This paper proposes a collapsible method for estimating causal effects that maintains the estimator's consistency before and after marginalization over some variables in completed partially directed acyclic graphs (CPDAGs). We first introduce the estimate collapsibility for CPDAGs and characterize the minimal collapsible sets as strong d-convex hulls. An efficient algorithm is devised to obtain such sets in DAGs and is generalized to CPDAGs. Then, we combine the graph reduction procedure with the IDA framework.

This work introduces a novel technique, named structural dimension reduction, to collapse a Bayesian network onto a minimum and localized one while ensuring that probabilistic inferences between the original and reduced networks remain consistent. To this end, we propose a new combinatorial structure in directed acyclic graphs called the directed convex hull, which has turned out to be equivalent to their minimum localized Bayesian networks. An efficient polynomial-time algorithm is devised to identify them by determining the unique directed convex hulls containing the variables of interest from the original networks. Experiments demonstrate that the proposed technique has high dimension reduction capability in real networks, and the efficiency of probabilistic inference based on directed convex hulls can be significantly improved compared with traditional methods such as variable elimination and belief propagation algorithms. The code of this study is open at \href{https://github.com/Balance-H/Algorithms}{https://github.com/Balance-H/Algorithms} and the proofs of the results in the main body are postponed to the appendix.

Inspired by the digital twinning systems, a novel real-time digital double framework is developed to enhance robot perception of the terrain conditions. Based on the very same physical model and motion control, this work exploits the use of such simulated digital double synchronized with a real robot to capture and extract discrepancy information between the two systems, which provides high dimensional cues in multiple physical quantities to represent differences between the modelled and the real world. Soft, non-rigid terrains cause common failures in legged locomotion, whereby visual perception solely is insufficient in estimating such physical properties of terrains. We used digital double to develop the estimation of the collapsibility, which addressed this issue through physical interactions during dynamic walking. The discrepancy in sensory measurements between the real robot and its digital double are used as input of a learning-based algorithm for terrain collapsibility analysis. Although trained only in simulation, the learned model can perform collapsibility estimation successfully in both simulation and real world. Our evaluation of results showed the generalization to different scenarios and the advantages of the digital double to reliably detect nuances in ground conditions.

Inspired by human behavior when traveling over unknown terrain, this study proposes the use of probing strategies and integrates them into a traversability analysis framework to address safe navigation on unknown rough terrain. Our framework integrates collapsibility information into our existing traversability analysis, as vision and geometric information alone could be misled by unpredictable non-rigid terrains such as soft soil, bush area, or water puddles. With the new traversability analysis framework, our robot has a more comprehensive assessment of unpredictable terrain, which is critical for its safety in outdoor environments. The pipeline first identifies the terrain's geometric and semantic properties using an RGB-D camera and desired probing locations on questionable terrains. These regions are probed using a force sensor to determine the risk of terrain collapsing when the robot steps over it. This risk is formulated as a collapsibility metric, which estimates an unpredictable region's ground collapsibility. Thereafter, the collapsibility metric, together with geometric and semantic spatial data, is combined and analyzed to produce global and local traversability grid maps. These traversability grid maps tell the robot whether it is safe to step over different regions of the map. The grid maps are then utilized to generate optimal paths for the robot to safely navigate to its goal. Our approach has been successfully verified on a quadrupedal robot in both simulation and real-world experiments.

Historically used in settings where the outcome is rare or data collection is expensive, outcome-dependent sampling is relevant to many modern settings where data is readily available for a biased sample of the target population, such as public administrative data. Under outcome-dependent sampling, common effect measures such as the average risk difference and the average risk ratio are not identified, but the conditional odds ratio is. Aggregation of the conditional odds ratio is challenging since summary measures are generally not identified. Furthermore, the marginal odds ratio can be larger (or smaller) than all conditional odds ratios. This so-called non-collapsibility of the odds ratio is avoidable if we use an alternative aggregation to the standard arithmetic mean. We provide a new definition of collapsibility that makes this choice of aggregation method explicit, and we demonstrate that the odds ratio is collapsible under geometric aggregation. We describe how to partially identify, estimate, and do inference on the geometric odds ratio under outcome-dependent sampling. Our proposed estimator is based on the efficient influence function and therefore has doubly robust-style properties.