Jiji Zhang

Much of the recent work on the epistemology of causation has centered on two assumptions, known as the Causal Markov Condition and the Causal Faithfulness Condition. Philosophical discussions of the latter condition have exhibited situations in which it is likely to fail. This paper studies the Causal Faithfulness Condition as a conjunction of weaker conditions. We show that some of the weaker conjuncts can be empirically tested, and hence do not have to be assumed a priori. Our results lead to two methodologically significant observations: (1) some common types of counterexamples to the Faithfulness condition constitute objections only to the empirically testable part of the condition; and (2) some common defenses of the Faithfulness condition do not provide justification or evidence for the testable parts of the condition. It is thus worthwhile to study the possibility of reliable causal inference under weaker Faithfulness conditions. As it turns out, the modification needed to make standard procedures work under a weaker version of the Faithfulness condition also has the practical effect of making them more robust when the standard Faithfulness condition actually holds. This, we argue, is related to the possibility of controlling error probabilities with finite sample size (“uniform consistency”) in causal inference.

The framework of causal Bayes nets, currently influential in several scientific disciplines, provides a rich formalism to study the connection between causality and probability from an epistemological perspective. This article compares three assumptions in the literature that seem to constrain the connection between causality and probability in the style of Occam's razor. The trio includes two minimality assumptions—one formulated by Spirtes, Glymour, and Scheines (SGS) and the other due to Pearl—and the more well-known faithfulness or stability assumption. In terms of logical strength, it is fairly obvious that the three form a sequence of increasingly stronger assumptions. The focus of this article, however, is to investigate the nature of their relative strength. The comparative analysis reveals an important sense in which Pearl's minimality assumption is as strong as the faithfulness assumption and identifies a useful condition under which it is as safe as SGS's relatively secure minimality assumption. Both findings have notable implications for the theory and practice of causal inference. 1 Introduction2 Background: Inference of Causal Structure in Markovian Causal Models3 Three Assumptions of Simplicity4 A Comparison of P-minimality and Faithfulness5 A Comparison of P-minimality and SGS-minimality6 Methodological Formulations and Prior Knowledge of Causal Order7 Conclusion.

We argue that current discussions of criteria for actual causation are ill-posed in several respects. (1) The methodology of current discussions is by induction from intuitions about an infinitesimal fraction of the possible examples and counterexamples; (2) cases with larger numbers of causes generate novel puzzles; (3) "neuron" and causal Bayes net diagrams are, as deployed in discussions of actual causation, almost always ambiguous; (4) actual causation is (intuitively) relative to an initial system state since state changes are relevant, but most current accounts ignore state changes through time; (5) more generally, there is no reason to think that philosophical judgements about these sorts of cases are normative; but (6) there is a dearth of relevant psychological research that bears on whether various philosophical accounts are descriptive. Our skepticism is not directed towards the possibility of a correct account of actual causation; rather, we argue that standard methods will not lead to such an account. A different approach is required

In this paper, we take another look at the reasons for which the causal criterion of event identity has been abandoned. We argue that the reasons are not strong. First of all, there is a criterion in the neighborhood of the causal criterion—the counterfactual criterion—that is not vulnerable to any of the putative counterexamples brought up in the literature. Secondly, neither the causal criterion nor the counterfactual criterion suffers from any form of vicious circularity. Nonetheless, we do not recommend adopting either the causal or the counterfactual criterion because, given a sufficiently lax principle of event composition, neither criterion can be applied to complex events. This we regard as a (prima facie) undesirable restriction on their applicability.

