Causal Analytics for Applied Risk Analysis Louis Anthony Cox, Jr


Ashcroft, M. (2013) Performing decision-theoretic inference in Bayesian network ensemble models. In: Twelfth Scandinavian Conference on Artificial Intelligence. Jaeger M, NielsenTD, Viappiani P (Eds).



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Chapter 2

Causal Concepts, Principles, and Algorithms


It is an important truism that association is not causation. For example, people living in low-income areas may have higher levels of exposure to an environmental hazard and also higher levels of some adverse health effect than people living in wealthier areas. Yet this observed association, no matter how strong, consistent, statistically significant, biologically plausible, and well documented by multiple independent teams, does not necessarily tell a policy maker anything about whether or by how much a proposed costly reduction in exposure would reduce adverse health effects. Perhaps only increasing income, or something that income can buy, would reduce adverse health effects. Or maybe factors that cannot be changed by policy interventions increase both the probability of living in low-income areas and the probability of adverse health effects. Whatever the truth is about opportunities to improve health by changing policy variables, it typically cannot be determined by studying correlations, regression coefficients, relative risks, or other measures of association between exposures and health effects (Pearl, 2009). Observed associations between variables can contain both causal and non-causal (“spurious”) components. In general, the effects of policy changes on outcomes of interest can only be predicted and evaluated correctly by modeling the network of causal relationships by which effects of exogenous changes propagate among variables. The chapter reviews current causal concepts, principles, and algorithms for carrying out such causal modeling and compares them to other approaches.

Many different concepts of causality were proposed in the twentieth century and earlier by philosophers (Suppes, 1970; Hausman and Woodward, 1999), geneticists (Wright, 1921), statisticians and social statisticians (Neyman, 1923; Campbell and Stanley, 1963; Blalock, 1964; Rubin, 1974); epidemiologists (Robins and Greenland, 1992), mathematicians and physicists (Wiener, 1956; Schreiber, 2000), economists and econometricians (Simon,1953; Granger, 1969), artificial intelligence and machine learning researchers, and computer scientists (Charniak,1991; Druzdzel and Simon, 1993). They expressed, with varying degrees of rigor and precision, intuitions such as that effects regularly and predictably follow their causes; that causes make their effects different from what they otherwise would have been; that causes are informative about and help to predict their effects; that expected values or probability distributions for effect sizes can be determined from the values of their causes; and that changing causes changes the probabilities of their effects. By the year 2000, these strands of thought on how to define, measure, and estimate causal relationships and effects had largely been unified in a framework that emphasizes the use of diagrams with nodes representing variables and arrows between nodes representing causal dependencies (Pearl, 2009). This framework includes the popular “directed acyclic graph” (DAG) models introduced in Chapter 1, as well as more general models with cycles and undirected arcs (representing dependency with unknown causal direction) allowed. We shall use the DAG models in the following sections.

This chapter explores what it means to say that one thing causes another and reviews key ideas about causality that have proved useful in interpreting a broad variety of data and estimating causal impacts of interventions on outcomes. It discusses how to represent different types of causal knowledge using diagrams and mathematical, statistical, and computational models to facilitate explanation, communication, and computation of causal inferences. Finally, this chapter surveys principles and algorithms for using causal models to answer practical questions requiring causal inference. These include questions of attribution and diagnosis, prediction and prognosis, explanation, prescriptive optimization of decisions, and evaluation of their impacts.

Learning goals for this chapter are as follows:



  • Distinguish between (a) statistical associations, inferences, and models; and (b) causal models to support/evaluate/improve policy decisions 

  • Introduce, explain, and show how to apply several different types or concepts of causality to improve predictions, decisions, and learning. The main types discussed in this chapter are associational, attributive, counterfactual, structural (computational), predictive, manipulative, and mechanistic or explanatory causation.

  • Explain the main concepts and software tools currently available to solve causal analytics problems. These include techniques for identifying causal network models from data and for using them to predict, infer, attribute, and explain effects based on observations; optimize decisions; and quantify partial (“direct” and “indirect”) and total causal relationships.

