probability weighting formula, and double robust formula. We examine our identification framework through Monte Carlo simulations.

The paper is organized as follows. In Section 2 we present a literature review of related work. In Section 3, we define the notation and parameters of interest. Then we propose identification assumptions and generalize the well-known g-computation, inverse probability weighting, and double robust formulas. Additionally, we prove that our identification is nonparametric. We conduct Monte Carlo simulations to examine our framework in Section 4. Section 5 discusses future directions. While this paper builds a population-level framework with numerical results, it does not explore estimation or associated inference, which is beyond the scope of this study and left for future research.

## 2 Related Work

Causal Inference for Non-Functional Longitudinal Studies.
Current causal frameworks for longitudinal studies are inadequate for handling “functional longitudinal data.” They were mostly designed to accommodate “regular longitudinal studies” where time moves in specific fixed intervals (Greenland and Robins, ; Robins, ) or “irregular longitudinal studies.” where changes in data occur randomly yet discretely (Lok, ; Johnson and Tsiatis, ; Røysland, ; Hu and Hogan, ; Rytgaard et al., ), allowing only finite treatment-confounder feedbacks. In Figure 1 illustrates the relation among cross sectional data, regular, irregular, and functional longitudinal data. All literature designed for non-functional longitudinal data fall short to be directly applied onto functional longitudinal data.

Figure: Figure 1: Relation among cross sectional data, regular, irregular, and functional longitudinal data. Cross section data without considering time scale are special cases of all longitudinal data. Regular longitudinal data are special cases of irregular longitudinal data, which are special cases of functional longitudinal data.
Refer to caption: x1.png

Causal Inference for Functional Data.
Existing research on causal inference has explored the realm of functional data within observational studies, as noted in works by (Miao et al., ; Zhang et al., ; Tan et al., ). The data format these studies investigate is consistent with the framework of our analysis. Nonetheless, our work sets itself apart by emphasizing the temporal aspect inherent in longitudinal studies, in contrast to the primary focus on point exposure in the mentioned literature.

Existing Work for Functional Longitudinal Data.
The only exceptions that investigated causal inference for functional longitudinal data are Ying () and Sun and Crawford (). However, Ying () only investigated an end-of-study outcome, neither proving the nonparametric property nor conducting any numerical investigation. On the other hand, Sun and Crawford () imposed stochastic differential equations with stringent parametric assumptions. This situation highlights a significant gap in methodological advancements within the field.

## 3 Proposed Method

### 3.1 Preparation

Consider a longitudinal study spanning from time 0 to $\infty$:

- •
$A(t)$ and $L(t)$ are two stochastic processes denoting the treatment administered and the measured confounders, respectively, at any given time $t$. At any time, $A(t)$ and $L(t)$ could be binary, categorical, continuous, or even functional. We denote $\bar{A}(t)=\{A(s):0\leq s\leq t\}$ and $\bar{L}(t)=\{L(s):0\leq s\leq t\}$, with $\bar{A}$ and $\bar{L}$ representing the collections of treatments and confounders over the entire study.
- •
We are interested in an outcome of interest $Y(t)$, as a subset of $L(t)$, that is, $Y(t)\subset L(t)$.
- •
Let $T$ be a time-to-event endpoint, for instance, death, and $C$ be the right censoring time. Define $X=\min(T,C)$ as the censored event time and $\Delta=\mathbbm{1}(T\leq C)$ the event indicator. Therefore when $\Delta=1$, $X=T$ and when $\Delta=0$, $X=C$.
- •
Write the counterfactual time-to-event endpoint $T_{\bar{a}}$ and counterfactual covariates $L_{\bar{a}}(t)$, for any $\bar{a}\in\mathcal{A}$, where $\mathcal{A}$ encompasses all possible values of $\bar{a}$. Therefore we have $X_{\bar{a}}=\min(T_{\bar{a}},C)$ and $\Delta_{\bar{a}}=\mathbbm{1}(T_{\bar{a}}<C)$. We assume that the future cannot affect the past, that is, $\mathbbm{1}(T_{\bar{a}}\geq t)=\mathbbm{1}(T_{\bar{a}^{\prime}}\geq t)$ and $L_{\bar{a}}(t)=L_{\bar{a}^{\prime}}(t)$ whenever $\bar{a}(t)=\bar{a}^{\prime}(t)$. We also write $T_{\mathcal{A}}=\{T_{\bar{a}}\}_{\bar{a}\in\mathcal{A}}$ and $\bar{L}_{\mathcal{A}}=\{\bar{L}_{\bar{a}}\}_{\bar{a}\in\mathcal{A}}$.
- •
The full data are $\{\bar{A},C,T_{\mathcal{A}},\bar{L}_{\mathcal{A}}\}$ and the observed data are $\{\bar{A},X,\Delta,\bar{L}\}$. Note that on the observed data level, $A(t)$ and $L(t)$ are not observed for $t\leq X$ or defined for $t\leq T$. For easier notation in this paper, we offset $A(t)=A(X)$ and $L(t)=L(X)$ for observed data whenever $t>X$. In this way, the stochastic processes $A(t)$ and $L(t)$ are well defined at any $t>0$.
- •
Define $\mathscr{F}_{t}=\sigma\{A(s),L(s),\mathbbm{1}(X\leq s),\mathbbm{1}(X\leq s)
\Delta:\forall s\leq t\}$ as a filtration of information observed up to time $t$. Also we write $\mathscr{F}_{t-}=\sigma(\cup_{0\leq s<t}\mathscr{F}_{t})$ and $\mathscr{G}_{t}=\sigma(\mathscr{F}_{t-},A(t))$. We define $\mathscr{G}_{\infty+}=\mathscr{F}_{\infty}$. We write $\mathscr{F}_{0-}$ and $\mathscr{G}_{0-}$ as the trivial sigma algebra for convenience. Note that $X$ is a stopping time with respect to $\mathscr{F}_{t}$, with $\mathscr{F}_{\infty}=\mathscr{F}_{X}=\sigma\{\bar{A},X,\Delta,\bar{L}\}$.
- •
To denote the distribution on the path space induced by the stochastic processes and the measure $\mathbb{P}$ on $\Omega$, we also use $\mathbb{P}(\mathrm{d}\bar{a}\mathrm{d}\bar{l})$ (Bhattacharya and Waymire, ; Durrett, ; Gill and Robins, ). Note that this is not a density function.
- •
Assuming the event space is Polish enables the selection of regular conditional probabilities. Conditional distribution is understood as a function over a sigma algebra combined with a path set, for instance, $\mathbb{P}(\mathrm{d}\bar{a}|\bar{l})$, and is defined almost surely.
- •
The total variation norm over the path space’s signed measure space is represented by $\|\cdot\|_{\textup{TV}}$.
- •
A partition $\Delta_{K}[0,\infty]$ over $[0,\infty]$ is a finite sequence of $K+1$ numbers of the form $0=t_{0}<\cdots<t_{K}=\infty$. The mesh $|\Delta_{K}[0,\infty]|$ of a partition $\Delta_{K}[0,\infty]$ is defined as $\max\{\max_{i=0,\cdots,K-}(t_{j+1}-t_{j}),1/t_{K-1}\}$, representing the maximum gap length of the partition.

We are interested in learning a marginal mean of transformed potential outcomes including a time-to-event outcome and an outcome process under a user-specified treatment regime in the absence of censoring,

$$ $\int_{\mathcal{A}}\operatorname{\mathbb{E}}\{\nu(T_{\bar{a}},\bar{Y}_{\bar{a}} )\}\mathbb{G}(\textup{d}\bar{a}),$ (1) $$

where $\nu$ is some user-specified function and $\mathbb{G}$ is a priori defined (signed) measure on $\mathcal{A}$. We assume $\operatorname{\mathbb{E}}\{\nu(T_{\bar{a}},\bar{Y}_{\bar{a}})\}$ is integrable against $\mathbb{G}$. This exploration encompasses marginal means under static treatment regimes, as discussed in various literature (Rytgaard et al., ; Cain et al., ; Young et al., ; Hernán and Robins, ). This quantity can be seen as the mean of counterfactual outcomes under an idealized randomized world, where $\bar{a}$ is randomized to follow $\mathbb{G}$.

### 3.2 Identification assumptions

We have defined the parameter of interest (1). Intuitively if treating treatment process $\bar{A}$ as a selection process (Heitjan and Rubin, ), (1) is the mean of $\nu(T_{\bar{a}},\bar{Y}_{\bar{a}})$ when $\bar{A}$ were to follow $\mathbb{G}$. To create such a pseudo-population, note that for any sequences of partitions $\{\Delta_{K}[0,\infty]\}_{K=1}^{\infty}$, we have the following decomposition

$$ $\displaystyle\operatorname{\mathbb{P}}(\textup{d}x\textup{d}\delta\textup{d} \bar{a}\textup{d}\bar{l})=\prod_{j=0}^{K-1}$ $\displaystyle\operatorname{\mathbb{P}}\{T\leq t_{j+1}|\bar{a}(t_{j+1}), \mathbbm{1}(t_{j}<x\leq t_{j+1},\delta=0),\mathscr{F}_{t_{j}}\}^{\mathbbm{1}(t _{j}<x\leq t_{j+1},\delta=1)}$ (2) $\displaystyle\operatorname{\mathbb{P}}\{T>t_{j+1}|\bar{a}(t_{j+1}),\mathbbm{1} (t_{j}<x\leq t_{j+1},\delta=0),\mathscr{F}_{t_{j}}\}^{1-\mathbbm{1}(t_{j}<x \leq t_{j+1},\delta=1)}$ (3) $\displaystyle\operatorname{\mathbb{P}}\{C\leq t_{j+1}|\bar{a}(t_{j+1}), \mathscr{F}_{t_{j}}\}^{\mathbbm{1}(t_{j}<x\leq t_{j+1},\delta=0)}$ (4) $\displaystyle\operatorname{\mathbb{P}}\{C>t_{j+1}|\bar{a}(t_{j+1}),\mathscr{F} _{t_{j}}\}^{1-\mathbbm{1}(t_{j}<x\leq t_{j+1},\delta=0)}$ (5) $\displaystyle\operatorname{\mathbb{P}}\{\textup{d}\bar{l}(t_{j+1})|\bar{a}(t_{ j+1}),\mathscr{F}_{t_{j}}\}\operatorname{\mathbb{P}}\{\textup{d}\bar{a}(t_{j+1 })|\mathscr{F}_{t_{j}}\}.$ (6) $$