In the artificial intelligence literature a promising approach to counterfactual reasoning is to interpret counterfactual conditionals based on causal models. Different logics of such causal counterfactuals have been developed with respect to different classes of causal models. In this paper I characterize the class of causal models that are Lewisian in the sense that they validate the principles in Lewis’s well-known logic of counterfactuals. I then develop a system sound and complete with respect to this class. The resulting logic is the weakest logic of causal counterfactuals that respects Lewis’s principles, sits in between the logic developed by Galles and Pearl and the logic developed by Halpern, and stands to Galles and Pearl’s logic in the same fashion as Lewis’s stands to Stalnaker’s

We examine a formal semantics for counterfactual conditionals due to Judea Pearl, which formalizes the interventionist interpretation of counterfactuals central to the interventionist accounts of causation and explanation. We show that a characteristic principle validated by Pearl’s semantics, known as the principle of reversibility, states a kind of irreversibility: counterfactual dependence (in David Lewis’s sense) between two distinct events is irreversible. Moreover, we show that Pearl’s semantics rules out only mutual counterfactual dependence, not cyclic dependence in general. This, we argue, suggests that Pearl’s logic is either too weak or too strong

We clarify the status of the so-called causal minimality condition in the theory of causal Bayesian networks, which has received much attention in the recent literature on the epistemology of causation. In doing so, we argue that the condition is well motivated in the interventionist (or manipulability) account of causation, assuming the causal Markov condition which is essential to the semantics of causal Bayesian networks. Our argument has two parts. First, we show that the causal minimality condition, rather than an add-on methodological assumption of simplicity, necessarily follows from the substantive interventionist theses, provided that the actual probability distribution is strictly positive. Second, we demonstrate that the causal minimality condition can fail when the actual probability distribution is not positive, as is the case in the presence of deterministic relationships. But we argue that the interventionist account still entails a pragmatic justification of the causal minimality condition. Our argument in the second part exemplifies a general perspective that we think commendable: when evaluating methods for inferring causal structures and their underlying assumptions, it is relevant to consider how the inferred causal structure will be subsequently used for counterfactual reasoning

In the causal inference framework of Spirtes, Glymour, and Scheines, inferences about causal relationships are made from samples from probability distributions and a number of assumptions relating causal relations to probability distributions. The most controversial of these assumptions is the Causal Faithfulness Assumption, which roughly states that if a conditional independence statement is true of a probability distribution generated by a causal structure, it is entailed by the causal structure and not just for particular parameter values. In this paper we show that the addition of the Causal Faithfulness Assumption plays three quite different roles in the SGS framework: it reduces the degree of underdetermination of causal structure by probability distribution; computationally, it justifies reliable causal inference algorithms that would otherwise have to be slower in order to be reliable; and statistically, it implies that those algorithms reliably obtain the correct answer at smaller sample sizes than would otherwise be the case. We also consider a number of variations on the Causal Faithfulness Assumption, and show how they affect each of these three roles

We argue that current discussions of criteria for actual causation are ill-posed in several respects. (1) The methodology of current discussions is by induction from intuitions about an infinitesimal fraction of the possible examples and counterexamples; (2) cases with larger numbers of causes generate novel puzzles; (3) “neuron” and causal Bayes net diagrams are, as deployed in discussions of actual causation, almost always ambiguous; (4) actual causation is (intuitively) relative to an initial system state since state changes are relevant, but most current accounts ignore state changes through time; (5) more generally, there is no reason to think that philosophical judgements about these sorts of cases are normative; but (6) there is a dearth of relevant psychological research that bears on whether various philosophical accounts are descriptive. Our skepticism is not directed towards the possibility of a correct account of actual causation; rather, we argue that standard methods will not lead to such an account. A different approach is required. Once upon a time a hungry wanderer came into a village. He filled an iron cauldron with water, built a fire under it, and dropped a stone into the water. “I do like a tasty stone soup” he announced. Soon a villager added a cabbage to the pot, another added some salt and others added potatoes, onions, carrots, mushrooms, and so on, until there was a meal for all.

A main message from the causal modelling literature in the last several decades is that under some plausible assumptions, there can be statistically consistent procedures for inferring (features of) the causal structure of a set of random variables from observational data. But whether we can control the error probabilities with a finite sample size depends on the kind of consistency the procedures can achieve. It has been shown that in general, under the standard causal Markov and Faithfulness assumptions, the procedures can only be pointwise but not uniformly consistent without substantial background knowledge. This implies the impossibility of choosing a finite sample size to control the worst case error probabilities. In this paper, I consider the simpler task of inferring causal directions when the skeleton of the causal structure is known, and establish a similarly negative result concerning the possibility of controlling error probabilities. Although the result is negative in form, it has an interesting positive implication for causal discovery methods.