  • Introduce algorithms and principles for identifying approximately correct causal models from data using relatively objective (assumption-free, investigator-independent) machine-learning methods where possible, together with knowledge-based constraints where necessary (e.g., that effects do not precede their causes, or that weather can be a cause but not an effect of illnesses).

  • Illustrate how to use freely available software for applying causal analytics methods and specific causal discovery and inference algorithms to data.  Air pollution health effects research is used as an example for illustrating state-of-the-art causal analytics algorithms.

The chapter is relatively long and introduces many technical concepts and terms needed to take advantage of current causal analytics methods and software. By the end of this chapter, the reader will be conversant with the main ideas and methods of modern causal analytics and will understand their potential and limitations for practical applications in risk analysis. To minimize the burden on readers who are mainly interested in applications, subsequent chapters briefly recapitulate key concepts and techniques where they are used, leaving a fuller exposition of concepts and methods to this chapter. On the other hand, for readers who wish to delve further into the technical methods surveyed in this chapter, an extensive and up-to-date list of references gives access to the primary research literature and to several outstanding surveys, tutorials, and software packages. As in so much of the current practice of data science, the exposition here is targeted mainly at readers who seek to understand technical concepts and methods well enough to use them correctly and effectively and to provide a relatively accessible point of entry to the large and exciting recent technical and research literatures that are transforming how arrtifical intelligence, machine learning, and data science are being used to learn about cause and effect and to improve understanding and control of the behaviors of a broad range of uncertain systems that affect human health and wellbeing.


Multiple Meanings of “Cause”
The claim that one event or condition causes another has meant different things to different people and organizations. Modern causal analysis clarifies these different meanings, allowing more precise expression of what questions a causal study addresses and how the answers should be interpreted. For example, in public and occupational health risk analysis, the causal claim “Each extra unit of exposure to substance X increases rates of an adverse health effect (e.g., lung cancer, heart attack deaths, asthma attacks, etc.) among exposed people by R additional expected cases per person-year” can be interpreted in at least the following ways:

  1. Probabilistic causation (Suppes,1970): The conditional probability of the health response or effect occurring in a given interval of time is greater among individuals with more exposure compared to otherwise similar-seeming individuals with less exposure; in this sense, probability of response (or age-specific hazard rate for occurrence of response) increases with exposure. On average, there are R extra cases per person-year per unit of exposure. The main intuition is that causes (exposures) make their effects (responses) more likely to occur within a given time interval, or increase their occurrence rates.

  2. Associational causation (IARC, 2006): Higher levels of exposure have been observed in conjunction with higher risks, and this association is judged to be strong, consistent across multiple studies and locations, and biologically plausible. The slope of a regression line between these historical observations in the exposed population of interest is R extra cases per person-year per unit of exposure. The main intuition is that causes are associated with their effects. Relative risk (RR) ratios – the ratios of responses per person per year in exposed compared to unexposed populations – and quantities derived from RR, such as burden-of-disease metrics, population attributable fractions, probability of causation formulas, and closely related metrics, are widely used in epidemiology and public health to quantify associational causation.

  3. Attributive causation (Murray and Lopez, 2013): Authorities attribute R extra cases per person-year per unit of exposure to X; equivalently, they blame exposure to X for R extra cases per person-year per unit of exposure. In practice, such attributions are usually made based on measures of association such as the ratio or difference of estimated risks between populations with higher and lower levels of exposure. Differences in risks between the populations are attributed to their differences in exposures without further analysis of other possible explanations. The main idea is that if people with higher exposures have higher risks for any reason, then the increased risk can be attributed to the higher exposure. (If many risk factors differ between low-risk and high-risk groups, then the difference in risks can be attributed to each of them separately; there is no consistency constraint preventing multiples of the total difference in risks from being attributed to the various factors.)

  4. Counterfactual and potential outcomes causation (Höfler, 2005; Glass et al., 2013; Lok, 2017; Li et al., 2017): In a hypothetical world (or maybe in all conceivable counterfactual worlds) with 1 unit less of exposure to X, expected cases per person-year in the exposed population would also be less by R. Usually, such counterfactual numbers are derived from modeling assumptions, and explanations for the counterfactual reduction in exposure are not discussed. The main intuition is that differences in causes make their effects different from what they otherwise would have been.