We intervene treatment distribution at each time $t_{j}$ to approximate the pseudo-population where $\bar{A}$ were to follow $\mathbb{G}$ as:

$$ $\displaystyle\operatorname{\mathbb{P}}_{\Delta_{K}[0,\infty],\mathbb{G}}( \textup{d}x\textup{d}\delta\textup{d}\bar{a}\textup{d}\bar{l})=\prod_{j=0}^{K-1}$ $\displaystyle\operatorname{\mathbb{P}}\{T\leq t_{j+1}|C>t_{j+1},\bar{a}(t_{j+1 }),\mathscr{F}_{t_{j}}\}^{\mathbbm{1}(t_{j}<x\leq t_{j+1},\delta=1)}$ (7) $\displaystyle\operatorname{\mathbb{P}}\{T>t_{j+1}|C>t_{j+1},\bar{a}(t_{j+1}), \mathscr{F}_{t_{j}}\}^{1-\mathbbm{1}(t_{j}<x\leq t_{j+1},\delta=1)}$ (8) $\displaystyle\operatorname{\mathbb{P}}\{\textup{d}\bar{l}(t_{j+1})|C>t_{j+1}, \bar{a}(t_{j+1}),\mathscr{F}_{t_{j}}\}$ (9) $\displaystyle\{1-\mathbbm{1}(x\leq t_{j+1},\delta=0)\}\mathbb{G}\{\textup{d} \bar{a}(t_{j+1})|\bar{a}(t_{j})\}.$ (10) $$

To eliminate confounder bias, we need to make sure there is no unmeasured confounders. We adapt the commonly known “coarsening at random” (Heitjan and Rubin, ) assumption into:

###### Assumption 1 (Full conditional randomization) .

The treatment assignment is independent of the all potential outcomes and covariates given history, in the sense that there exists a bounded function $\varepsilon(t,\eta)>0$ with $\int_{0}^{\infty}\varepsilon(t,\eta)dt\to 0$ as $\eta\to 0$, such that for any $t\in[0,\infty]$, $\eta>0$,

$$ $\sup_{\bar{a}\in\mathcal{A}}\operatorname{\mathbb{E}}(\|\operatorname{\mathbb{ P}}\{\textup{d}t_{\bar{a}}\textup{d}\bar{l}_{\bar{a}}|\bar{A}(t+\eta),\mathscr {F}_{t}\}-\operatorname{\mathbb{P}}\{\textup{d}t_{\bar{a}}\textup{d}\bar{l}_{ \bar{a}}|\mathscr{F}_{t}\}\|_{\textup{TV}})<\varepsilon(t,\eta).$ (11) $$

This assumption claims that, the treatment distribution, or equally, the probability of coarsening, in a small period of time around $t$, only depends on the observed data up to time $t$ and independent of further part of counterfactuals. This assumption says in a approximating sense that there is no common cause between treatment decision between time $[t,t+\eta]$ and all future counterfactual confounders.

We also need an assumption over the censoring mechanism to eliminate the censoring bias. We consider the well-known conditionally independent censoring assumption (Tsiatis, ; Andersen et al., ). Define the full data censoring time hazard function as

$$ $\lambda_{C}(t|T,\bar{A},\bar{L})=\lim_{dt\to 0}\operatorname{\mathbb{P}}(C\leq t +\textup{d}t|C>t,T,\bar{A},\bar{L})/dt.$ (12) $$

The following assumption requires that the full data censoring time hazard at time $t$ only depends on the observed data up to time $t$.

###### Assumption 2 (Conditional independent censoring) .

The censoring mechanism is said to be conditionally independent if

$$ $\lambda_{C}(t|T,\bar{A},\bar{L})=\lim_{dt\to 0}\operatorname{\mathbb{P}}\{C \leq t+\textup{d}t|C>t,T>t,\bar{A}(t),\bar{L}(t)\}\mathbbm{1}(T>t)/dt.$ (13) $$

Note that in order to overcome the continuous-time issue, here we impose Assumption 1 over an infinitesimal period of time. This type of idea is also adopted in Assumption 2. Note that how Assumption 2 is given on the intensity process whereas Assumption 1 is on the conditioning event. This is because one does not have intensity process for a general stochastic process.

With Assumptions 1 and 2, we are able to show that whenever $|\Delta_{K}[0,\infty]|\to 0$, $\operatorname{\mathbb{P}}_{\Delta_{K}[0,\infty],\mathbb{G}}$ approximates a pseudo-measure where treatment distribution are intervened by uncountable times into following $\mathbb{G}$:

###### Proposition 1 (Intervenable) .

Under Assumptions 1 and 2, the measures $\operatorname{\mathbb{P}}_{\Delta_{K}[0,\infty],\mathbb{G}}$ converges to the same (signed) measure $\operatorname{\mathbb{P}}_{\mathbb{G}}:=\operatorname{\mathbb{P}}(\textup{d}x_
{\bar{a}}\textup{d}l_{\bar{a}})\mathbb{G}(\textup{d}\bar{a})\delta_{\bar{a}}$ in the total variation norm on the path space, regardless of the choices of partitions,

$$ $\|\operatorname{\mathbb{P}}_{\Delta_{K}[0,\infty],\mathbb{G}}(\textup{d}x \textup{d}\delta\textup{d}\bar{a}\textup{d}\bar{l})-\operatorname{\mathbb{P}}( \textup{d}x_{\bar{a}}\textup{d}\bar{l}_{\bar{a}})\mathbb{G}(\textup{d}\bar{a}) \delta_{\bar{a}}\|_{\textup{TV}}\to 0.$ (14) $$

We refer $\operatorname{\mathbb{P}}_{\mathbb{G}}$ as the target distribution. The following assumption links the observed variable with the counterfactuals.

###### Assumption 3 (Full consistency) .

For any $t$,

$$ $T=T_{\bar{A}},L(t)=L_{\bar{A}}(t).$ (15) $$

The full consistency assumption links the observed outcome and the potential outcome via the treatment actually received. It says that if an individual receives the treatment $\bar{A}=\bar{a}$, then his/her observed outcome $Y$ matches $Y_{\bar{a}}$.

The following assumption ensures that the observed data can identify the target distribution.

###### Assumption 4 (Positivity) .

$$ $\operatorname{\mathbb{P}}_{\mathbb{G}}\ll\operatorname{\mathbb{P}}.$ (16) $$

With the above assumptions, we are able to generalize the well-known identification formulas: g-computation, inverse probability weighting, and double robust formulas, into functional longitudinal data. Note that our assumptions can be weaker but chosen for ease to interpret.

###### Assumption 1 (Full conditional randomization) .

###### Assumption 2 (Conditional independent censoring) .

###### Proposition 1 (Intervenable) .

###### Assumption 3 (Full consistency) .

###### Assumption 4 (Positivity) .

### 3.3 Identification formulas

###### Definition 1 (G-computation process) .

Under Assumptions 1 and 2, define

$$ $H_{\mathbb{G}}(t)=\operatorname{\mathbb{E}}_{\mathbb{G}}\{\nu(X,\bar{Y})| \mathscr{G}_{t}\},$ (17) $$

as a projection process, which is apparently a $\operatorname{\mathbb{P}}_{\mathbb{G}}$-martingale. We call $H_{\mathbb{G}}(t)$ the g-computation process. Note that

$$ $H_{\mathbb{G}}(\infty)=\nu(X,\bar{Y}),~{}~{}~{}H_{\mathbb{G}}(0-)= \operatorname{\mathbb{E}}_{\mathbb{G}}\{\nu(X,\bar{Y})\}.$ (18) $$

The g-computation process intuitively serves as a consecutive adjustment of the target $\nu(X,\bar{Y})$ from $\infty$ to 0 0. It represents a mix of original conditional distributions of covariate process together with the intervened treatment process $\mathbb{G}$, from end of study to the beginning. Following this adjustment to the beginning of study, we have:

###### Theorem 1 (G-computation formula) .

Under Assumptions 1, 2, 3, and 4, (1) is identified via a g-computation formula as

$$ $\int_{\mathcal{A}}\operatorname{\mathbb{E}}\{\nu(T_{\bar{a}},\bar{Y}_{\bar{a}} )\}\mathbb{G}(\textup{d}\bar{a})=H_{\mathbb{G}}(0-).$ (19) $$

###### Definition 2 (Inverse probability weighting process) .

Under Assumptions 1 and 2, define

$$ $Q_{\mathbb{G}}(t)=\operatorname{\mathbb{E}}\left\{\frac{\textup{d} \operatorname{\mathbb{P}}_{\mathbb{G}}}{\textup{d}\operatorname{\mathbb{P}}} \bigg{|}\mathscr{G}_{t}\right\},$ (20) $$

as the Radon-Nikodym derivative at any time $t$, which is apparently a $\operatorname{\mathbb{P}}$-martingale. We call $Q_{\mathbb{G}}(t)$ the inverse probability weighting process. Note that

$$ $Q_{\mathbb{G}}(\infty)=\operatorname{\mathbb{E}}\left\{\frac{\textup{d} \operatorname{\mathbb{P}}_{\mathbb{G}}}{\textup{d}\operatorname{\mathbb{P}}} \bigg{|}\mathscr{G}_{\infty}\right\}=\frac{\textup{d}\operatorname{\mathbb{P}} _{\mathbb{G}}}{\textup{d}\operatorname{\mathbb{P}}},~{}~{}~{}Q_{\mathbb{G}}(0- )=1.$ (21) $$

The IPW process intuitively serves as a continuous adjustment of the treatment process $\bar{A}$ from 0 0 to $\infty$, using which as weights one may create a pseudo population as if the whole process were to follow $\operatorname{\mathbb{P}}_{\mathbb{G}}$. It reweights the observed data distribution $\operatorname{\mathbb{P}}$ into $\operatorname{\mathbb{P}}_{\mathbb{G}}$ from the beginning of the study to the end. Following this reweighting throughout the longitudinal study, we have:

###### Theorem 2 (Inverse probability weighting formula) .

Under Assumptions 1, 2, 3, and 4, (1) is identified via an inverse probability weighting formula as

$$ $\int_{\mathcal{A}}\operatorname{\mathbb{E}}\{\nu(T_{\bar{a}},\bar{Y}_{\bar{a}} )\}\mathbb{G}(\textup{d}\bar{a})=\operatorname{\mathbb{E}}\left\{Q_{\mathbb{G} }(\infty)\nu(X,\bar{Y})\right\}.$ (22) $$

