Reversible MCMC on Markov equivalence classes of sparse directed acyclic graphs

Graphical models are popular statistical tools which are used to represent dependent or causal complex systems. Statistically equivalent causal or directed graphical models are said to belong to a Markov equivalent class. It is of great interest to describe and understand the space of such classes. However, with currently known algorithms, sampling over such classes is only feasible for graphs with fewer than approximately 20 vertices. In this paper, we design reversible irreducible Markov chains on the space of Markov equivalent classes by proposing a perfect set of operators that determine the transitions of the Markov chain. The stationary distribution of a proposed Markov chain has a closed form and can be computed easily. Specifically, we construct a concrete perfect set of operators on sparse Markov equivalence classes by introducing appropriate conditions on each possible operator. Algorithms and their accelerated versions are provided to efficiently generate Markov chains and to explore properties of Markov equivalence classes of sparse directed acyclic graphs (DAGs) with thousands of vertices. We find experimentally that in most Markov equivalence classes of sparse DAGs, (1) most edges are directed, (2) most undirected subgraphs are small and (3) the number of these undirected subgraphs grows approximately linearly with the number of vertices. The article contains supplement arXiv:1303.0632, http://dx.doi.org/10.1214/13-AOS1125SUPPComment: Published in at http://dx.doi.org/10.1214/13-AOS1125 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Evaluation of 1:5 Soil to Water Extract Electrical Conductivity Methods and Comparison to Electrical Conductivity of Saturated Paste Extract

Conducting a 1 :5 soil:water extract to measure electrical conductivity (EC) is an approach to assess salinity and is the preferred method used in Australia. However, the influence of salinity on plant growth is predominantly based on saturated paste extract electrical conductivity (ECe) and ECe is recommended as a general method for estimating soil salinity internationally, so it is necessary to convert EC1:s to ECe, The objectives of this research were to 1) compare methods of agitation (shaking plus centrifuging (shaking/centrifuging), shaking, and stirring) for determining EC1: 5; 2) determine optimal times for equilibration for each method across a range of salinity levels determined from saturated paste extracts (ECe) (objectives 1 and 2 are for paper 1); and 3) develop predictive models to convert ECu data to ECe based on four different 1 :5 extraction methods listed above and a USDA-NRCS equilibration technique ( objective 3 is for paper 2). The soils evaluated for the two studies were from north central North Dakota, USA, where 20 soil samples having ECe values ranging from 0.96 to 21 dS m-1were used for the first study (objectives 1 and 2), and 100 samples having ECe values ranging from 0.30 to 17.9 dS m-1were used in the second study (objective 3). In the first study, for each method, nine equilibrium times were used up to 48 hrs. In the second study, a uniform agitation time (8 hrs) was applied to the first three agitation methods, and 1 hr was also used for the USDA-NRCS method. For the first study, significant relationships (p < 0.05) existed between values ofEC1:s and agitation time across the three methods. Agitation methods were significantly different (p S 0.05) from each other for 65% of the soils and shaking/centrifuging was significantly different (p < 0.05) from stirring for all soils. In addition, for 75% of the soils, shaking/centrifuging was significantly different (p :S 0.05) from shaking. Based on these results, methods were analyzed separately for optimal equilibration times. The agitation times required for the three methods to reach 95 and 98% of equilibration were a function of the level of soil salinity. For soils with ECe values less than 4 dS m?1, over 24 hrs was needed to obtain both 95 and 98% of equilibration for the three methods. However, less than 3 and 8 hrs were needed to reach 95 and 98% equilibration, respectively, across methods for soils having ECe values greater than 4 dS m?1. These results indicate that establishing a standard method is necessary to help reduce variation across EC1:s measurements. In the second study, the value ofECe was highly correlated with EC1:s (p < 0.0001) across four agitation methods in non-transformed, log10- transformed, and dilution ratio models through regression analysis. The values of coefficient of determination (r2 ) were greatly improved and average about 0.87 using log10- transformation compared to other two models (r2 values of about 0.68 for the nontransformed models and 0.69 for the dilution ratio models). Since agitation methods were determined to be highly correlated with each other, any regression model determined under the four agitation methods were applicable for the estimation of ECe from another method. The results from this research indicate that comparing data across studies should be done with caution because both agitation method and time can influence results. Also, estimation ofECe from EC1:5 can be done with confidence, but models may not be transferrable across different soil orders or across various salt types

Mediation pathway selection with unmeasured mediator-outcome confounding

Causal mediation analysis aims to investigate how an intermediary factor, called a mediator, regulates the causal effect of a treatment on an outcome. With the increasing availability of measurements on a large number of potential mediators, methods for selecting important mediators have been proposed. However, these methods often assume the absence of unmeasured mediator-outcome confounding. We allow for such confounding in a linear structural equation model for the outcome and further propose an approach to tackle the mediator selection issue. To achieve this, we firstly identify causal parameters by constructing a pseudo proxy variable for unmeasured confounding. Leveraging this proxy variable, we propose a partially penalized method to identify mediators affecting the outcome. The resultant estimates are consistent, and the estimates of nonzero parameters are asymptotically normal. Motivated by these results, we introduce a two-step procedure to consistently select active mediation pathways, eliminating the need to test composite null hypotheses for each mediator that are commonly required by traditional methods. Simulation studies demonstrate the superior performance of our approach compared to existing methods. Finally, we apply our approach to genomic data, identifying gene expressions that potentially mediate the impact of a genetic variant on mouse obesity.Comment: 35 page