  5. Predictive causation (Granger, 1969; Kleinberg and Hripcsak, 2011; Papana et al., 2017): In the absence of interventions, time series data show that the observation that exposure has increased or decreased is predictably followed, perhaps after a lag, by the observation that average cases per person-year have also increased or decreased, respectively, by an average of R cases per unit of change in exposure. The main intuition is that causes help to predict their effects, and changes in causes help to predict changes in their effects. More generally, causes are informative about their effects, so effects can be predicted better with information about their causes than without it.

  6. Structural causation (Simon, 1953; Simon and Iwasaki, 1988; Hoover, 2012): In a valid mathematical or computational simulation model (or possibly in all valid simulation models), the number of cases per person-year is derived at least in part from the value of exposure. Thus, the value of exposure must be determined before the value of yearly case count can be determined. Moreover, the average calculated or simulated value of the case count per person-year decreases by R for each exogenously specified unit decrease in exposure. The main intuition is that effects depend on, and are calculated from, their causes.

  7. Manipulative causation (Voortman et al., 2010; Hoover, 2012; Simon and Iwasaki, 1988): Reducing exposure by one unit reduces expected cases per person-year by R. The main intuition is that changing causes changes their effects.

  8. Explanatory/mechanistic causation (Menzies, 2012; Simon and Iwasaki, 1988): Increasing exposure by one unit causes changes to propagate through a biological network of causal mechanisms. When all changes have finished propagating, the new expected value for case count per person-year in the exposed population will be R more than before exposure was increased. The main intuition is that changes in causes propagate through a network of law-like causal mechanisms to produce changes in their effects. Causal mechanisms are usually represented mathematically by structural equations or by conditional probability tables (CPTs) that are invariant across settings, as discussed in Chapter 1 (Pearl, 2009).

For risk managers and policy makers, manipulative causation is key, since the goal of decision-making is to choose acts that cause desired outcomes, in the sense of making them more probable. Manipulative causation is implied by mechanistic causation – if there is a network of mechanisms by which acts change the probabilities of outcomes (mechanistic causation), then taking the acts will indeed change the probabilities of outcomes (manipulative causation). But neither one is implied by associational, attributive, counterfactual, or predictive concepts of causation (Pearl, 2009). Understanding and appropriately applying these distinctions among concepts of causation, and making sure that associational concepts are not misrepresented or misunderstood as manipulative causal ones in policy deliberations and supporting epidemiological analyses, provides a crucial first step toward improving current practice in epidemiology (Petitti, 1991).