For any two $\mathscr{G}_{t}$-adapted processes $H(t)$ and $Q(t)$, and a partition $\Delta_{K}[0,\infty]$, we write

$$ $\displaystyle\Xi_{\Delta_{K}[0,\infty]}(H,Q)=\sum_{j=0}^{K}Q(t_{j})\left[\int H (t_{j+1})\mathbb{G}\{\textup{d}\bar{a}(t_{j+1})|\bar{A}(t_{j})\}-H(t_{j}) \right]+\int H(0)\mathbb{G}\{\textup{d}\bar{a}(0)\}.$ (23) $$

We also define $\Xi(H,Q)$ as the limit of $\Xi_{\Delta_{K}[0,\infty]}(H,Q)$ in probability whenever it exists. We have:

###### Theorem 3 (Doubly robust formula) .

Under Assumptions 1, 2, 3, and 4, for any $\mathscr{G}_{t}$-adapted processes $H(t)$ and $Q(t)$ at the law where $\Xi(H,Q)$, as the limit of $\Xi_{\Delta_{K}[0,\infty]}(H,Q)$ in probability, exists and

$$ $\lim_{|\Delta_{K}[0,\infty]|\to 0}\operatorname{\mathbb{E}}\{\Xi_{\Delta_{K}[0 ,\infty]}(H,Q)\}=\operatorname{\mathbb{E}}\{\Xi(H,Q)\},$ (24) $$

we have

$$ $\int_{\mathcal{A}}\operatorname{\mathbb{E}}\{\nu(T_{\bar{a}},\bar{Y}_{\bar{a}} )\}\mathbb{G}(\textup{d}\bar{a})=\operatorname{\mathbb{E}}\left\{\Xi(H,Q) \right\},$ (25) $$

provided that either $H=H_{\mathbb{G}}$ or $Q=Q_{\mathbb{G}}$.

As one can see, the doubly robust formula provides extra protection against possible misspecification on either the g-computation process or the IPW process.

###### Definition 1 (G-computation process) .

###### Theorem 1 (G-computation formula) .

###### Definition 2 (Inverse probability weighting process) .

###### Theorem 2 (Inverse probability weighting formula) .

###### Theorem 3 (Doubly robust formula) .

### 3.4 No restrictions on the observed data distribution: A Nonparametric Framework

In this subsection, we demonstrate that our identification framework imposes no restrictions on the observed data. This property is advantageous for researchers and practitioners because nonparametric frameworks are flexible and require minimal assumptions, making them robust and adaptable to diverse datasets. We demonstrate this by proving that, for any given observed data distribution, we can identify a sequence of full data distributions—each satisfying Assumptions 1, 2, 3, and 4—such that their corresponding distributions on the observed data closely approximate the initial observed data distribution. That is, we write the set of all observed data distribution as $\mathcal{P}$ and its subset satisfying Assumptions 1, 2, 3, and 4 as $\mathcal{M}$, then we show that $\mathcal{M}$ is a dense subset of $\mathcal{P}$ in the total variation norm.

Up to now, we have used $\operatorname{\mathbb{P}}$ to represent both the distribution on the sample space and the path space. In this subsection, we use $\operatorname{\mathbb{P}}$ to denote the distribution on the observed data $(\bar{A},X,\Delta,\bar{L})$ and $\operatorname{\mathbb{P}}^{F}$ to denote the distribution on the full data $(\bar{A},C,T_{\mathcal{A}},\bar{L}_{\mathcal{A}})$. We have

###### Theorem 4 .

When the path space consists of all piece-wise continuous processes, for any measure $\operatorname{\mathbb{P}}$ over the observed data $(\bar{A},X,\Delta,\bar{L})$, there exists a sequence of measures $\operatorname{\mathbb{P}}_{n}^{F}$ over the full data $(\bar{A},C,T_{\mathcal{A}},\bar{L}_{\mathcal{A}})$ satisfying Assumptions 1, 2, 3, 4, whose inductions on the observed data converges to $\operatorname{\mathbb{P}}$ in the total variation norm.

Technically, we have not achieved full nonparametric paradigm. However, we deem that the regularity condition “the path space is piece-wise continuous processes” is general enough for practical considerations. For example, both multivariate counting processes and continuous processes like Brownian process satisfy this regularity condition. It is noteworthy that achieving this “almost nonparametric” nature is the best one can hope for. This realization was confirmed in (Gill et al., , Section 9) for “coarsening at random” assumption, even though our framework exhibits certain distinctions.

###### Theorem 4 .

## 4 Experiment result

In this section, we employ Monte Carlo simulations to empirically assess how the identification works. We decide to evaluate the performance of the g-computation formula only, for two reasons:

- 1.
The g-computation formula is the only one that can be easily approximated through raw simulated data, whereas inverse probability weighting (and hence the doubly robust formula) cannot be directly approximated without estimation or computation;
- 2.
On the population level, the values of the three formulas are equal. Therefore, approximating g-computation formula is sufficient for our purposes.

To that end, we need to go through 4 steps:

- 1.
Come up with a reasonable data generating process;
- 2.
Compute the parameter of interest (1) (or equivalently, the left-hand side of g-computation formula in Theorem 1) according to this data generating process;
- 3.
Simulate according to this data generating process;
- 4.
Approximate the right-hand side of g-computation formula in Theorem 1 using the simulated data.

Step 1: To sharp the focus and ease the computation, we consider a simple setting where there is no mortality or censoring ($T=C\equiv\infty$), or other measured confounding process, except for the outcome process itself. We take glucose levels as the outcome and insulin levels as the treatment. Both glucose and insulin levels exhibit smooth, continuous changes over time. Gaussian processes are particularly well-suited for modeling such smooth and continuous temporal processes. For $t\in[0,1]$, consider a potential outcome process $Y_{\bar{a}}(t)$ capturing potential logarithm of glucose levels, following a Gaussian process with mean process as

$$ $\operatorname{\mathbb{E}}\{Y_{\bar{a}}(t)\}=-a(t),$ (26) $$

and covariance process as

$$ $\operatorname{Cov}\{Y_{\bar{a}}(t),Y_{\bar{a}}(s)\}=e^{-|t-s|},~{}\forall t,s \in[0,1].$ (27) $$

This ensures the joint dependence among $Y_{\bar{a}}(t)$ and negative treatment effect of logarithm of insulin level $\bar{a}$. For instance, $Y_{\bar{a}}(t)$ can be log of blood glucose level. Define $\nu(T_{\bar{a}},\bar{Y}_{\bar{a}})$ as the integral of $\bar{Y}_{\bar{a}}$ over time $t\in[0,1]$, that is,

$$ $\nu(T_{\bar{a}},\bar{Y}_{\bar{a}})=\int_{0}^{1}\bar{Y}_{\bar{a}}(t)dt.$ (28) $$

Suppose the targeted treatment regime $\mathbb{G}$ is a Gaussian measure with mean process $t-0.5$ and jointly independent normal variables at any time points. That is, the intervened $A$ follows a Gaussian process with a mean process

$$ $\operatorname{\mathbb{E}}\{A(t)\}=t-0.5,$ (29) $$

and covariance process

$$ $\operatorname{Cov}\{A(t),A(s)\}=e^{-|t-s|},~{}\forall t,s\in[0,1],$ (30) $$

representing an increase of insulin level, possibly due to some insulin intake.

Step : Then we can show that (1) (or equivalently, the left-hand side of g-computation formula in Theorem 1) equals zero, that is,

$$ $\displaystyle\int\operatorname{\mathbb{E}}\{\nu(T_{\bar{a}},\bar{Y}_{\bar{a}}) \}\mathbb{G}(\textup{d}\bar{a})=\int\operatorname{\mathbb{E}}\left\{\int_{0}^{ 1}\bar{Y}_{\bar{a}}(t)dt\right\}\mathbb{G}(\bar{a})$ (31) $\displaystyle=\int\int_{0}^{1}\operatorname{\mathbb{E}}\{\bar{Y}_{\bar{a}}(t) \}dt\mathbb{G}(\bar{a})=\int\int_{0}^{1}a(t)dt\mathbb{G}(\bar{a})$ (32) $\displaystyle=\int_{0}^{1}\int a(t)\mathbb{G}\{\bar{a}(t)\}dt=\int_{0}^{1}(t-0 .5)dt=0.$ (33) $$

Step 3: In practice, we observe a stochastic process at finite points. We consider evenly splitting $t\in[0,1]$ into a grid of size $K+1$: $\Delta_{K}[0,1]=\{t_{0}=0,t_{1}=1/K,\cdots,t_{K-1}=(K-1)/K,t_{K}=1\}$, and for $1\leq i\leq n$, according to $\mathbb{G}$ specified in Step 1, we simulate i.i.d. samples $A_{i}(t)$ according to $\mathbb{G}$ specified in Step 1 at $\Delta_{K}[0,1]$ as

$$ $\displaystyle\begin{pmatrix}A_{i}(t_{0})\\ A_{i}(t_{1})\\ \cdots\\ A_{i}(t_{K-1})\\ A_{i}(t_{K})\\ \end{pmatrix}$ $\displaystyle\sim\mathcal{N}\begin{bmatrix}\begin{pmatrix}t_{0}-0.5\\ t_{1}-0.5\\ \cdots\\ t_{K-1}-0.5\\ t_{K}-0.5\\ \end{pmatrix}\!\!,&\begin{pmatrix}1&e^{-|t_{1}-t_{0}|}&\cdots&e^{-|t_{K-1}-t_{ 0}|}&e^{-|t_{K}-t_{0}|}\\ e^{-|t_{1}-t_{0}|}&1&\cdots&e^{-|t_{K-1}-t_{1}|}&e^{-|t_{K}-t_{1}|}\\ \cdots&\cdots&\cdots&\cdots&\cdots\\ e^{-|t_{K-1}-t_{0}|}&e^{-|t_{K-1}-t_{1}|}&\cdots&1&e^{-|t_{K}-t_{K-1}|}\\ e^{-|t_{K}-t_{0}|}&e^{-|t_{K}-t_{1}|}&\cdots&e^{-|t_{K}-t_{K-1}|}&1\end{ pmatrix}\end{bmatrix}.$ $$

By according to the distribution of $Y_{\bar{a}}(t)$ specified in Step 1 and consistency, we generate $Y_{i}(t)$ at $\Delta_{K}[0,1]$ as

$$ $\displaystyle\begin{pmatrix}Y_{i}(t_{0})\\ Y_{i}(t_{1})\\ \cdots\\ Y_{i}(t_{K-1})\\ Y_{i}(t_{K})\\ \end{pmatrix}$ $\displaystyle\sim\mathcal{N}\begin{bmatrix}\begin{pmatrix}A_{i}(t_{0})\\ A_{i}(t_{1})\\ \cdots\\ A_{i}(t_{K-1})\\ A_{i}(t_{K})\\ \end{pmatrix}\!\!,&\begin{pmatrix}1&e^{-|t_{1}-t_{0}|}&\cdots&e^{-|t_{K-1}-t_{ 0}|}&e^{-|t_{K}-t_{0}|}\\ e^{-|t_{1}-t_{0}|}&1&\cdots&e^{-|t_{K-1}-t_{1}|}&e^{-|t_{K}-t_{1}|}\\ \cdots&\cdots&\cdots&\cdots&\cdots\\ e^{-|t_{K-1}-t_{0}|}&e^{-|t_{K-1}-t_{1}|}&\cdots&1&e^{-|t_{K}-t_{K-1}|}\\ e^{-|t_{K}-t_{0}|}&e^{-|t_{K}-t_{1}|}&\cdots&e^{-|t_{K}-t_{K-1}|}&1\end{ pmatrix}\end{bmatrix}.$ $$

Step 4: The integral of $Y_{i}(t_{k})$ over $[0,1]$ is $\sum_{k=0}^{K}Y_{i}(t_{k})/(K+1)$. The approximate of the right-hand side of g-computation formula is $\sum_{i=1}^{n}\sum_{k=0}^{K}Y_{i}(t_{k})/(K+1)/n$.