Identifying Causal Effects Using Instrumental Variables from the Auxiliary Population

Instrumental variable approaches have gained popularity for estimating causal effects in the presence of unmeasured confounding. However, the availability of instrumental variables in the primary population is often challenged due to stringent and untestable assumptions. This paper presents a novel method to identify and estimate causal effects in the primary population by utilizing instrumental variables from the auxiliary population, incorporating a structural equation model, even in scenarios with nonlinear treatment effects. Our approach involves using two datasets: one from the primary population with joint observations of treatment and outcome, and another from the auxiliary population providing information about the instrument and treatment. Our strategy differs from most existing methods by not depending on the simultaneous measurements of instrument and outcome. The central idea for identifying causal effects is to establish a valid substitute through the auxiliary population, addressing unmeasured confounding. This is achieved by developing a control function and projecting it onto the function space spanned by the treatment variable. We then propose a three-step estimator for estimating causal effects and derive its asymptotic results. We illustrate the proposed estimator through simulation studies, and the results demonstrate favorable performance. We also conduct a real data analysis to evaluate the causal effect between vitamin D status and BMI.Comment: 19 page

On the Representation of Causal Background Knowledge and its Applications in Causal Inference

Causal background knowledge about the existence or the absence of causal edges and paths is frequently encountered in observational studies. The shared directed edges and links of a subclass of Markov equivalent DAGs refined due to background knowledge can be represented by a causal maximally partially directed acyclic graph (MPDAG). In this paper, we first provide a sound and complete graphical characterization of causal MPDAGs and give a minimal representation of a causal MPDAG. Then, we introduce a novel representation called direct causal clause (DCC) to represent all types of causal background knowledge in a unified form. Using DCCs, we study the consistency and equivalency of causal background knowledge and show that any causal background knowledge set can be equivalently decomposed into a causal MPDAG plus a minimal residual set of DCCs. Polynomial-time algorithms are also provided for checking the consistency, equivalency, and finding the decomposed MPDAG and residual DCCs. Finally, with causal background knowledge, we prove a sufficient and necessary condition to identify causal effects and surprisingly find that the identifiability of causal effects only depends on the decomposed MPDAG. We also develop a local IDA-type algorithm to estimate the possible values of an unidentifiable effect. Simulations suggest that causal background knowledge can significantly improve the identifiability of causal effects

Identification and Estimation of Causal Effects Using non-Gaussianity and Auxiliary Covariates

Assessing causal effects in the presence of unmeasured confounding is a challenging problem. Although auxiliary variables, such as instrumental variables, are commonly used to identify causal effects, they are often unavailable in practice due to stringent and untestable conditions. To address this issue, previous researches have utilized linear structural equation models to show that the causal effect can be identifiable when noise variables of the treatment and outcome are both non-Gaussian. In this paper, we investigate the problem of identifying the causal effect using auxiliary covariates and non-Gaussianity from the treatment. Our key idea is to characterize the impact of unmeasured confounders using an observed covariate, assuming they are all Gaussian. The auxiliary covariate can be an invalid instrument or an invalid proxy variable. We demonstrate that the causal effect can be identified using this measured covariate, even when the only source of non-Gaussianity comes from the treatment. We then extend the identification results to the multi-treatment setting and provide sufficient conditions for identification. Based on our identification results, we propose a simple and efficient procedure for calculating causal effects and show the $\sqrt{n}$-consistency of the proposed estimator. Finally, we evaluate the performance of our estimator through simulation studies and an application.Comment: 16 papges, 7 Figure

Low Rank Directed Acyclic Graphs and Causal Structure Learning

Despite several important advances in recent years, learning causal structures represented by directed acyclic graphs (DAGs) remains a challenging task in high dimensional settings when the graphs to be learned are not sparse. In particular, the recent formulation of structure learning as a continuous optimization problem proved to have considerable advantages over the traditional combinatorial formulation, but the performance of the resulting algorithms is still wanting when the target graph is relatively large and dense. In this paper we propose a novel approach to mitigate this problem, by exploiting a low rank assumption regarding the (weighted) adjacency matrix of a DAG causal model. We establish several useful results relating interpretable graphical conditions to the low rank assumption, and show how to adapt existing methods for causal structure learning to take advantage of this assumption. We also provide empirical evidence for the utility of our low rank algorithms, especially on graphs that are not sparse. Not only do they outperform state-of-the-art algorithms when the low rank condition is satisfied, the performance on randomly generated scale-free graphs is also very competitive even though the true ranks may not be as low as is assumed

A sensor with coating Pt/WO3 powder with an Erbium-doped fiber amplifier to detect the hydrogen concentration

A highly sensitive hydrogen sensor coated with Pt/WO3 powder with an Erbium-doped fibre amplifier (EDFA) is proposed and experimentally demonstrated. The sensing head is constructed by splicing a short section of tapered small diameter coreless fiber (TSDCF diameter of 62.5 μm, and tapered to 14.5 μm) between two single-mode fibres. The Pt/WO3 powder adheres to the surface of PDMS film coated on the TSDCF structure, which is sensitive to hydrogen. An EDFA is introduced into the sensor system to improve the quality factor of the output spectrum and thus improve the sensor’s resolution. As the hydrogen concentration varies from 0 to 1.44, the measured maximum light intensity variation and the sensor’s sensitivity are -32.41 dB and -21.25 dB/, respectively. The sensor demonstrates good stability with the light intensity fluctuation of < 1.26 dB over a 30-minute duration