Forster presented some interesting examples having to do with distinguishing the direction of causal influence between two variables, which he argued are counterexamples to the likelihood theory of evidence. In this paper, we refute Forster's arguments by carefully examining one of the alleged counterexamples. We argue that the example is not convincing as it relies on dubious intuitions that likelihoodists have forcefully criticized. More importantly, we show that contrary to Forster's contention, the consilience-based methodology he favored is accountable within the framework of the LTE.

One conception of underdetermination is that it corresponds to the impossibility of reliable inquiry. In other words, underdetermination is defined to be the situation where, given a set of background assumptions and a space of hypotheses, it is logically impossible for any hypothesis selection method to meet a given reliability standard. From this perspective, underdetermination in a given subject of inquiry is a matter of interplay between background assumptions and reliability or success criteria. In this paper I discuss underdetermination in causal inference along this line. In particular I will analyze several success criteria that can be applied to causal inference from statistical regularities. The criteria center on the notions of consistency in mathematical statistics. For each criterion I present its epistemic implication in terms of simple conditions under which the criterion cannot possibly be met. I then investigate which of the familiar principles and their variants in the literature, if adopted as background assumptions, are sufficient to overcome different levels of underdetermination induced by the success criteria. “亚决定性”是知识论和科学哲学中一个重要的概念。对这个概念的一种阐释是把它对应于可靠探索的不可能性。就是说，在一个（经验）问题中，给定一些公设和一些供选择的理论或假说，如果逻辑上不可能找到一种理论选择的方法能满足一定的可靠或成功标准，那么相对于这个标准就存在亚决定性。从这个观点看，亚决定性总是相对于一个问题设定，尤其是公设和成功标准而言的。本文从这个角度对近来的统计因果推理研究作一番梳理。首先，基于数理统计中的一致性概念，我会讨论和分析一系列可应用于因果推理的成功标准。对每一个标准，我会用一个相对简单的条件来刻画它对应的亚决定性。然后我对文献里一部分重要的结果作一个综述，以澄清什么样的公设可以消除什么样的亚决定性。

Causal reasoning is primarily concerned with what would happen to a system under external interventions. In particular, we are often interested in predicting the probability distribution of some random variables that would result if some other variables were forced to take certain values. One prominent approach to tackling this problem is based on causal Bayesian networks, using directed acyclic graphs as causal diagrams to relate post-intervention probabilities to pre-intervention probabilities that are estimable from observational data. However, such causal diagrams are seldom fully testable given observational data. In consequence, many causal discovery algorithms based on data-mining can only output an equivalence class of causal diagrams. This paper is concerned with causal reasoning given an equivalence class of causal diagrams, represented by a ancestral graph. We present two main results. The first result extends Pearl 's celebrated do-calculus to the context of ancestral graphs. In the second result, we focus on a key component of Pearl's calculus---the property of invariance under interventions, and give stronger graphical conditions for this property than those implied by the first result. The second result also improves the earlier, similar results due to Spirtes et al.

Most causal discovery algorithms in the literature exploit an assumption usually referred to as the Causal Faithfulness or Stability Condition. In this paper, we highlight two components of the condition used in constraint-based algorithms, which we call “Adjacency-Faithfulness” and “Orientation- Faithfulness.” We point out that assuming Adjacency-Faithfulness is true, it is possible to test the validity of Orientation- Faithfulness. Motivated by this observation, we explore the consequence of making only the Adjacency-Faithfulness assumption. We show that the familiar PC algorithm has to be modified to be correct under the weaker, Adjacency-Faithfulness assumption. The modified algorithm, called Conservative PC (CPC), checks whether Orientation- Faithfulness holds in the orientation phase, and if not, avoids drawing certain causal conclusions the PC algorithm would draw. Howtion: ever, if the stronger, standard causal Faith-.