We vary the grid sizes ($K=10,50,250$) to examine how a denser grid improves the approximation. This approach simulates the scenario where the mesh $|\Delta_{K}[0,1]|$ is shrunk to zero. Additionally, we vary the sample sizes ($n=100,500,2500$) to explore how larger samples enhance the approximation, leveraging the law of large numbers to better approximate the right-hand side of the g-computation formula. We repeat the process $R=10{,}000$ times. The resulting $10{,}000$ approximations of $\sum_{i=1}^{n}\sum_{k=0}^{K}Y_{i}(t_{k})/(K+1)/n$ are presented in boxplots in Figure 2, where we append biases.

Figure: Figure 2: Simulation Results of using g-computation formula by varying grid sizes in $K=10,50,250$ and sample sizes in $n=100,500,2500$, for $R=10000$ repeats. We plot boxplots and give biases.
Refer to caption: x2.png

The simulation results demonstrate that the g-computation formula can adequately approximate (1) even with moderate sample and grid sizes. Increasing the sample size while keeping the grid size fixed enhances the accuracy and reduces the variance of the approximation. In contrast, increasing the grid size while keeping the sample size fixed does not consistently improve accuracy or reduce variance. However, simultaneously increasing both the sample and grid sizes significantly improves accuracy and reduces variance in the approximation.

## 5 Conclusion

In this work, we proposed on a novel theoretical framework for causal inference under functional longitudinal studies. We introduced three methodological paradigms for causal identification: the g-computation formula, inverse probability weighting formula, and doubly robust formula. This framework, noted for nonparametric foundation, substantiates and expands upon the estimand-based causal framework introduced by Ying (). It incorporates considerations for time-varying outcomes and addresses complexities such as death and right censoring, marking a significant advancement in the analysis of functional longitudinal data and enhancing the toolkit for causal inference in this area.

There are significant theoretical and methodological opportunities, given the limited investigation on functional longitudinal data, for the machine learning, functional data analysis and causal inference communities. To list a few, first, adapting our framework to accommodate scenarios where Assumption 1 may not hold, including contexts involving time-dependent instrumental variables and time-dependent proxies (Ying et al., ), warrants rigorous exploration. Second, the positivity Assumption 4 in longitudinal studies faces practical challenges due to the potential scarcity of subjects adhering to specific treatment regimes within observed populations. One might consider using semiparametric models such as marginal structural models (Robins, ; Røysland, ) and structural nested models (Robins, ; Lok, ). Other solutions include dynamic treatment regimes (Fitzmaurice et al., ; Young et al., ; Rytgaard et al., ) and incremental interventions (Kennedy, ). Third, establishing the efficiency bound for our quantity of interest by leveraging semiparametric theory, represents an engaging challenge. Fourth, partial identification using discrete-time observations is a promising direction. Finally, developing a comprehensive estimation framework remains of ultimate interest.

## Appendix A Proofs

### A.1 Proof of Proposition 1

We temporarily define $\mathscr{F}_{/C,t}=\sigma\{L(s),A(s),\mathbbm{1}(T\leq s):\forall s\leq t\}$ as the censoring free filtration and $\mathscr{F}_{\bar{a},t}=\sigma\{L_{\bar{a}}(s),\mathbbm{1}(T_{\bar{a}}\leq s):
\forall s\leq t\}$ as the counterfactual filtration:

$$ $\displaystyle\left\|\operatorname{\mathbb{P}}_{\Delta_{K}[0,\infty],\mathbb{G} }(\textup{d}x\textup{d}\delta\textup{d}\bar{a}\textup{d}\bar{l})-\operatorname {\mathbb{P}}(\textup{d}x_{\bar{a}}\textup{d}l_{\bar{a}})\mathbb{G}(\textup{d} \bar{a})\delta_{\bar{a}}\right\|_{\textup{TV}}$ $\displaystyle=\Bigg{\|}\prod_{j=0}^{K-1}\mathbb{G}\{\textup{d}\bar{a}(t_{j+1}) |\bar{a}(t_{j})\}\{1-\mathbbm{1}(x\leq t_{j+1},\delta=0)\}\operatorname{ \mathbb{P}}\{T\leq t_{j+1}|\bar{a}(t_{j+1}),\mathscr{F}_{t_{j}}\}^{\mathbbm{1} (t_{j}<x\leq t_{j+1})}$ $\displaystyle~{}~{}~{}~{}~{}~{}\operatorname{\mathbb{P}}\{T>t_{j+1}|\bar{a}(t_ {j+1}),\mathscr{F}_{t_{j}}\}^{1-\mathbbm{1}(t_{j}<x\leq t_{j+1})}\operatorname {\mathbb{P}}\{\textup{d}\bar{l}(t_{j+1})|\bar{a}(t_{j}),\mathscr{F}_{t_{j}}\}$ $\displaystyle~{}~{}~{}~{}~{}~{}-\operatorname{\mathbb{P}}(\textup{d}x_{\bar{a} }\textup{d}l_{\bar{a}})\mathbb{G}(\textup{d}\bar{a})\delta_{\bar{a}}\Bigg{\|}_ {\textup{TV}}$ $\displaystyle\leq\Bigg{\|}\prod_{j=0}^{K-1}\mathbb{G}\{\textup{d}\bar{a}(t_{j+ 1})|\bar{a}(t_{j})\}\{1-\mathbbm{1}(x\leq t_{j+1},\delta=0)\}$ $\displaystyle~{}~{}~{}~{}~{}~{}\operatorname{\mathbb{P}}\{T\leq t_{j+1}| \mathscr{F}_{/C,t_{j}}\}^{\mathbbm{1}(t_{j}<x\leq t_{j+1})}\operatorname{ \mathbb{P}}\{T>t_{j+1}|\mathscr{F}_{/C,t_{j}}\}^{1-\mathbbm{1}(t_{j}<x\leq t_{ j+1})}\operatorname{\mathbb{P}}\{\textup{d}\bar{l}(t_{j+1})|\mathscr{F}_{/C,t_ {j}}\}$ $\displaystyle~{}~{}~{}~{}~{}~{}-\operatorname{\mathbb{P}}(\textup{d}x_{\bar{a} }\textup{d}l_{\bar{a}})\mathbb{G}(\textup{d}\bar{a})\delta_{\bar{a}}\Bigg{\|}_ {\textup{TV}}+o(1)$ $\displaystyle=\Bigg{\|}\prod_{j=0}^{K-1}\mathbb{G}\{\textup{d}\bar{a}(t_{j+1}) |\bar{a}(t_{j})\}\{1-\mathbbm{1}(x_{\bar{a}}\leq t_{j+1},\delta_{\bar{a}}=0)\} \operatorname{\mathbb{P}}\{T_{\bar{a}}\leq t_{j+1}|\bar{a}(t_{j+1}),\mathscr{F }_{\bar{a},t_{j}}\}^{\mathbbm{1}(t_{j}<x_{\bar{a}}\leq t_{j+1},\delta_{\bar{a} }=1)}$ $\displaystyle~{}~{}~{}~{}~{}~{}\operatorname{\mathbb{P}}\{T_{\bar{a}}>t_{j+1}| \bar{a}(t_{j+1}),\mathscr{F}_{\bar{a},t_{j}}\}^{1-\mathbbm{1}(t_{j}<x_{\bar{a} }\leq t_{j+1},\delta_{\bar{a}}=1)}\operatorname{\mathbb{P}}\{\textup{d}\bar{l} _{\bar{a}}(t_{j+1})|\bar{a}(t_{j+1}),\mathscr{F}_{\bar{a},t_{j}}\}$ $\displaystyle~{}~{}~{}~{}~{}~{}-\operatorname{\mathbb{P}}(\textup{d}x_{\bar{a} }\textup{d}l_{\bar{a}})\mathbb{G}(\textup{d}\bar{a})\delta_{\bar{a}}\Bigg{\|}_ {\textup{TV}}$ $\displaystyle=\Bigg{\|}\prod_{j=0}^{K-1}\mathbb{G}\{\textup{d}\bar{a}(t_{j+1}) |\bar{a}(t_{j})\}\{1-\mathbbm{1}(x_{\bar{a}}\leq t_{j+1},\delta_{\bar{a}}=0)\}$ $\displaystyle~{}~{}~{}~{}~{}~{}(\operatorname{\mathbb{P}}\{T_{\bar{a}}\leq t_{ j+1}|C>t_{j+1},\bar{a}(t_{j+1}),\mathscr{F}_{\bar{a},t_{j}}\}^{\mathbbm{1}(t_{ j}<x_{\bar{a}}\leq t_{j+1},\delta_{\bar{a}}=1)}$ $\displaystyle~{}~{}~{}~{}~{}~{}\operatorname{\mathbb{P}}\{T_{\bar{a}}>t_{j+1}| C>t_{j+1},\bar{a}(t_{j+1}),\mathscr{F}_{\bar{a},t_{j}}\}^{1-\mathbbm{1}(t_{j}< x_{\bar{a}}\leq t_{j+1},\delta_{\bar{a}}=1)}\operatorname{\mathbb{P}}\{\textup {d}\bar{l}_{\bar{a}}(t_{j+1})|C>t_{j},\bar{a}(t_{j+1}),\mathscr{F}_{\bar{a},t_ {j}}\}$ $\displaystyle~{}~{}~{}~{}~{}~{}-\operatorname{\mathbb{P}}\{T_{\bar{a}}\leq t_{ j+1}|\mathscr{F}_{\bar{a},t_{j}}\}^{\mathbbm{1}(t_{j}<x_{\bar{a}}\leq t_{j+1}) }\operatorname{\mathbb{P}}\{T_{\bar{a}}>t_{j+1}|\mathscr{F}_{\bar{a},t_{j}}\}^ {1-\mathbbm{1}(t_{j}<x_{\bar{a}}\leq t_{j+1})}\operatorname{\mathbb{P}}\{ \textup{d}\bar{l}_{\bar{a}}(t_{j+1})|\mathscr{F}_{\bar{a},t_{j}}\})\Bigg{\|}_{ \textup{TV}}\to 0.$ $$

