Causal Analysis in Theory and Practice

January 4, 2023

Causal Inference (CI) − A year in review

2022 has witnessed a major upsurge in the status of CI, primarily in its general recognition as an independent and essential component in every aspect of intelligent decision making. Visible evidence of this recognition were several prestigious prizes awarded explicitly to CI-related research accomplishments. These include (1) the Nobel Prize in economics, awarded to David Card, Joshua Angrist, and Guido Imbens for their works on cause and effect relations in natural experiments (2) The BBVA Frontiers of Knowledge Award to Judea Pearl for “laying the foundations of modern AI” and (3) The Rousseeuw Prize for Statistics to Jamie Robins, Thomas Richardson, Andrea Rotnitzky, Miguel Hernán, and Eric Tchetchgen Tchetchgen, for their “pioneering work on Causal Inference with applications in Medicine and Public Health”
My acceptance speech at the BBVA award can serve as a gentle summary of the essence of causal inference, its basic challenges and major achievements:
It is not a secret that I have been critical of the approach Angrist and Imbens are taking in econometrics, for reasons elaborated here, and mainly here I nevertheless think that their selection to receive the Nobel Prize in economics is a positive step for CI, in that it calls public attention to the problems that CI is trying to solve and will eventually inspire curious economists to seek a more broad-minded approach to these problems, so as to leverage the full arsenal of tools that CI has developed.
Coupled with these highlights of recognition, 2022 has seen a substantial increase in CI activities on both the academic and commercial fronts. The number of citations to CI related articles has reached a record high of over 10,200 citations in 2022, , showing positive derivatives in all CI categories. Dozens, if not hundreds of seminars, workshops and symposia have been organized in major conferences to disseminate progress in CI research. New results on individualized decision making were prominently featured in these meetings (e.g., Several commercial outfits have come up with platforms for CI in their areas of specialization, ranging from healthcare to finance and marketing. (Company names such as #causallense, and Vianai Systems come to mind:, Naturally, these activities have led to increasing demands for trained researchers and educators, versed in the tools of CI; jobs openings explicitly requiring experience in CI have become commonplace in both industry and academia.
I am also happy to see CI becoming an issue of contention in AI and Machine Learning (ML), increasingly recognized as an essential capability for human-level AI and, simultaneously, raising the question of whether the data-fitting methodologies of Big Data and Deep Learning could ever acquire these capabilities. In I’ve answered this question in the negative, though various attempts to dismiss CI as a species of “inductive bias” (e.g., or “missing data problem” (e.g., are occasionally being proposed as conceptualizations that could potentially replace the tools of CI. The Ladder of Causation tells us what extra-data information would be required to operationalize such metaphorical aspirations.
Researchers seeking a gentle introduction to CI are often attracted to multi-disciplinary forums or debates, where basic principles are compiled and where differences and commonalities among various approaches are compared and analyzed by leading researchers. Not many such forums were published in 2022, perhaps because the differences and commonalities are now well understood or, as I tend to believe, CI and its Structural Causal Model (SCM) unifies and embraces all other approaches. I will describe two such forums in which I participated.
(1) In March of 2022, the Association for Computing Machinery (ACM) has published an anthology containing highlights of my works (1980-2020) together with commentaries and critics from two dozens authors, representing several disciplines. The Table of Content can be seen here: It includes 17 of my most popular papers, annotated for context and scope, followed by 17 contributed articles of colleagues and critics. The ones most relevant to CI in 2022 are in Chapters 21-26.
Among these, I consider the causal resolution of Simpson’s paradox (Chapter 22, to be one of the crown achievements of CI. The paradox lays bare the core differences between causal and statistical thinking, and its resolution brings an end to a century of debates and controversies by the best philosophers of our time. It is also related to Lord’s Paradox (see − a qualitative version of Simpson’s Paradox which became a focus of endless debates with statisticians and trialists throughout 2022 (on Twitter @yudapearl). I often cite Simpson’s paradox as a proof that our brain is governed by causal, not statistical, calculus.
This question − causal or statistical brain − is not a cocktail party conversation but touches on the practical question of choosing an appropriate language for casting the knowledge necessary for commencing any CI exercise. Philip Dawid − a proponent of counterfactual-free statistical languages − has written a critical essay on the topic ( and my counterfactual-based rebuttal,, clarifies the issues involved.
(2) The second forum of inter-disciplinary discussions can be found in a special issue of the Journal Observational Studies (edited by Ian Shrier, Russell Steele, Tibor Schuster and Mireille Schnitzer) in a form of interviews with Don Rubin, Jamie Robins, James Heckman and myself.
In my interview,, I compiled aspects of CI that I normally skip in scholarly articles. These include historical perspectives of the development of CI, its current state of affairs and, most importantly for our purpose, the lingering differences between CI and other frameworks. I believe that this interview provides a fairly concise summary of these differences, which have only intensified in 2022.
Most disappointing to me are the graph-avoiding frameworks of Rubin, Angrist, Imbens and Heckman, which still dominate causal analysis in economics and some circles of statistics and social science. The reasons for my disappointments are summarized in the following paragraph:
Graphs are new mathematical objects, unfamiliar to most researchers in the statistical sciences, and were of course rejected as “non-scientific ad-hockery” by top leaders in the field [Rubin, 2009]. My attempts to introduce causal diagrams to statistics [Pearl, 1995; Pearl, 2000] have taught me that inertial forces play at least as strong a role in science as they do in politics. That is the reason that non-causal mediation analysis is still practiced in certain circles of social science [Hayes, 2017], “ignorability” assumptions still dominate large islands of research [Imbens and Rubin, 2015], and graphs are still tabooed in the econometric literature [Angrist and Pischke, 2014]. While most researchers today acknowledge the merits of graph as a transparent language for articulating scientific information, few appreciate the computational role of graphs as “reasoning engines,” namely, bringing to light the logical ramifications of the information used in their construction. Some economists even go to great pains to suppress this computational miracle [Heckman and Pinto, 2015; Pearl, 2013].
My disagreements with Heckman go back to 2007 when he rejected the do-operator for metaphysical reasons (see and then to 2013, when he celebrated the do-operator after renaming it “fixing” but remained in denial of d-separation (see In this denial he retreated 3 decades in time while castrating graphs from their inferential power. Heckman’s 2022 interview in Observational Studies continues his on-going crusade to prove that econometrics has nothing to learn from neighboring fields. His fundamental mistake lies in assuming that the rules of do-calculus lie “outside of formal statistics”; they are in fact logically derivable from formal statistics, REGARDLESS of our modeling assumptions but (much like theorems in geometry) once established, save us the labor of going back to the basic axioms.
My differences with Angrist, Imbens and Rubin go even deeper (see, for they involve not merely the avoidance of graphs but also the First Law of Causal Inference ( hence issues of transparency and credibility. These differences are further accentuated in Imbens’s Nobel lecture which treats CI as a computer science creation, irrelevant to “credible” econometric research. In, as well as in my book Causality, I present dozens of simple problems that economists need, but are unable to solve, lacking the tools of CI.
It is amazing to watch leading researchers, in 2022, still resisting the benefits of CI while committing their respective fields to the tyranny of outdatedness.
To summarize, 2022 has seen an unprecedented upsurge in CI popularity, activity and stature. The challenge of harnessing CI tools to solve critical societal problems will continue to inspire creative researchers from all fields, and the aspirations of advancing towards human-level artificial intelligence will be pursued with an accelerated pace in 2023.