The following sections examine these different concepts of causality more closely and discuss how they are related. Probabilistic causal models, which are common to all of these concepts of causation, are emphasized. In particular, we explain how Bayesian network (BN) models can be used to represent probabilistic dependencies among variables, manipulate probabilities to make predictions, and draw probabilistic inferences. They also provide a useful unifying framework and generalization of many well-known probabilistic risk assessment (PRA) and decision analysis techniques. Modern software makes it relatively easy to build and use BNs. Several examples show how to use current BN software to create simple BN models and use them to draw inferences and make predictions. BN algorithms can also be extended to networks with decisions, i.e., influence diagrams (IDs), and used prescriptively to solve for optimal statistical decisions; additional examples illustrate these methods. The final sections of the chapter consider how to learn causal models from data and conclude with a brief description of selected milestones in the historical development of modern causal analysis.
Probabilistic Causation and Bayesian Networks (BN)
Perhaps the simplest intuition relating probability and causation is that causes make their effects more probable. To sharpen this intuition and use it to draw quantitative inferences, it is necessary to be more explicit about how one observation, action, or event can make another more probable. The assumed technical background for this discussion is elementary probability theory, especially the concept of a random variable and the definitions and notations for joint, marginal, and conditional probability distributions.
Technical Background: Probability Concepts, Notation and Bayes’ Rule
Uncertain quantities in this chapter are represented by random variables. Most of this chapter assumes that the random variables in question are discrete. The notation P(x) will be used as an abbreviation for the probability that random variable X has specific value x. Thus, P(x) is a short-hand for P(X = x), or, as it is sometimes more explicitly denoted, PX(x), where the subscript shows the particular random variable for which probabilities of values are being given. P(x) is often called the probability mass function, or, for continuous random variables, the probability density function of the random variable X. When X is just one of several random variables being considered, P(x) is also called its marginal distribution. In such a multivariate context, where the particular random variable being referred to might be unclear, the notation PX(x) for the marginal distribution of X can preserve clarity. We will use the simpler P(x) when the random variable being referred to is clear from context. Likewise, P(x, y) will denote the joint probability that random variable X has specific value x and that random variable Y has specific value y; thus, P(x, y) is short for P(X = x, Y = y) and for the more explicit notation PX,Y(x, y) for the joint probability that X = x and Y = y. The conditional probability that X = x, given that Y = y, will usually be written as P(x | y) in preference to the longer and more explicit notations P(X = x | Y = y) or PX|Y(x | y). Recall the definition of conditional probability:
P(x | y) = P(x, y)/P(y) (2.1)
when the denominator is greater than zero. This definition follows by rearranging the identity
P(x, y) = P(y)P(x | y) (2.2)
i.e., the probability that both X = x and Y = y is the probability that Y = y times the conditional probability that X = x given that Y = y. With equal validity, the joint probability P(x, y) can be factored as a product of a marginal and a conditional probability in a different way, as follows:
P(x, y) = P(x)P(y | x) (2.3)
The marginal distribution for a random variable X can always be calculated from its conditional probabilities, given each of the values of one or more other variables, and from the marginal probabilities of those values, via the law of total probability. This states that the total probability of an event (such as that X has specific value x) is the sum of the probabilities of all of the ways in which it can occur in conjunction with each of a set of mutually exclusive, collectively exhaustive events (such as that Y has each of its possible specific values).

Applying the law of total probability to two random variables X and Y to obtain the marginal distribution of Y from the marginal distribution of X and the conditional probability distributions of Y given each value of X yields the following prediction formula for Y values::


P(y) = xP(y | x)P(x) (2.4)
Here, the sum is taken over each of the distinct possible values, x, of X; if X is a continuous random variable, then the sum must be replaced by an integral. Equating the right-hand sides of equations (2.2) and (2.3), since they both equal P(x, y), yields

P(y)P(x | y) = P(x)P(y | x) (2.5)
Dividing both sides by P(y) (assuming it is non-zero) gives the identity
P(x | y) = P(x)P(y | x)/P(y) (2.6)
Then, expanding P(y) via the law of total probability (2.4), yields Bayes’ Rule:
P(x | y) = P(x)P(y | x)/x’P(y | x’)P(x’) (2.7)
(The primes on the x values in the denominator are inserted to make clear that x’ is simply an index for the values of X being summed over, not to be confused with the specific, fixed value x in the numerator and on the left side of the equation.) P(x) is called the prior probability that X = x, and P(x | y) is called the posterior probability that X = x, given the observation or data that Y = y. Bayes’ Rule allows data on the marginal probabilities of X values and on the conditional probabilities of Y values given X values to be used to infer conditional probabilities of X values given Y values. We assume familiarity with these aspects of probability theory throughout the remainder of this chapter.
Example: Joint, Marginal, and Conditional Probabilities for Answering Queries
Table 2.1 shows the 9 joint probabilities for all possible combinations (i.e., pairs) of values for two discrete random variables, each with three possible values: X with possible values 1, 2, and 3; and Y with possible values 4, 8, and 16. Such a joint probability table can be used to answer any question about the probabilities that the values of X and Y fall in specified sets or satisfy specified constraints.
Table 2.1 A joint probability distribution for two random variables: X with possible values 1, 2, and 3; and Y with possible values 4, 8, and 16. For example, P(x, y) = (3,16) = 0.3.




X values

Y values

1

2

3

4

0.0

0.25

0.15

8

0.05

0.15

0.0

16

0.1

0.0

0.3


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