### A.2 Proof of Theorem 1

Since $\operatorname{\mathbb{P}}_{\mathbb{G}}(\textup{d}x\textup{d}\delta\textup{d}
\bar{a}\textup{d}\bar{l})$ is a limit of measures in total variation norm of

$$ $\operatorname{\mathbb{P}}_{\Delta_{K}[0,\infty],\mathbb{G}}(\textup{d}x\textup {d}\delta\textup{d}\bar{a}\textup{d}\bar{l}),$ (34) $$

whenever $|\Delta_{K}[t,\infty]|\to 0$, we have

$$ $\int f(x,\delta,\bar{a},\bar{l})\operatorname{\mathbb{P}}_{\Delta_{K}[0,\infty ],\mathbb{G}}(\textup{d}x\textup{d}\delta\textup{d}\bar{a}\textup{d}\bar{l}) \to\operatorname{\mathbb{E}}_{\mathbb{G}}\{f(X,\Delta,\bar{A},\bar{L})\},$ (35) $$

for any bounded functions $f(x,\delta,\bar{a},\bar{l})$. Therefore, we have

$$ $\displaystyle\left|H_{\mathbb{G}}(0-)-\int\operatorname{\mathbb{E}}\{\nu(T_{ \bar{a}},Y_{\bar{a}})\}\mathbb{G}(\textup{d}\bar{a})\right|$ $\displaystyle\leq\left|\int\nu(x,\bar{y})\operatorname{\mathbb{P}}(\textup{d}x \textup{d}\delta\textup{d}\bar{a}\textup{d}\bar{l})-\int\nu(x,\bar{y}) \operatorname{\mathbb{P}}_{\Delta_{K}[0,\infty],\mathbb{G}}(\textup{d}x\textup {d}\delta\textup{d}\bar{a}\textup{d}\bar{l})\right|$ $\displaystyle+\left|\int\nu(x,\bar{y})\operatorname{\mathbb{P}}_{\Delta_{K}[0, \infty],\mathbb{G}}(\textup{d}x\textup{d}\delta\textup{d}\bar{a}\textup{d}\bar {l})-\int\operatorname{\mathbb{E}}\{\nu(T_{\bar{a}},Y_{\bar{a}})\}\mathbb{G}( \textup{d}\bar{a})\right|$ $\displaystyle=\left|\int\nu(x,\bar{y})\operatorname{\mathbb{P}}_{\Delta_{K}[0, \infty],\mathbb{G}}(\textup{d}x\textup{d}\delta\textup{d}\bar{a}\textup{d}\bar {l})-\int\operatorname{\mathbb{E}}\{\nu(T_{\bar{a}},Y_{\bar{a}})\}\mathbb{G}( \textup{d}\bar{a})\right|+o(1)$ $\displaystyle=\Bigg{|}\int\nu(x,\bar{y})\prod_{j=0}^{K-1}\operatorname{\mathbb {P}}\{T\leq t_{j+1}|C>t_{j+1},\bar{a}(t_{j+1}),\mathscr{F}_{t_{j}}\}^{\mathbbm {1}(t_{j}<x\leq t_{j+1},\delta=1)}$ $\displaystyle\operatorname{\mathbb{P}}\{T>t_{j+1}|C>t_{j+1},\bar{a}(t_{j+1}), \mathscr{F}_{t_{j}}\}^{1-\mathbbm{1}(t_{j}<x\leq t_{j+1},\delta=1)} \operatorname{\mathbb{P}}\{\textup{d}\bar{l}(t_{j+1})|C>t_{j+1},\bar{a}(t_{j+1 }),\mathscr{F}_{t_{j}}\}$ $\displaystyle\{1-\mathbbm{1}(x\leq t_{j+1},\delta=0)\}\mathbb{G}\{\textup{d} \bar{a}(t_{j+1})|\bar{a}(t_{j})\}-\int\operatorname{\mathbb{E}}\{\nu(T_{\bar{a }},Y_{\bar{a}})\}\mathbb{G}(\textup{d}\bar{a})\Bigg{|}+o(1),$ $$

where $o(1)$ converges to zero when $\Delta_{K}[0,\infty]\to 0$. By Assumptions 1, 2, 3, and 4, the above term is less than or equal to

$$ $\displaystyle\Bigg{|}\int\nu(x,\bar{y})\operatorname{\mathbb{P}}\{T\leq t_{K}| \bar{a}(t_{K}),\mathscr{F}_{t_{K-1}}\}^{\mathbbm{1}(t_{K-1}<x\leq t_{K},\delta =1)}$ $\displaystyle\operatorname{\mathbb{P}}\{T>t_{K}|\bar{a}(t_{K}),\mathscr{F}_{t_ {K-1}}\}^{1-\mathbbm{1}(t_{K-1}<x\leq t_{K},\delta=1)}\operatorname{\mathbb{P} }\{\textup{d}\bar{l}(t_{K})|\bar{a}(t_{K}),\mathscr{F}_{t_{K-1}}\}$ $\displaystyle\{1-\mathbbm{1}(x\leq t_{K},\delta=0)\}\mathbb{G}\{\textup{d}\bar {a}(t_{K})|\bar{a}(t_{K-1})\}\prod_{j=0}^{K-}\operatorname{\mathbb{P}}\{T\leq t _{j+1}|\bar{a}(t_{j+1}),\mathscr{F}_{/C,t_{j}}\}^{\mathbbm{1}(t_{j}<x\leq t_{j +1},\delta=1)}$ $\displaystyle\operatorname{\mathbb{P}}\{T>t_{j+1}|\bar{a}(t_{j+1}),\mathscr{F} _{/C,t_{j}}\}^{1-\mathbbm{1}(t_{j}<x\leq t_{j+1},\delta=1)}\operatorname{ \mathbb{P}}\{\textup{d}\bar{l}(t_{j+1})|\bar{a}(t_{j+1}),\mathscr{F}_{/C,t_{j}}\}$ $\displaystyle\{1-\mathbbm{1}(x\leq t_{j+1},\delta=0)\}\mathbb{G}\{\textup{d} \bar{a}(t_{j+1})|\bar{a}(t_{j})\}-\int\operatorname{\mathbb{E}}\{\nu(T_{\bar{a }},Y_{\bar{a}})\}\mathbb{G}(\textup{d}\bar{a})\Bigg{|}+o(1),$ $$

which by Assumptions 1, 2, 3, and 4, equals

$$ $\displaystyle\Bigg{|}\int\nu(x,\bar{y})\operatorname{\mathbb{P}}\{T_{\bar{a}} \leq t_{K}|\bar{a}(t_{K}),\mathscr{F}_{t_{K-1}}\}^{\mathbbm{1}(t_{K-1}<x_{\bar {a}}\leq t_{K},\delta_{\bar{a}}=1)}$ $\displaystyle\operatorname{\mathbb{P}}\{T_{\bar{a}}>t_{K}|\bar{a}(t_{K}), \mathscr{F}_{t_{K-1}}\}^{1-\mathbbm{1}(t_{K-1}<x_{\bar{a}}\leq t_{K},\delta_{ \bar{a}}=1)}\operatorname{\mathbb{P}}\{\textup{d}\bar{l}_{\bar{a}}(t_{K})|\bar {a}(t_{K}),\mathscr{F}_{t_{K-1}}\}$ $\displaystyle\{1-\mathbbm{1}(x_{\bar{a}}\leq t_{K},\delta_{\bar{a}}=0)\} \mathbb{G}\{\textup{d}\bar{a}(t_{K})|\bar{a}(t_{K-1})\}\prod_{j=0}^{K-} \operatorname{\mathbb{P}}\{T\leq t_{j+1}|\bar{a}(t_{j+1}),\mathscr{F}_{/C,t_{j }}\}^{\mathbbm{1}(t_{j}<x\leq t_{j+1},\delta=1)}$ $\displaystyle\operatorname{\mathbb{P}}\{T>t_{j+1}|\bar{a}(t_{j+1}),\mathscr{F} _{/C,t_{j}}\}^{1-\mathbbm{1}(t_{j}<x\leq t_{j+1},\delta=1)}\operatorname{ \mathbb{P}}\{\textup{d}\bar{l}(t_{j+1})|\bar{a}(t_{j+1}),\mathscr{F}_{/C,t_{j}}\}$ $\displaystyle\{1-\mathbbm{1}(x\leq t_{j+1},\delta=0)\}\mathbb{G}\{\textup{d} \bar{a}(t_{j+1})|\bar{a}(t_{j})\}-\int\operatorname{\mathbb{E}}\{\nu(T_{\bar{a }},Y_{\bar{a}})\}\mathbb{G}(\textup{d}\bar{a})\Bigg{|}+o(1),$ $$

which by Assumptions 1, 2, 3, and 4, is less than or equal to

$$ $\displaystyle\Bigg{|}\int\nu(x,\bar{y})\operatorname{\mathbb{P}}\{T_{\bar{a}} \leq t_{K}|\mathscr{F}_{t_{K-1}}\}^{\mathbbm{1}(t_{K-1}<x_{\bar{a}}\leq t_{K}, \delta_{\bar{a}}=1)}$ $\displaystyle\operatorname{\mathbb{P}}\{T_{\bar{a}}>t_{K}|\mathscr{F}_{t_{K-1} }\}^{1-\mathbbm{1}(t_{K-1}<x_{\bar{a}}\leq t_{K},\delta_{\bar{a}}=1)} \operatorname{\mathbb{P}}\{\textup{d}\bar{l}_{\bar{a}}(t_{K})|\mathscr{F}_{t_{ K-1}}\}$ $\displaystyle\{1-\mathbbm{1}(x_{\bar{a}}\leq t_{K},\delta_{\bar{a}}=0)\} \mathbb{G}\{\textup{d}\bar{a}(t_{K})|\bar{a}(t_{K-1})\}\prod_{j=0}^{K-} \operatorname{\mathbb{P}}\{T\leq t_{j+1}|\bar{a}(t_{j+1}),\mathscr{F}_{/C,t_{j }}\}^{\mathbbm{1}(t_{j}<x\leq t_{j+1},\delta=1)}$ $\displaystyle\operatorname{\mathbb{P}}\{T>t_{j+1}|\bar{a}(t_{j+1}),\mathscr{F} _{/C,t_{j}}\}^{1-\mathbbm{1}(t_{j}<x\leq t_{j+1},\delta=1)}\operatorname{ \mathbb{P}}\{\textup{d}\bar{l}(t_{j+1})|\bar{a}(t_{j+1}),\mathscr{F}_{/C,t_{j}}\}$ $\displaystyle\{1-\mathbbm{1}(x\leq t_{j+1},\delta=0)\}\mathbb{G}\{\textup{d} \bar{a}(t_{j+1})|\bar{a}(t_{j})\}-\int\operatorname{\mathbb{E}}\{\nu(T_{\bar{a }},Y_{\bar{a}})\}\mathbb{G}(\textup{d}\bar{a})\Bigg{|}+o(1).$ $$