I refute Bailey's claim that his argument for incompatibilism is immune to Campbell's No Past Objection. In my refutation I stress a simple point, that nomological necessitation by future world states does not undermine one's freedom with respect to the present world state. My analysis reveals that the No Past Objection challenges van Inwagen's second consequence argument about as much as it does the others, and suggests that the (uncompromising) incompatibilist must pursue some of the options that Bailey regarded as costly in order to overcome the No Past Objection

Compared to constraint-based causal discovery, causal discovery based on functional causal models is able to identify the whole causal model under appropriate assumptions [Shimizu et al. 2006; Hoyer et al. 2009; Zhang and Hyvärinen 2009b]. Functional causal models represent the effect as a function of the direct causes together with an independent noise term. Examples include the linear non-Gaussian acyclic model, nonlinear additive noise model, and post-nonlinear model. Currently, there are two ways to estimate the parameters in the models: dependence minimization and maximum likelihood. In this article, we show that for any acyclic functional causal model, minimizing the mutual information between the hypothetical cause and the noise term is equivalent to maximizing the data likelihood with a flexible model for the distribution of the noise term. We then focus on estimation of the PNL causal model and propose to estimate it with the warped Gaussian process with the noise modeled by the mixture of Gaussians. As a Bayesian nonparametric approach, it outperforms the previous one based on mutual information minimization with nonlinear functions represented by multilayer perceptrons; we also show that unlike the ordinary regression, estimation results of the PNL causal model are sensitive to the assumption on the noise distribution. Experimental results on both synthetic and real data support our theoretical claims.

Spirtes, Glymour and Scheines [Causation, Prediction, and Search Springer] described a pointwise consistent estimator of the Markov equivalence class of any causal structure that can be represented by a directed acyclic graph for any parametric family with a uniformly consistent test of conditional independence, under the Causal Markov and Causal Faithfulness assumptions. Robins et al. [Biometrika 90 491–515], however, proved that there are no uniformly consistent estimators of Markov equivalence classes of causal structures under those assumptions. Subsequently, Kalisch and B¨uhlmann [J. Mach. Learn. Res. 8 613–636] described a uniformly consistent estimator of the Markov equivalence class of a linear Gaussian causal structure under the Causal Markov and Strong Causal Faithfulness assumptions. However, the Strong Faithfulness assumption may be false with high probability in many domains. We describe a uniformly consistent estimator of both the Markov equivalence class of a linear Gaussian causal structure and the identifiable structural coefficients in the Markov equivalence class under the Causal Markov assumption and the considerably weaker k-Triangle-Faithfulness assumption.

This paper has two main parts. In the first part, we motivate a kind of indeterminate, suppositional credences by discussing the prospect for a subjective interpretation of a causal Bayesian network, an important tool for causal reasoning in artificial intelligence. A CBN consists of a causal graph and a collection of interventional probabilities. The subjective interpretation in question would take the causal graph in a CBN to represent the causal structure that is believed by an agent, and interventional probabilities in a CBN to represent suppositional credences. We review a difficulty noted in the literature with such an interpretation, and suggest that a natural way to address the challenge is to go for a generalization of CBN that allows indeterminate credences. In the second part, we develop a decision-theoretic foundation for such indeterminate suppositional credences, by generalizing a theory of coherent choice functions to accommodate some form of act-state dependence. The upshot is a decision-theoretic framework that is not only rich enough to, so to speak, ground the probabilities in a subjectively interpreted causal network, but also interesting in its own right, in that it accommodates both act-state dependence and imprecise probabilities.

Causal discovery becomes especially challenging when the possibility of latent confounding and/or selection bias is not assumed away. For this task, ancestral graph models are particularly useful in that they can represent the presence of latent confounding and selection effect, without explicitly invoking unobserved variables. Based on the machinery of ancestral graphs, there is a provably sound causal discovery algorithm, known as the FCI algorithm, that allows the possibility of latent confounders and selection bias. However, the orientation rules used in the algorithm are not complete. In this paper, we provide additional orientation rules, augmented by which the FCI algorithm is shown to be complete, in the sense that it can, under standard assumptions, discover all aspects of the causal structure that are uniquely determined by facts of probabilistic dependence and independence. The result is useful for developing any causal discovery and reasoning system based on ancestral graph models

