Gang Fang's Blog

What does modeling really mean in statistical studies?

Answer after reading: modeling is the process of representing a real-world phenomenon of interest with a mathematical model, usually simplified, in order to understand the phenomenon, answer questions and make predictions. The crucial steps include: identify the problem, collect data, select model, fit model (parameters) to data, validate model, interpret results, refine model and apply model/make predictions.

Critique: Simple QA, considered accurate.

What is the meaning of underdetermination of causal models? And why is that caused by the hierarchy of equivalence relations (i.e., Markov equivalence, statistical equivalence and causal equivalence)?

My answer: underdetermination of causal models means the effects of manipulations can’t be determined or predicted precisely because the set of causal models best describing the causal relationships can’t be found for such predictions.

The hierarchy of equivalence relations leads to that because current available techniques only allow the identifications of Markov equivalent (which is easier) and statistically equivalent class of models. Since Markov equivalent class is a superset of statistically equivalent class, which in turn is a superset of causally equivalent class, we have to consider all models’ results from models in the statistically equivalent class, consisting of both inside the causal equivalence class and outside (i.e., the best set of causal models and the other). This causes underdetermination.

Critique after reading ChatGPT’s critique: 2 points –

Definition of underdetermination is a broader philosophical issue than the sole explanation of the hierarchy of equivalence relations. Fundamentally, it means a causal phenomenon can be described by multiple models a given set of evidence can be explained by multiple, equally valid theories or models. And that implies even within the causal equivalent class, underdetermination still exists due to the multiplicity of models. Hence, the true causal relationships that produce the observational data can be identified.

The most important implication of the equivalence hierarchy goes beyond the nested structure of the hierarchy. It focuses on the difficulty in going from statistical equivalence to causal equivalence, which is a central challenge in causal inference because causal equivalence requires not just observational, but also interventional data (or strong assumptions) to distinguish between models that are statistically equivalent but differ causally.