By iterating the above process for $0\leq j\leq K-2$, we arrive the conclusion.

### A.3 Proof of Theorem 2

The proof is immediate by noting that

$$ $\displaystyle\operatorname{\mathbb{E}}\left\{Q_{\mathbb{G}}(\infty)\nu(X,\bar{ Y})\right\}=\operatorname{\mathbb{E}}_{\mathbb{G}}\{\nu(X,\bar{Y})\}=\int_{ \mathcal{A}}\operatorname{\mathbb{E}}\{\nu(T_{\bar{a}},\bar{Y}_{\bar{a}})\} \mathbb{G}(\textup{d}\bar{a}),$ (36) $$

by Theorem 1.

### A.4 Proof of Theorem 3

We first prove the theorem when $H=H_{\mathbb{G}}$. Indeed, as the limit and expectation can interchange, we can show that

$$ $\displaystyle\left|\operatorname{\mathbb{E}}\left\{\Xi_{\textup{out},\Delta_{K }[0,\infty]}(H_{\mathbb{G}},Q)\right\}\right|$ $\displaystyle=\bigg{|}\operatorname{\mathbb{E}}\left(Q(t_{K})[\nu(X,\bar{Y})-H _{\mathbb{G}}(t_{K})\}]\right)$ $\displaystyle+\sum_{j=1}^{K-1}\operatorname{\mathbb{E}}\left(Q(t_{j})\left[ \int H_{\mathbb{G}}(t_{j+1})\mathbb{G}\{\textup{d}\bar{a}(t_{j+1})|\bar{A}(t_{ j})\}-H_{\mathbb{G}}(t_{j})\right]\right)\bigg{|}$ $\displaystyle=\Bigg{|}0+\sum_{j=1}^{K-1}\operatorname{\mathbb{E}}\left(Q(t_{j} )\left[\int H_{\mathbb{G}}(t_{j+1})\mathbb{G}\{\textup{d}\bar{a}(t_{j+1})|\bar {A}(t_{j})\}-H_{\mathbb{G}}(t_{j})\right]\right)\Bigg{|}$ $\displaystyle\leq\sum_{j=1}^{K-1}\Bigg{|}\operatorname{\mathbb{E}}\left(Q(t_{j })\left[\int H_{\mathbb{G}}(t_{j+1})\mathbb{G}\{\textup{d}\bar{a}(t_{j+1})| \bar{A}(t_{j})\}-H_{\mathbb{G}}(t_{j})\right]\right)\Bigg{|}$ $\displaystyle\leq\sum_{j=1}^{K-1}\Bigg{|}\operatorname{\mathbb{E}}\left(Q(t_{j })\left[\int H_{\mathbb{G}}(t_{j+1})\mathbb{G}\{\textup{d}\bar{a}(t_{j+1})| \bar{A}(t_{j})\}-\operatorname{\mathbb{E}}_{\mathbb{G}}\left\{H_{\mathbb{G}}(t _{j+1})|\mathscr{G}_{t_{j}}\right\}\right]\right)\Bigg{|}$ $\displaystyle\leq\sum_{j=0}^{K}\kappa\|H_{\mathbb{G}}(t_{j})Q(t_{j})\|_{1}(t_{ j+1}-t_{j})^{\alpha}$ $\displaystyle\leq\kappa\sup_{t}\|H_{\mathbb{G}}(t)Q(t)\|_{1}\sum_{j=1}^{K-1}(t _{j+1}-t_{j})^{\alpha}\to 0,$ $$

when $|\Delta_{K}[0,\infty]|\to 0$, where we have used the fact that $H_{\mathbb{G}}(t)$ is a $\operatorname{\mathbb{P}}_{\mathbb{G}}$-martingale and Assumptions 1, 2.

We now proceed to the case when $Q=Q_{\mathbb{G}}$. Indeed, as the limit and expectation can interchange, we can show that

$$ $\displaystyle\left|\operatorname{\mathbb{E}}\left\{\Xi_{\textup{trt},\Delta_{K }[0,\infty]}(H,Q_{\mathbb{G}})\right\}\right|$ $\displaystyle=\Bigg{|}\operatorname{\mathbb{E}}\left(\sum_{j=0}^{K}\left[Q_{ \mathbb{G}}(t_{j})H(t_{j})-Q_{\mathbb{G}}(t_{j-1})\int H(t_{j})\mathbb{G}\{ \textup{d}\bar{a}(t_{j})|\bar{A}(t_{j-1})\}\right]\right)$ $\displaystyle-\operatorname{\mathbb{E}}\left[Q_{\mathbb{G}}(0)H(0)-\int H(0) \mathbb{G}\{\textup{d}\bar{a}(0)\}\right]\Bigg{|}$ $\displaystyle=\Bigg{|}\operatorname{\mathbb{E}}\left(\sum_{j=0}^{K}\left[Q_{ \mathbb{G}}(t_{j})H(t_{j})-Q_{\mathbb{G}}(t_{j-1})\int H(t_{j})\mathbb{G}\{ \textup{d}\bar{a}(t_{j})|\bar{A}(t_{j-1})\}\right]\right)+0\Bigg{|}$ $\displaystyle\leq\sum_{j=0}^{K}\left|\operatorname{\mathbb{E}}\left[Q_{\mathbb {G}}(t_{j})H(t_{j})-Q_{\mathbb{G}}(t_{j-1})\int H(t_{j})\mathbb{G}\{\textup{d} \bar{a}(t_{j})|\bar{A}(t_{j-1})\}\right]\right|$ $\displaystyle=\sum_{j=0}^{K}\bigg{|}\operatorname{\mathbb{E}}\bigg{[}Q_{ \mathbb{G}}(t_{j-1})\operatorname{\mathbb{E}}_{\mathbb{G}}\{H(t_{j})|\mathcal{ G}_{t_{j-1}}\}$ $\displaystyle-Q_{\mathbb{G}}(t_{j-1})\int H(t_{j})\mathbb{G}\{\textup{d}\bar{a }(t_{j})|\bar{A}(t_{j-1})\}\operatorname{\mathbb{P}}\{\textup{d}\bar{l}(t_{j}) |\mathcal{G}_{t_{j-1}}\}\bigg{]}\bigg{|}$ $\displaystyle\leq\kappa\sum_{j=0}^{K}\|H(t_{j-1})Q_{\mathbb{G}}(t_{j-1})\|_{1} (t_{j}-t_{j-1})^{\alpha}$ $\displaystyle\leq\kappa\sup_{t}\|H(t)Q_{\mathbb{G}}(t)\|_{1}\sum_{j=0}^{K}(t_{ j}-t_{j-1})^{\alpha}\to 0,$ $$

when $|\Delta_{K}[0,\infty]|\to 0$, where we have used the fact that $Q_{\mathbb{G}}(t)$ is a $\operatorname{\mathbb{P}}$-martingale and Assumptions 1, 2.

### A.5 Proof of Theorem 4

We first simplify our setting by ignoring censoring and absorbing event time $T$ into $\bar{L}$ as well. This is because the conditional independent censoring assumption (Assumption 2) is known to be nonparametric. Our observed data become $(\bar{A},\bar{L})$ and full data become $(\bar{A},\bar{L}_{\mathcal{A}})$.

We first proves that Assumption 1 does not have restrictions on the observed data. We proceed with a constructive proof. For any partition $\Delta_{K}[0,\infty]$, we define a measure on the full data path space. In fact, one has the knowledge on the decomposition

$$ $\displaystyle\operatorname{\mathbb{P}}(\textup{d}\bar{a}\textup{d}\bar{l})$ $\displaystyle=\prod_{j=0}^{K-1}\left[\operatorname{\mathbb{P}}\{\textup{d}\bar {a}(t_{j+1})|\mathscr{F}_{t_{j}}\}\operatorname{\mathbb{P}}\{\textup{d}\bar{l} (t_{j+1})|\bar{a}(t_{j+1}),\mathscr{F}_{t_{j}}\}\right]$ (37) $\displaystyle=\prod_{j=0}^{K-1}\left[\operatorname{\mathbb{P}}\{\textup{d}\bar {a}(t_{j+1})|\bar{a}(t_{j}),\bar{l}_{\bar{a}}(t_{j})\}\operatorname{\mathbb{P} }\{\textup{d}\bar{l}_{\bar{a}}(t_{j+1})|\bar{a}(t_{j+1}),\bar{l}_{\bar{a}}(t_{ j})\}\right].$ (38) $$

