Learning Time-shared Hidden Heterogeneity for Counterfactual Outcome Forecast

We are grateful for your insightful suggestions and constructive feedbacks. Below, we outlined the major contributions of our paper and address your concerns accordingly.

It is noteworthy that our major contribution is not improving the capability of model architecture in dealing with long time-series, but leveraging the property of time series (treatment & response interaction) data to recover hidden factors and address hidden heterogeneity under weaker assumptions than previous methods.

Previous research[4] has revealed the importance of hidden heterogeneity induced by hidden factors. The previous methods utilize the information in high-dimensional treatments and proxies for the recovery of latent factors. However, in most real world scenarios, such as marketing promotions, medicine treatment effect, financial lending decision, treatments are mostly single-dimensional, rendering previous methods inapplicable. To adapt the idea of recovering hidden factors into more real applications, we proposed method that can capture hidden factors under single-dimensional treatment with time-series data, and we argue that time-series data such as e-commerce marketing, patient trajectories and financial records are ubiquitous in real world scenario. Specifically, we designed a novel mechanism that leverages the inherent advantage of possessing multiple outcome across time steps in time series to recover time-shared latent factors. Therefore, our contribution does not focus on improving the capability of model architecture in dealing with long time series (e.g. Causal Transformer leverage the powerful ability of Transformers in capturing the complex and long-range dependencies, to improve the model capability for counterfactual forecast). Instead, our paper focus on addressing the hidden heterogeneity problem brought by hidden factors. These two perspectives are orthogonal and complementary.

To address the hidden heterogeneity problem, our proposed THLTS method acting as a flexible component instead of a new model architecture to recover the hidden factors, which can be combined with many off-the-shelf counterfactual forecast model. Therefore, in our implementations, we adopt the backbone model architecture same to the previous works, such as CRN, Causal Transformer. Further exploration on incorporating periodicity and seasonality to improve backbone model architecture can be left to future work.

Weakness 1: The paper lacks novelty. The idea of recovering hidden factors has been widely explored in previous research.

Response: We have illustrated the major contribution of our paper above, specifically, we consider the new opportunity and challenge brought by the temporal structure and design the corresponding solution. To be concrete, the hidden factors can vary across many time steps. Hence, the entire solution space of hidden factors can be extremely large which makes straightforwardly learning the hidden factors of each time step difficult. Moreover, the limited supervision information (i.e. the outcome variable is of few-dimension) of single time step further exacerbate the difficulty in learning hidden factors.

To alleviate this issue, we propose a new strategy which leverages the multiple outcome across time steps to learn the time-shared latent factors exclusive to each sample and consequently constrain the over-flexible space of latent factor learning. Although it sacrifices the flexibility in modeling time-varying dynamics of latent factors, this design constrains the model flexibility and acts like a regularizer to improve forecast performance. Taking these into consideration, our work does not simply adopt the existing idea of recovering hidden factors, but give the design leveraging the exclusive problem property. Therefore, our contribution is sufficient.

Weakness 2: It does not introduce any specialized structures to capture unique features of long time series

Response: Thanks for your valuable suggestions. Since the major motivation of our paper is to resolve the hidden heterogeneity induced by hidden factors, we propose a flexible component THLTS that can be combined with off-the-shell counterfactual forecast models. The combined backbone use sequence models, such as LSTM and Transformers to capture the dependencies among time steps. Based on these, our THLTS method complement the design for temporal structure by setting the prior distribution at the $(t+1)^{th}$ step as the obtained posterior distribution at the $t^{th}$ step of the same sample according to the Bayes’ Theorem. The idea behind this design is that the observation and information of early stages can serves as the evidence for inference at the later stages in time series.

We appreciate your valuable comments and constructive feedbacks on our manuscript.

Weakness 1: In Proposition 4.1, it would be helpful for the authors to explain more about when the prediction model $g$ is Lipschitz with respect to $e$.

Response: It is not a restricted hypothesis for the prediction module $g$ to be Lipschitz w.r.t latent factors $e$. The model $g$ is designed as a Multilayer Perceptron (MLP) comprised of fully-connected layers, and can be viewed as the composition of linear functions and non-linear activation functions $z_i=\sigma(W_iz_{i-1}+\mathbf{b}_i), 1\leq i \leq K$. The linear functions $Wz+\mathbf{b}$ are $\alpha$-Lipschitz continuous function w.r.t the input, where the constant $\alpha$ is the spectral norm of the weight matrix $\mathbf{W}$. The typical activation functions, including Sigmoid, ReLu and Tanh, are also Lipschitz continuous function, as the derivatives of them are bounded. For example, the Lipschitz constant of ReLu function is 1, and that of sigmoid function is $\frac{1}{4}$. Therefore, the composition of these Lipschitz continuous functions (i.e. the prediction model $g$) is also a Lipschitz continuous function.

Moreover, the hypothesis of Lipschitz function is broadly adopted in other related works of causal inference [1,2,3], where the loss function is assumed to be Lipschitz w.r.t the input covariates.

In summary, it is not a restricted hypothesis that the model $g$ is Lipschitz with respect to the part of input vector $e$.

Weakness 2: It would be beneficial to provide some analysis regarding the identifiability of your method.