Wishing you a productive new year,


August 14, 2019

A Crash Course in Good and Bad Control

Filed under: Back-door criterion,Bad Control,Econometrics,Economics,Identification — Judea Pearl @ 11:26 pm

Carlos Cinelli, Andrew Forney and Judea Pearl

Update: check the updated and extended version of the crash course here.


If you were trained in traditional regression pedagogy, chances are that you have heard about the problem of “bad controls”. The problem arises when we need to decide whether the addition of a variable to a regression equation helps getting estimates closer to the parameter of interest. Analysts have long known that some variables, when added to the regression equation, can produce unintended discrepancies between the regression coefficient and the effect that the coefficient is expected to represent. Such variables have become known as “bad controls”, to be distinguished from “good controls” (also known as “confounders” or “deconfounders”) which are variables that must be added to the regression equation to eliminate what came to be known as “omitted variable bias” (OVB).

Recent advances in graphical models have produced a simple criterion to distinguish good from bad controls, and the purpose of this note is to provide practicing analysts a concise and visible summary of this criterion through illustrative examples. We will assume that readers are familiar with the notions of “path-blocking” (or d-separation) and back-door paths. For a gentle introduction, see d-Separation without Tears

In the following set of models,  the target of the analysis is the average causal effect (ACE) of a treatment X on an outcome Y, which stands for the expected increase of Y per unit of a controlled increase in X. Observed variables will be designated by black dots and unobserved variables by white empty circles. Variable Z (highlighted in red) will represent the variable whose inclusion in the regression is to be decided, with “good control” standing for bias reduction, “bad control” standing for bias increase and “netral control” when the addition of Z does not increase nor reduce bias. For this last case, we will also make a brief remark about how Z could affect the precision of the ACE estimate.