Intuitively $\operatorname{\mathbb{P}}\{\textup{d}\bar{l}_{\bar{a}}(t_{j+1})|\bar{a}(t_{j+1
}),\bar{l}_{\bar{a}}(t_{j})\}$ are close to $\operatorname{\mathbb{P}}\{\textup{d}\bar{l}_{\bar{a}}(t_{j+1})|\bar{l}_{\bar{
a}}(t_{j})\}$, whereas the other term $\operatorname{\mathbb{P}}\{\textup{d}\bar{a}(t_{j+1})|\bar{a}(t_{j}),\bar{l}(t
_{j})\}$ is close to $\operatorname{\mathbb{P}}\{\textup{d}\bar{a}(t_{j+1})|\bar{a}(t_{j}),\bar{l}_{
\mathcal{A}}\}$. Therefore, one may define a measure on the full data path space by

$$ $\operatorname{\mathbb{P}}_{\Delta_{K}[0,\infty]}^{F}(\textup{d}\bar{l}_{\bar{a }}):=\prod_{j=0}^{K-1}\operatorname{\mathbb{P}}\{\textup{d}\bar{l}_{\bar{a}}(t _{j+1})|\bar{a}(t_{j+1}),\bar{l}_{\bar{a}}(t_{j})\}=\prod_{j=0}^{K-1} \operatorname{\mathbb{P}}\{\textup{d}\bar{l}_{\bar{a}}(t_{j+1})|\bar{a}(t_{j+1 }),\mathscr{F}_{t_{j}}\}.$ (39) $$

Then without loss of generality one may construct $\operatorname{\mathbb{P}}_{\Delta_{K}[0,\infty]}^{F}(\bar{l}_{\mathcal{A}})$ by assuming joint independence among $\bar{l}_{\mathcal{A}}$. One can also define

$$ $\operatorname{\mathbb{P}}_{\Delta_{K}[0,\infty]}^{F}(\textup{d}\bar{a}|\bar{l} _{\mathcal{A}}):=\prod_{j=0}^{K-1}\operatorname{\mathbb{P}}\{\textup{d}\bar{a} (t_{j+1})|\mathscr{F}_{t_{j}}\}.$ (40) $$

Then for any sequences of partitions with the mesh going to zero, one may construct a sequence of measures and show that this sequence of measures is Cauchy by a triangular inequality and Assumption 1, following a similar logic as previous proofs. Therefore the sequence converge to a measure $\operatorname{\mathbb{P}}^{F}$, which is independent of the choice of partitions.

Next we need to show that $\operatorname{\mathbb{P}}^{F}$ induces $\operatorname{\mathbb{P}}$ on the observed data and $\operatorname{\mathbb{P}}^{F}$ satisfies Assumption 1. The first is trivial because any $\operatorname{\mathbb{P}}_{\Delta_{K}[0,\infty]}^{F}$ induces $\operatorname{\mathbb{P}}$ on the observed data, then so is their limit. To prove the second, for any time $t$ and $\varepsilon>0$, one might smartly choose a partition $\Delta_{K}[0,\infty]$ with $\operatorname{\mathbb{P}}_{\Delta_{K}[0,\infty]}^{F}$ close enough to $\operatorname{\mathbb{P}}^{F}$ and $t,t+\eta\in\Delta_{K}[0,\infty]$. This can be done because the convergence point is independent of the choice of partitions. We have

$$ $\displaystyle\operatorname{\mathbb{E}}^{F}(\|\operatorname{\mathbb{P}}^{F}\{ \textup{d}\bar{l}_{\mathcal{A}}|\mathscr{F}_{t}\}-\operatorname{\mathbb{P}}^{F }\{\textup{d}\bar{l}_{\mathcal{A}}|\bar{A}(t+\eta),\mathscr{F}_{t}\}\|_{ \textup{TV}})$ $\displaystyle\leq\operatorname{\mathbb{E}}^{F}(\|\operatorname{\mathbb{P}}^{F} \{\textup{d}\bar{l}_{\mathcal{A}}|\mathscr{F}_{t}\}-\operatorname{\mathbb{P}}_ {\Delta_{K}[0,\infty]}^{F}\{\textup{d}\bar{l}_{\mathcal{A}}|\mathscr{F}_{t}\} \|_{\textup{TV}})$ $\displaystyle+\operatorname{\mathbb{E}}^{F}(\|\operatorname{\mathbb{P}}_{ \Delta_{K}[0,\infty]}^{F}\{\textup{d}\bar{l}_{\mathcal{A}}|\mathscr{F}_{t}\}- \operatorname{\mathbb{P}}_{\Delta_{K}[0,\infty]}^{F}\{\textup{d}\bar{l}_{ \mathcal{A}}|\bar{A}(t+\eta),\mathscr{F}_{t}\}\|_{\textup{TV}})$ $\displaystyle+\operatorname{\mathbb{E}}^{F}(\|\operatorname{\mathbb{P}}_{ \Delta_{K}[0,\infty]}^{F}\{\textup{d}\bar{l}_{\mathcal{A}}|\bar{A}(t+\eta), \mathscr{F}_{t}\}-\operatorname{\mathbb{P}}^{F}\{\textup{d}\bar{l}_{\mathcal{A }}|\bar{A}(t+\eta),\mathscr{F}_{t}\}\|_{\textup{TV}})$ $\displaystyle\leq 2\|\operatorname{\mathbb{P}}^{F}-\operatorname{\mathbb{P}}_{ \Delta_{K}[0,\infty]}^{F}\|_{\textup{TV}}$ $\displaystyle+\operatorname{\mathbb{E}}^{F}(\|\operatorname{\mathbb{P}}_{ \Delta_{K}[0,\infty]}^{F}\{\textup{d}\bar{l}_{\mathcal{A}}|\mathscr{F}_{t}\}- \operatorname{\mathbb{P}}_{\Delta_{K}[0,\infty]}^{F}\{\textup{d}\bar{l}_{ \mathcal{A}}|\bar{A}(t+\eta),\mathscr{F}_{t}\}\|_{\textup{TV}})$ $\displaystyle\leq 4\|\operatorname{\mathbb{P}}^{F}-\operatorname{\mathbb{P}}_{ \Delta_{K}[0,\infty]}^{F}\|_{\textup{TV}}$ $\displaystyle+\operatorname{\mathbb{E}}_{\Delta_{K}[0,\infty]}^{F}(\| \operatorname{\mathbb{P}}_{\Delta_{K}[0,\infty]}^{F}\{\textup{d}\bar{l}_{ \mathcal{A}}|\mathscr{F}_{t}\}-\operatorname{\mathbb{P}}_{\Delta_{K}[0,\infty] }^{F}\{\textup{d}\bar{l}_{\mathcal{A}}|\bar{A}(t+\eta),\mathscr{F}_{t}\}\|_{ \textup{TV}}).$ $$

The first term can be chosen to be sufficiently small. We rewrite the second term

$$ $\displaystyle\operatorname{\mathbb{E}}_{\Delta_{K}[0,\infty]}^{F}(\| \operatorname{\mathbb{P}}_{\Delta_{K}[0,\infty]}^{F}\{\textup{d}\bar{l}_{ \mathcal{A}}|\mathscr{F}_{t}\}-\operatorname{\mathbb{P}}_{\Delta_{K}[0,\infty] }^{F}\{\textup{d}\bar{l}_{\mathcal{A}}|\bar{A}(t+\eta),\mathscr{F}_{t}\}\|_{ \textup{TV}})$ $\displaystyle=\sup_{f:\|f\|_{1}=1}\int f(\bar{l}_{\mathcal{A}},\{y_{\bar{a}^{ \prime}}\}_{\bar{a}^{\prime}\in\mathcal{A}})[\operatorname{\mathbb{P}}_{\Delta _{K}[0,\infty]}^{F}\{\textup{d}\bar{l}_{\mathcal{A}}|\mathscr{F}_{t}\}$ $\displaystyle~{}~{}-\operatorname{\mathbb{P}}_{\Delta_{K}[0,\infty]}^{F}\{ \textup{d}\bar{l}_{\mathcal{A}}|\bar{a}(t+\eta),\mathscr{F}_{t}\}] \operatorname{\mathbb{P}}\{\textup{d}\bar{a}(t+\eta)\textup{d}\mathscr{F}_{t}\}$ $\displaystyle\leq\sup_{f:\|f\|_{1}=1}\sup_{g:\|g\|_{1}=1}\int f(\bar{l}_{ \mathcal{A}})g(\bar{a}(t+\eta))[\operatorname{\mathbb{P}}_{\Delta_{K}[0,\infty ]}^{F}\{\textup{d}\bar{l}_{\mathcal{A}}\textup{d}\bar{a}(t+\eta)|\mathscr{F}_{ t}\}$ $\displaystyle~{}~{}-\operatorname{\mathbb{P}}_{\Delta_{K}[0,\infty]}^{F}\{ \textup{d}\bar{l}_{\mathcal{A}}|\mathscr{F}_{t}\}\operatorname{\mathbb{P}}_{ \Delta_{K}[0,\infty]}^{F}\{\textup{d}\bar{a}(t+\eta)|\mathscr{F}_{t}\}] \operatorname{\mathbb{P}}\{\textup{d}\mathscr{F}_{t}\}$ $\displaystyle=\sup_{f:\|f\|_{1}=1}\sup_{g:\|g\|_{1}=1}\int f(\bar{l}_{\mathcal {A}})g(\bar{a}(t+\eta))\operatorname{\mathbb{P}}_{\Delta_{K}[0,\infty]}^{F}\{ \textup{d}\bar{l}_{\mathcal{A}}|\mathscr{F}_{t}\}$ $\displaystyle~{}~{}[\operatorname{\mathbb{P}}_{\Delta_{K}[0,\infty]}^{F}\{ \textup{d}\bar{a}(t+\eta)|\bar{a}(t),\bar{l}_{\mathcal{A}}\}-\operatorname{ \mathbb{P}}_{\Delta_{K}[0,\infty]}^{F}\{\textup{d}\bar{a}(t+\eta)|\mathscr{F}_ {t}\}]\operatorname{\mathbb{P}}\{\textup{d}\mathscr{F}_{t}\}$ $\displaystyle=\sup_{f:\|f\|_{1}=1}\sup_{g:\|g\|_{1}=1}\int f(\bar{l}_{\mathcal {A}})g(\bar{a}(t+\eta))\operatorname{\mathbb{P}}_{\Delta_{K}[0,\infty]}^{F}\{ \textup{d}\bar{l}_{\mathcal{A}}|\mathscr{F}_{t}\}$ $\displaystyle~{}~{}\left[\prod_{j=0}^{K-1}\operatorname{\mathbb{P}}\{\textup{d }\bar{a}(t+\eta)|\mathscr{F}_{t}\}-\operatorname{\mathbb{P}}\{\textup{d}\bar{a }(t+\eta)|\mathscr{F}_{t}\}\right]\operatorname{\mathbb{P}}\{\textup{d} \mathscr{F}_{t}\}$ $\displaystyle\leq\operatorname{\mathbb{E}}\left\|\prod_{j=0}^{K-1} \operatorname{\mathbb{P}}\{\textup{d}\bar{a}(t+\eta)|\mathscr{F}_{t}\}- \operatorname{\mathbb{P}}\{\textup{d}\bar{a}(t+\eta)|\mathscr{F}_{t}\}\right\| _{\textup{TV}}$ $\displaystyle\leq\varepsilon(t,\eta).$ $$

Assumption 3 is irrelevant here because it is imposed on the stochastic process but not the distributions.

