Local Causal Discovery with the MMB-by-MMB Algorithm

Causal Discovery is the process of identifying relationships between variables in observational data. This is important for understanding how different factors affect each other and for making predictions about how changes to one variable might influence another. However, finding these Causal Relationships can be tough, especially when there are hidden or unmeasured variables, also known as Latent Variables. These hidden variables can interfere with our ability to understand the true relationships among the measured variables.

#The Challenge of Latent Variables

Latent variables are those that we cannot directly observe or measure. They may influence the variables we do measure and can lead to incorrect conclusions if not accounted for. For example, if we are studying the relationship between exercise and weight loss, a latent variable could be a person's metabolism, which affects both exercise effectiveness and weight loss. If we ignore this hidden factor, we might fail to accurately identify how exercise impacts weight.

#Current Methods and Their Limitations

Many existing methods for causal discovery assume that we have access to all relevant variables. This assumption is known as causal sufficiency. While some techniques have been developed to handle situations where there are latent variables, these often aim to identify the entire causal graph involving all variables. In many practical cases, researchers are more interested in understanding the local causal relationships related to a specific variable of interest.

For instance, if we want to know how exercise affects weight loss, we might only care about the relationships involving these two variables instead of the full network of related factors. Some methods exist, like the Local Causal Discovery (LCD) algorithm, which focus on subsets of variables. However, these often still assume we have measured all relevant factors, which is not always the case in real-world situations.

#Our Approach to Local Causal Structure Learning

In light of the challenges presented by latent variables, we propose a new method called the MMB-by-MMB algorithm. This algorithm aims to identify the direct causes and effects of a specific variable, even when there are hidden variables involved. By focusing on local structures, our method can provide clearer insights into the relationships surrounding a target variable, without the need to know the entire causal graph.

#Key Ideas of the MMB-by-MMB Algorithm

The MMB-by-MMB algorithm works in a sequential manner, identifying the local causal structure around a target variable. We start with a set of relevant nodes and iteratively refine our understanding of the causal relationships by checking for potential edges and directional relationships between these nodes.

In each step of the process, we focus on learning the Markov Blanket of the target variable. The Markov Blanket consists of the parents (causes), children (effects), and spouses (other connected nodes that are neither parents nor children) of the target. By identifying this blanket, we can better understand the local influences affecting our target variable.

#Steps of the Algorithm

1. Initialization: We begin by defining the target variable and setting up initial lists of nodes to check.
2. Learning the Markov Blanket: We learn the causal structure around the target variable by determining which nodes are connected to it and how they influence each other.
3. Updating Causal Information: After learning the Markov Blanket, we use this information to identify true causal relationships and update our list of relevant nodes.
4. Orientation of Edges: We orient the edges based on the identified relationships, distinguishing between causes and effects.
5. Stopping Criteria: The algorithm continues until specific criteria are met, indicating that we have sufficiently identified the causal structure around our target variable.

#Validation of the MMB-by-MMB Algorithm

To ensure that our method works correctly, we provide theoretical evidence that the MMB-by-MMB algorithm can accurately identify the direct causes and effects of a target variable. Under certain assumptions, such as having enough observational data and no selection bias, our algorithm is shown to yield the same results as global learning methods.

We also validate our approach through experiments utilizing synthetic data and real-world datasets. In various scenarios, our algorithm successfully identified causal relationships and demonstrated better performance than existing methods, particularly in situations involving latent variables.

#Experimental Results

We conducted extensive experiments to compare the MMB-by-MMB algorithm against both global and local learning methods. This involved testing on different networks and datasets, varying in complexity and size.

In our experiments, we measured performance using several metrics, including precision (how many of the identified edges are truly causal), recall (how many actual causal edges were identified), F1 score (the balance between precision and recall), and the number of conditional independence tests performed.

Our results showed that the MMB-by-MMB algorithm consistently outperformed other methods across different metrics and datasets. This indicates that it is more effective at identifying Local Causal Structures, especially when latent variables are involved.

#Application in Gene Expression Data

To further illustrate the utility of our method, we applied the MMB-by-MMB algorithm to gene expression datasets. In this context, it is crucial to understand how different genes interact with each other and how they may be influenced by unseen factors such as environmental conditions.

For example, we examined genes involved in isoprenoid synthesis in plants, focusing on how specific genes influence one another. By applying our algorithm, we were able to identify meaningful causal relationships among the genes, which aligned with existing biological knowledge.

#Conclusion and Future Directions

The MMB-by-MMB algorithm presents a practical approach to local causal discovery in the presence of latent variables. By focusing on local structures, we can derive insights that are relevant to specific questions without needing a complete understanding of all causal relationships in a complex system.

However, we acknowledge that there are still challenges in causal discovery, particularly when it comes to fully understanding the effects of latent variables. Future work will look into leveraging background knowledge and integrating different approaches, such as combining observational and experimental data, to improve our ability to identify causal relationships.

Overall, our research enhances the tools available for causal discovery, providing a clearer methodology for analyzing the complex interactions present in many real-world systems. The potential applications of this research span across various fields, including social sciences, epidemiology, and biology, where understanding causal relationships is essential for effective decision-making and intervention strategies.