Models 1, 2 and 3 – Good Controls 

In model 1,  Z stands for a common cause of both X and Y. Once we control for Z, we block the back-door path from X to Y, producing an unbiased estimate of the ACE. 

In models 2 and 3, Z is not a common cause of both X and Y, and therefore, not a traditional “confounder” as in model 1. Nevertheless, controlling for Z blocks the back-door path from X to Y due to the unobserved confounder U, and again, produces an unbiased estimate of the ACE.

Models 4, 5 and 6 – Good Controls

When thinking about possible threats of confounding, one needs to keep in mind that common causes of X and any mediator (between X and Y) also confound the effect of X on Y. Therefore, models 4, 5 and 6 are analogous to models 1, 2 and 3 — controlling for Z blocks the backdoor path from X to Y and produces an unbiased estimate of the ACE.

Model 7 – Bad Control

We now encounter our first “bad control”. Here Z is correlated with the treatment and the outcome and it is also a “pre-treatment” variable. Traditional econometrics textbooks would deem Z a “good control”. The backdoor criterion, however, reveals that Z is a “bad control”. Controlling for Z will induce bias by opening the backdoor path X ← U1→ Z← U2→Y, thus spoiling a previously unbiased estimate of the ACE.

Model 8 – Neutral Control (possibly good for precision)

Here Z is not a confounder nor does it block any backdoor paths. Likewise, controlling for Z does not open any backdoor paths from X to Y. Thus, in terms of bias, Z is a “neutral control”. Analysis shows, however, that controlling for Z reduces the variation of the outcome variable Y, and helps improve the precision of the ACE estimate in finite samples.

Model 9 – Neutral control (possibly bad for precision)

Similar to the previous case, here Z is “neutral” in terms of bias reduction. However, controlling for Z will reduce the variation of treatment variable X and so may hurt the precision of the estimate of the ACE in finite samples.  

Model 10 – Bad control

We now encounter our second “pre-treatment” “bad control”, due to a phenomenon called “bias amplification” (read more here). Naive control for Z in this model will not only fail to deconfound the effect of X on Y, but, in linear models, will amplify any existing bias.

Models 11 and 12 – Bad Controls

If our target quantity is the ACE, we want to leave all channels through which the causal effect flows “untouched”.

In Model 11, Z is a mediator of the causal effect of X on Y. Controlling for Z will block the very effect we want to estimate, thus biasing our estimates. 

In Model 12, although Z is not itself a mediator of the causal effect of X on Y, controlling for Z is equivalent to partially controlling for the mediator M, and will thus bias our estimates.

Models 11 and 12 violate the backdoor criterion, which excludes controls that are descendants of the treatment along paths to the outcome.

Model 13 – Neutral control (possibly good for precision)

At first look, model 13 might seem similar to model 12, and one may think that adjusting for Z would bias the effect estimate, by restricting variations of the mediator M. However, the key difference here is that Z is a cause, not an effect, of the mediator (and, consequently, also a cause of Y). Thus, model 13 is analogous to model 8, and so controlling for Z will be neutral in terms of bias and may increase precision of the ACE estimate in finite samples.

Model 14 – Neutral controls (possibly helpful in the case of selection bias)

Contrary to econometrics folklore, not all “post-treatment” variables are inherently bad controls. In models 14 and 15 controlling for Z does not open any confounding paths between X and Y. Thus, Z is neutral in terms of bias. However, controlling for Z does reduce the variation of the treatment variable X and so may hurt the precision of the ACE estimate in finite samples. Additionally, in model 15, suppose one has only samples with W = 1 recorded (a case of selection bias). In this case, controlling for Z can help obtaining the W-specific effect of X on Y, by blocking the colliding path due to W.

Model 16 – Bad control

Contrary to Models 14 and 15, here controlling for Z is no longer harmless, since it opens the backdoor path X → Z ← U → Y and so biases the ACE.

Model 17 – Bad Control

Here, Z is not a mediator, and one might surmise that, as in Model 14, controlling for Z is harmless. However, controlling for the effects of the outcome Y will induce bias in the estimate of the ACE, making Z a “bad control”. A visual explanation of this phenomenon using “virtual colliders” can be found here.

Model 17 is usually known as a “case-control bias” or “selection bias”. Finally, although controlling for Z will generally bias numerical estimates of the ACE, it does have an exception when X has no causal effect on Y. In this scenario, X is still d-separated from Y even after conditioning on Z. Thus, adjusting for Z is valid for testing whether the effect of X on Y is zero.

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