Counting and Sampling from Markov Equivalent DAGs Using Clique Trees

A directed acyclic graph (DAG) is the most common graphical model for representing causal relationships among a set of variables. When restricted to using only observational data, the structure of the ground truth DAG is identifiable only up to Markov equivalence, based on conditional independence relations among the variables. Therefore, the number of DAGs equivalent to the ground truth DAG is an indicator of the causal complexity of the underlying structure--roughly speaking, it shows how many interventions or how much additional information is further needed to recover the underlying DAG. In this paper, we propose a new technique for counting the number of DAGs in a Markov equivalence class. Our approach is based on the clique tree representation of chordal graphs. We show that in the case of bounded degree graphs, the proposed algorithm is polynomial time. We further demonstrate that this technique can be utilized for uniform sampling from a Markov equivalence class, which provides a stochastic way to enumerate DAGs in the equivalence class and may be needed for finding the best DAG or for causal inference given the equivalence class as input. We also extend our counting and sampling method to the case where prior knowledge about the underlying DAG is available, and present applications of this extension in causal experiment design and estimating the causal effect of joint interventions

A Double Regression Method for Graphical Modeling of High-dimensional Nonlinear and Non-Gaussian Data

Graphical models have long been studied in statistics as a tool for inferring conditional independence relationships among a large set of random variables. The most existing works in graphical modeling focus on the cases that the data are Gaussian or mixed and the variables are linearly dependent. In this paper, we propose a double regression method for learning graphical models under the high-dimensional nonlinear and non-Gaussian setting, and prove that the proposed method is consistent under mild conditions. The proposed method works by performing a series of nonparametric conditional independence tests. The conditioning set of each test is reduced via a double regression procedure where a model-free sure independence screening procedure or a sparse deep neural network can be employed. The numerical results indicate that the proposed method works well for high-dimensional nonlinear and non-Gaussian data.Comment: 1 figur

How Rhodopsin Tunes the Equilibrium between Protonated and Deprotonated Forms of the Retinal Chromophore

Rhodopsin is a photoactive G-protein-coupled receptor (GPCR) that converts dim light into a signal for the brain, leading to eyesight. Full activation of this GPCR is achieved after passing through several steps of the protein's photoactivation pathway. Key events of rhodopsin activation are the initial cis-trans photoisomerization of the covalently bound retinal moiety followed by conformational rearrangements and deprotonation of the chromophore's protonated Schiff base (PSB), which ultimately lead to full activation in the meta II state. PSB deprotonation is crucial for achieving full activation of rhodopsin; however, the specific structural rearrangements that have to take place to induce this plc shift are not well understood. Classical molecular dynamics (MD) simulations were employed to identify intermediate states after the cis trans isomerization of rhodopsin's retinal moiety. In order to select the intermediate state in which PSB deprotonation is experimentally known to occur, the validity of the intermediate configurations was checked through an evaluation of the optical properties in comparison with experiment. Subsequently, the selected state was used to investigate the molecular factors that enable PSB deprotonation at body temperature to obtain a better understanding of the difference between the protonated and the deprotonated state of the chromophore. To this end, the deprotonation reaction has been investigated by applying QM/MM MD simulations in combination with thermodynamic integration. The study shows that, compared to the inactive 11-cis-retinal case, trans-retinal rhodopsin is able to undergo PSB deprotonation due to a change in the conformation of the retinal and a consequent alteration in the hydrogen-bond (HB) network in which PSB and the counterion.G1u113 are embedded. Besides the retinal moiety and Glu113, also two water molecules as well as Thr94 and Gly90 that are related to congenital night blindness are part of this essential HB network

The Resilience of Public Policies in Economic Development

This paper studies the resilience of public policies that governments design for catalyzing economic development. This property depends on the extent to which behavioral heuristics and spillover effects allow policymakers to attain their original goals when a particular policy cannot be funded as originally planned. This scenario takes place, for example, when unanticipated events such as natural disasters or political turmoil obstruct the use of resources to advance certain policy issues, e.g., infrastructure or labor reforms. Here, we analyze how the adaptive capacity of the policy-making process generates resilience in the face of disruptions. In order to estimate the allocation of resources across policies, we employ a computational model that accounts for diverse social mechanisms, for example, coevolutionary learning and network interdependencies. In our simulations, we use a data set of 117 countries on 79 development indicators over an 11-year period. Then, we calculate a resilience score corresponding to each development indicator via counter-factual analysis of policy disruptions. Next, we assess whether some development strategies produce resilient/fragile policy profiles. Finally, by studying the relationship between policy resilience and policy priority, we determine which issues are bottlenecks to economic development