Using the flexibility of recently developed methods for causal discovery based on Boolean satisfiability solvers, we encode a variety of assumptions that weaken the Faithfulness assumption. The encoding results in a number of SAT-based algorithms whose asymptotic correctness relies on weaker conditions than are standardly assumed. This implementation of a whole set of assumptions in the same platform enables us to systematically explore the effect of weakening the Faithfulness assumption on causal discovery. An important effect, suggested by simulation results, is that adopting weaker assumptions greatly alleviates the problem of conflicting constraints and substantially shortens solving time. As a result, SAT-based causal discovery is potentially more scalable under weaker assumptions.

We consider Geanakoplos and Polemarchakis’s generalization of Aumman’s famous result on “agreeing to disagree", in the context of imprecise probability. The main purpose is to reveal a connection between the possibility of agreeing to disagree and the interesting and anomalous phenomenon known as dilation. We show that for two agents who share the same set of priors and update by conditioning on every prior, it is impossible to agree to disagree on the lower or upper probability of a hypothesis unless a certain dilation occurs. With some common topological assumptions, the result entails that it is impossible to agree not to have the same set of posterior probabilities unless dilation is present. This result may be used to generate sufficient conditions for guaranteed full agreement in the generalized Aumman-setting for some important models of imprecise priors, and we illustrate the potential with an agreement result involving the density ratio classes. We also provide a formulation of our results in terms of “dilation-averse” agents who ignore information about the value of a dilating partition but otherwise update by full Bayesian conditioning.

The account of causal regularities in the influential INUS theory of causation has been refined in the recent developments of the regularity approach to causation and of the Boolean methods for inference of deterministic causal structures. A key element in the refinement is to strengthen the minimality or non-redundancy condition in the original INUS account. In this paper, we argue that the Boolean framework warrants a further strengthening of the minimality condition. We motivate our stronger condition by showing, first, that a rationale for strengthening the original minimality condition in the INUS theory is also applicable to our proposal to go further, and second, that the new element of the stronger condition is a counterpart to a well-established minimality condition for probabilistic causal models. We also compare the various minimality conditions in terms of the difference-making criteria they imply and argue for the criterion implied by our condition. Finally, we show that putative counterexamples to our proposal can be addressed in the same way that the Boolean theorists defend the current minimality conditions in their framework.

A primary object of causal reasoning concerns what would happen to a system under certain interventions. Specifically, we are often interested in estimating the probability distribution of some random variables that would result from forcing some other variables to take certain values. The renowned do-calculus gives a set of rules that govern the identification of such post-intervention probabilities in terms of pre-intervention probabilities, assuming available a directed acyclic graph that represents the underlying causal structure. However, a DAG causal structure is seldom fully testable given preintervention, observational data, since many competing DAG structures are equally compatible with the data. In this paper we extend the do-calculus to cover cases where the available causal information is summarized in a so-called partial ancestral graph that represents an equivalence class of DAG structures. The causal assumptions encoded by a PAG are significantly weaker than those encoded by a full-blown DAG causal structure, and are in principle fully testable by observed conditional independence relations.

Experimental results in Ultimatum, Trust and Social Dilemma games have been interpreted as showing that individuals are, by and large, not driven by selfish motives. But we do not need experiments to know that. In our view, what the experiments show is that the typical economic auxiliary hypothesis of non-tuism should not be generalized to other contexts. Indeed, we know that when the experimental situation is framed as a market interaction, participants will be more inclined to keep more money, share less, and disregard other participants’ welfare [Hoffman et al., 1994]. When the same game is framed as a fair division one, participants overall show a much greater concern for the other parties’ interests. The data thus indicate that the context of an interaction is of paramount importance in eliciting different motives. The challenge then is to model utility functions that are general enough to subsume a variety of motives and specific enough to allow for meaningful, interesting predictions to be made. For the sake of simplicity (and brevity), in what follows we will concentrate upon the results of experiments that show what appears to be individuals’ disposition to behave in a fair manner in a variety of circumstances [Camerer, 2003]., though what we are saying can be easily applied to other research areas. Such experimental results have been variously interpreted, each interpretation being accompanied by a specific utility function. We shall consider three such functions and the underlying interpretations that support them, and assess each one on the basis of what they claim to be able to explain and predict