For Assumption 4, a necessary condition for it to hold is that $\operatorname{\mathbb{P}}$ is well supported on the path space conditioning on any filtration. For this to happen, note that for any $\operatorname{\mathbb{P}}$, one can find a well-supported $\operatorname{\mathbb{P}}^{\prime}$ satisfying Assumptions 1, 2, 3, so that $(1-\varepsilon)\operatorname{\mathbb{P}}+\varepsilon\operatorname{\mathbb{P}}^
{\prime}$ is well-supported and hence satisfies Assumption 4. Since addition will not break Assumptions 1, 2, 3, we have found $(1-\varepsilon)\operatorname{\mathbb{P}}+\varepsilon\operatorname{\mathbb{P}}^
{\prime}$ satisfying Assumptions 1, 2, 3, and 4, and approximates $\operatorname{\mathbb{P}}$.

## Peer Reviews

confidence: 2
contribution: 2
initial_rating: 3
presentation: 3
questions: Line 141, authors mentioned "note this is not a density function". Then please specify what this is. 

Line 325, why is it sufficient to evaluate the approximation of the G-computation formula? I don't think on population level, the values of three formulas are numerically equal. Even if they are equal, they may have quite different finite-sample behaviors.
soundness: 2
strengths: Treatment effect on functional longitudinal data seems to be an understudied subject. This research nicely fills the gap of existing works. 

The resulting G-computation can be quite straightforwardly approximated using observations under simulation settings.
summary: In this paper, authors consider causal inference on time-varying data (functional longitudinal data). They generalize the classical g-computation, inverse probability weighting and doubly robust formulas to the time-varying setting subject to censoring and mortality. The g-computation formula is simulated using Monte-Carlo on a toy dataset, achieving promising results.
weaknesses: The way this paper is written obscures its main ideas (at least to a general, non-expert reader). There are many terms and phrases used without clear explanation (e.g., "g-computation", "counterfactual time-to-event endpoint"). This restricts the range of potential readers of this paper.  

**The experiments are limited to only simulation data and only validate the G-computation formula**. 

The literature review of this paper (section 2) does not seem to provide much information of existing works as without proper explanation, readers may be unclear what "temporal aspect", "point exposure" and "end-of-study outcome" means. I recommend removing Figure 1 and expanding on each of the subsections, providing more details of existing works. 

The preparation in Section 3.1 is quite long. Without concrete examples, it is hard for readers to understand what they actually mean. I suggest skip some unnecessary notations, and explain them as the paper progresses. 
    - Some symbols are better explained with examples. For instance, authors could give an example of nu and G, around equation (1).

---

confidence: 3
contribution: 3
initial_rating: 6
presentation: 3
questions: 1. Why is the interventional distribution ((7)-(10)) formulated in this way? Specifically, I’m curious about where the term $\{1 -   \mathbb{1}(x \leq t_{j+1}, \delta=0) }$ comes from.
2. Can this framework be extended to dependent censoring?
soundness: 3
strengths: This paper establishes a new causal identification framework for continuous-time longitudinal studies with functional data, and provides clear and concise theoretical demonstration. I believe that this framework will be of interest to causal inference and machine learning communities.
summary: This paper proposes a causal identification framework that bridges classical causal inference framework, continuous-time longitudinal analysis and functional data analysis. In this framework, the parameter of interest is the marginal mean of counterfactual outcomes under a measure that allows randomly assigned treatments, with absence of censoring. Leveraging the tools in stochastic process, the authors then demonstrate the identification results for three classical estimation strategies in causal inference: g-computation, IPW and doubly robust estimation. The authors further claims that the identification framework also has non-parametric property.
weaknesses: 1. The numerical experiment might be an over-simplification of the survival analysis scenario since neither mortality nor censoring are taken into consideration. 
2. What is the causal structure that the framework is focusing on? Specifically, why set $Y(t)$ (outcome of interest) to be a subset of $L(t)$ (measured confounders)? I might misunderstood but are we assuming that previous outcome will impact the current treatment assignment (since confounders, from my understanding, will impact treatment assignment)?
3. I guess it would be helpful to attract readers in a wider community if more intuitive explanation could be added after stating definitions/propositions.

---

confidence: 4
contribution: 2
initial_rating: 3
presentation: 1
questions: Could you elaborate on the situation when A(t) is a function? 

Why should Y(t) be a subset of L(t), and what does it mean?
soundness: 3
strengths: The approach is nonparametric and it accommodates functional treatment processes A(t) and functional confounders L(t), as well as functional response Y(t).
summary: The paper is challenging to follow, so I may have misunderstood some parts. My understanding is that the "functional longitudinal data" investigated here are conventional functional data, as described by Wang et al. (2016), which can be measured intensively, sparsely, or irregularly. However, this paper focuses solely on the ideal (hypothetical) setting where continuous-time measurements are available for each experimental subject, resulting in infinite-dimensional data. If this interpretation is correct, the goal of this paper is to explore causal identification for infinite-dimensional functional (time-varying) outcomes that are subject to mortality and censoring by generalizing the classical g-computation, inverse probability weighting, and doubly robust formulas.

Reference: 
Wang, Chiou and Müller (2016). Functional data analysis. Annual Review of Statistics ands its application.
weaknesses: The paper is hard to follow and the connection of the event-time T to the outcome Y(t) is unclear.

---

confidence: 2
contribution: 2
initial_rating: 3
presentation: 1
questions: Please provide additional explanations for the questions raised in the Weakness section.
soundness: 2
strengths: This paper addresses the problem of causality identification in complex scenarios inherent in functional longitudinal data, a highly general and advanced form of data. If it can overcome the criticisms and shortcomings pointed out by other reviewers and get published, it could serve as a milestone, taking one step forward in functional data analysis for causal inference.
summary: This paper presents a nonparametric causal identification framework for functional longitudinal data, exemplified by the MIMIC-IV dataset, which includes complexities such as events like death. By leveraging stochastic process theory and measure theory, the framework generalizes g-computation, inverse probability weighting, and doubly robust formulas, effectively handling time-varying outcomes with mortality and censoring. Monte Carlo simulations verifies it.
weaknesses: This paper is more abstract than necessary, making it harder to read than other papers. Even after spending more time on it than on other review papers, the preliminaries, equations, and theorems presented in the paper are not easily understood at once. As a result, I had to refer to the cited papers, and I found two papers that share many aspects with this one. One is "Causality for Functional Longitudinal Data," which is cited by (Ying, 2024) in this paper, and the other is "Causality for Complex Continuous-Time Functional Longitudinal Studies with Dynamic Treatment Regimes," which is not cited in this paper. The title of this paper is "Causal Identification for Complex Functional Longitudinal Studies," suggesting that the most significant update in this paper is the "complex" aspect in the title. The paper "Causality for Complex Continuous-Time Functional Longitudinal Studies with Dynamic Treatment Regimes" seems to have updated the dynamic treatment regimes element one step further.

Specifically, in the preparation section, the time-to-event endpoint $T$, $C$, $X$, and $\Delta$ are newly defined, assuming a more complex situation where the study may be forcibly terminated due to an event like death before the study is completed. This updates Theorems 1, 2, and 3 from the previous paper. I would like to ask about the academic and practical significance of solving the more complex problem introduced by those new variables. The most important contribution of this paper seems to be the introduction of the non-parametric property through Theorem 4. Why is the non-parametric property important in a functional data framework? The paper states that it makes the model more flexible and adaptable to various data, but doesn't the continuous functional data, which is more extensive, lead to increased computational and implementation complexity, a common drawback of non-parametric models? Doesn't this also create issues with the interpretability required for healthcare data analysis?

One of the most important equations in this paper seems to be Equation (1). The rest of the paper is dedicated to finding another representation of Equation (1). However, it is not easy for readers to immediately understand what Equation (1) means and why we should be interested. Additionally, it is not straightforward to grasp what $\mathbb{G}$ represents. By referring to the paper by Ying (2024), I could somewhat understand $\mathbb{G}$ through the following example:
*When the causal outcome under a specific regime $\bar{a}$ is of interest, for instance, all patients were under treatment, the point mass (delta) measure $\mathbb{G} = \mathbb{1}(\bar{A} = \bar{a})$ can be considered.*
Including this example in this paper would help in understanding. Furthermore, providing a concrete example of what $\nu$ represents would help comprehend Equation (1). Can Equation (1) be understood as a general expression representing the average treatment effect, the averaged treatment outcome, or a transformed form of these?

If the non-parametric property is a major contribution of the paper, it should be demonstrated through more concrete experimental examples, such as using the MIMIC-IV data mentioned in the introduction. The experimental section currently numerically verifies Theorem 1, which has already been proven in the Appendix, but a demonstration of Theorem 4 seems more necessary. However, under Theorem 4, it is only mentioned that "we have not achieved the full nonparametric paradigm."

Additionally, there is a need to clearly and definitively define loosely defined “functional” data. The abstract of this paper describes it as "characterized by continuous-time measurements," while another cited paper describes it as "characterized by continuous-time processes and high-dimensional measurements." I believe "continuous" alone is not sufficient to be called functional. What is the rationale for developing a framework that assumes functional continuity in the model even though real-world healthcare data does not have mathematically rigorous time continuity and does not observe over an infinite time? (The previous paper assumed up to time $\tau$, but this paper assumes up to $\infty$.) What is the justification for this assumption?

Meta-review: The paper provides steps towards causal adjustments with continuous-time data and situations with confounded feedback. While the discrete-time version is well-studied (including work by Robins dating back to the 80s), the continuous-time less so. The paper does provide advances towards this relevant problem, with an emphasis on a worked-out experimental example instead of a major benchmark - I think this was a good choice.

 It is fair to say that the novelty is relatively limited compared to recent progress, but on the other hand this line of work is not too well-known to the ICLR audience and bringing it to his audience that's a plus in that regard. The drawback is that even though terms will be familiar to experts in causal inference coming from a more statistical background, in general it is very dense for a more familiar ICLR audience.

Decision: Accept (Poster)