Response: It has been demonstrated by the previous literature [4,5] in this community that the identifiability of latent factor recovery is difficult to be theoretically guaranteed without restricted assumption. However, it does not hinder the practical value of it which has been justified by these works. As a complement, we conduct an empirically analysis to show that the learned latent factors can indicate the true hidden factors.

To be concrete, we train a predictor mapping the sampled latent factors to the true hidden factors. Lower prediction error implies that the learned latent factors is more closely related to the true hidden factors, which means higher degree of identifiablity. Specifically, we set the sequence length to 30 and employed the same hyperparameters to train our proposed model, which is based on the Causal Transformer architecture. We then derived the inferred latent factor $\bar{\mathbf{e}}^{(i)}$ at $t^{th}$ time stamp, meaning it relies solely on the information ($X_{1:t}^{(i)}, A_{1:t}^{(i)}, \mathbf{Y}_{1:t}^{(i)}$) available up to the $t^{th}$ time stamp. To note that for the time stamp $t = 0$, the sampled latent factor is initialised as $\bar{\mathbf{e}} ^{(i)} \sim \mathcal{N}(\mathbf{0}, \mathbf{I}_e)$ without any observation information. Subsequently, we utilized a Linear Regression model as the predictor backbone mapping the inferred $\bar{\mathbf{e}} ^{(i)}$ to the ground truth latent factor. The results are demonstrated in Table 1, the regression MSE decrease significantly from $t=0$ to $t=15$, and reach a significantly low level after $t=15$. This empirically validates that the latent factors learned by our method is closely related to the true hidden factors that we intend to find. In Figure 5 of our original paper, we can observe a similar phenomenon that the advantages of THLTS over backbone models become progressively more prominent since $t > 5$, which can be explained by the more precise recovery of latent factor with longer histories data.

We appreciate your insightful comments and positive feedback. In response to your suggestions, we have made the revised version of our manuscript to enhance its comprehensiveness and clarity.

Weakness 1: Clarity issues in some details

Response: Thank you for pointing out the presentation issues of our paper. We have addressed these concerns in the revised manuscript to minimize potential confusion for readers.

Specifically, we have revised Figure 1 and changed the sample indices for each latent factor to superscripts for better clarity. Besides, we have added the description of $m$ in Line 348-349. To further ensure the clear definition of symbols, we have added a table summarizing all used symbols in Section E of the Appendix.

Weakness 2: Insufficient background information

Response: We have supplemented the background information of VAE to provide readers with a clearer understanding of the underlying concepts. Hopefully this will make it easier for readers to grasp the model design and the pipeline of our proposed method. It can be found in the Section F of Appendix.

Weakness 3: The experiments are not sufficiently extensive

Response: Thank you for your feedback regarding the experiments. We will address your concerns as follows:

Reliance on synthetic dataset: Since counterfactual prediction aims to estimate the outcome of a unit under different treatments, which inherently requires groundtruth outcomes for all possible treatments, including the counterfactual treatments not observed in datasets. This makes it impractical to evaluate these methods in real-world scenarios [4]. For example, the commonly used dataset in temporal counterfactual learning literature is MIMIC-III, which records factual treatments and outcomes of patients. Specifically, the previous works in this field [1,2,3,4,5] mainly relies on the (semi-)synthetic datasets for empirical evaluation. In line with these works, we follow this established protocol to validate our proposed method. We use real-world features (collected patients status) to enhance the applicability and realism of our empirical results.

Comparison baselines appears to lack novelty: The most advanced baseline we have included in our original paper is Causal Transformer (it incorporates Transformer Architecture into Counterfactual Prediction task) and was published at ICML 2022. Additionally, we have incorporated the most recent baselines that leverage different causal learning mechanisms, such as G-Net (G-Computation, ML4H 2021), CRN (Invariant Learning, ICLR 2020), and RMSN (IPS-Weighted, NeurIPS 2018). These choices ensure that our experiments are both relevant and competitive within the current state of the field.

Question 1: Is there a more comprehensive explanation or theoretical justification to the superiority of Time-Shared Hidden Factors?

Response: On the one hand, the superior of modelling Time-Shared Hidden Factors is empirically justified in the experimental results. Specifically, the version of our method labeled THLTS$^{(v)}$ aims to capture the dynamic part of latent factors, which sets the prior distribution $\mathcal{N}(\mu^{pr}, \sigma^{pr})$ as a transformation of the posterior distribution $\mathcal{N}(\psi_{\mu}(\mu_{t-1}^{(i)}), Diag(\psi_{\sigma}(\sigma_{t-1}^{(i)})))$ instead of the original posterior distribution $\mathcal{N}(\mu_{t-1}^{(i)}, Diag(\sigma_{t-1}^{(i)}))$ of the previous time step. Our experimental results show that the THLTS model outperforms the THLTS$^{(v)}$ model, validating the effectiveness of learning time-shared latent factors.

The rationale behind this improvement can be attributed to the model regularization effect, which constrains the flexibility of the model. The traditional practice in machine learning have revealed that the excessively flexible model can suffer from overfitting problem. To mitigate this issue, various regularizers have been proposed to reduce overfitting and enhance predictive performance. Inspired by these findings, we design the mechanism of learning time-shared latent factors which constrains the solution space of latent factors and play a similar role to regularizers in learning temporal counterfactual outcome with latent factors. The more rigorous theoretical analysis is attractive and requires substantial effort. We decide to leave it to future work.