It is commonplace to encounter nonstationary or heterogeneous data, of which the underlying generating process changes over time or across data sets. Such a distribution shift feature presents both challenges and opportunities for causal discovery. In this paper we develop a principled framework for causal discovery from such data, called Constraint-based causal Discovery from Nonstationary/heterogeneous Data, which addresses two important questions. First, we propose an enhanced constraint-based procedure to detect variables whose local mechanisms change and recover the skeleton of the causal structure over observed variables. Second, we present a way to determine causal orientations by making use of independence changes in the data distribution implied by the underlying causal model, benefiting from information carried by changing distributions. Experimental results on various synthetic and real-world data sets are presented to demonstrate the efficacy of our methods.

A fundamental question in causal inference is whether it is possible to reliably infer the manipulation effects from observational data. There are a variety of senses of asymptotic reliability in the statistical literature, among which the most commonly discussed frequentist notions are pointwise consistency and uniform consistency (see, e.g. Bickel, Doksum [2001]). Uniform consistency is in general preferred to pointwise consistency because the former allows us to control the worst case error bounds with a finite sample size. In the sense of pointwise consistency, several reliable causal inference algorithms have been established under the Markov and Faithfulness assumptions [Pearl 2000, Spirtes et al. 2001]. In the sense of uniform consistency, however, reliable causal inference is impossible under the two assumptions when time order is unknown and/or latent confounders are present [Robins et al. 2000]. In this paper we present two natural generalizations of the Faithfulness assumption in the context of structural equation models, under which we show that the typical algorithms in the literature are uniformly consistent with or without modifications even when the time order is unknown. We also discuss the situation where latent confounders may be present and the sense in which the Faithfulness assumption is a limiting case of the stronger assumptions.

In “Flagpoles anyone? Causal and explanatory asymmetries”, James Woodward supplements his celebrated interventionist account of causation and explanation with a set of new ideas about causal and explanatory asymmetries, which he extracts from some cutting-edge methods for causal discovery from observational data. Among other things, Woodward draws interesting connections between observational causal discovery and interventionist themes that are inspired in the first place by experimental causal discovery, alluding to a sort of unity between observational and experimental causal discovery. In this paper, I make explicit what I take to be the implicated unity. Like experimental causal discovery, observational causal discovery also relies on interventions, albeit interventions that are not carried out by investigators and hence need to be detected as part of the inference. The observational patterns appealed to in observational causal discovery are not only surrogates for would-be interventions, as Woodward sometimes puts it; they also serve to mark relevant interventions that actually happen in the data generating process.

An important component in the interventionist account of causal explanation is an interpretation of counterfactual conditionals as statements about consequences of hypothetical interventions. The interpretation receives a formal treatment in the framework of functional causal models. In Judea Pearl’s influential formulation, functional causal models are assumed to satisfy a “unique-solution” property; this class of Pearlian causal models includes the ones called recursive. Joseph Halpern showed that every recursive causal model is Lewisian, in the sense that from the causal model one can construct a possible worlds model in David Lewis’s well-known semantics that satisfies the exact same formulas in a certain language. Moreover, he demonstrated that some Pearlian (non-recursive) models are not Lewisian in this sense. This raises the question regarding the exact contour of Lewisian causal models. In this paper, we provide a characterization of the class of Lewisian causal models and a complete axiomatization with respect to this class. Our results have philosophically interesting consequences, two of which are especially worth noting. First, the class of Stalnakerian causal models, a subclass of Lewisian causal models, is precisely the class of Pearlian models that do not contain any cycle of counterfactual dependence (in a sense of counterfactual dependence akin to Lewis’s famous relation between distinct events). Second, a more natural class of causal models is actually a superclass of Lewisian causal models, the logic of which respects only weak centering rather than centering.