# Causal Analysis in Theory and Practice

## September 12, 2016

### Logically equivalent yet way too different

Filed under: Uncategorized — bryantc @ 2:50 am

Contributor: Judea Pearl

In comparing the tradeoffs between the structural and potential outcome frameworks, I often state that the two are logically equivalent yet poles apart in terms of transparency and computational efficiency. (See Slide #34 of the JSM tutorial). Indeed, anyone who examines how the two frameworks solve a specific problem from begining to end (See, e.g., Slides #35-36 ) would find the differences astonishing.

The question naturally arises: How can two equivalent frameworks differ so substantially in actual use.

The answer is that epistemic equivalence does not mean representational equivalence. Two representations of the same information may highlight different aspects of the problem and thus differ substantially in how easy it is to solve a given problem.  This is a recurrent theme in complexity analysis, but is not generally appreciated outside computer science. We saw it in our discussions with Guido Imbens who could not accept the fact that the use of graphical models is a mathematical necessity not just a matter of taste. (http://causality.cs.ucla.edu/blog/index.php/2014/10/27/are-economists-smarter-than-epidemiologists-comments-on-imbenss-recent-paper/)

The examples usually cited in complexity analysis are combinatorial problems whose solution times depend critically on the initial representation. I hesitated from bringing up these examples, fearing that they will not be compelling to readers on this blog who are more familiar with classical mathematics.

Last week I stumbled upon a very simple example that demonstrates representational differences in no ambiguous terms; I would like to share it with readers.

Consider the age-old problem of finding a solution to an algebraic equation, say
y(x) = x3 + ax2 + bx + c = 0

This is a tough problem for those of us who do not remember Tartalia’s solution of the cubic.  (It can be made much tougher once we go to quintic equation.)

But there are many syntactic ways of representing the same function y(x) . Here is one equivalent representation:
y(x) = x(x2+ax) + b(x+c/b) = 0
and here is another:
y(x) = (x-x1)(x-x2)(x-x3) = 0,
where x1, x2, and x3 are some functions of a, b, c.

The last representation permits an immediate solution, which is:
x=x1, x=x2, x=x3.

The example may appear trivial, and some may even call it cheating, saying that finding x1, x2, and x3 is as hard as solving the original problem. This is true, but the purpose of the example was not to produce an easy solution to the cubic. The purpose was to demonstrate that different syntactic ways of representing the same information (i.e., the same polynomial) may lead to substantial differences in the complexity of computing an answer to a query (i.e., find a root).

A preferred representation is one that makes certain desirable aspects of the problem explicit, thus facilitating a speedy solution. Complexity theory is full of such examples.

Note that the complexity is query-dependent. Had our goal been to find a value x that makes the polynomial y(x) equal 4, not zero, the representation above y(x) = (x-x1)(x-x2)(x-x3) would offer no help at all. For this query, the representation
y(x) = (x-z1)(x-z2)(x-z3) + 4
would yield an immediate solution
x=z1, x=z2, x=z3,
where z1, z2, and z3 are the roots of another polynomial:
x3 + ax2 + bx + (c-4) = 0

This simple example demonstrates nicely the principle that makes graphical models more efficient than alternative representations of the same causal information, say a set of ignorability assumptions. What makes graphical models efficient is the fact that they make explicit the logical ramifications of the conditional-independencies conveyed by the model. Deriving those ramifications by algebraic or logical means takes substantially more work. (See http://ftp.cs.ucla.edu/pub/stat_ser/r396.pdf for the logic of counterfactual independencies)

A typical example of how nasty such derivations can get is given in Heckman and Pinto’s paper on “Causal Inference after Haavelmo” (Econometric Theory, 2015). Determined to avoid graphs at all cost, Heckman and Pinto derived conditional independence relations directly from Dawid’s axioms and the Markov condition (See https://en.wikipedia.org/wiki/Graphoid.) The results are pages upon pages of derivations of independencies that are displayed explicitly in the graph. http://ftp.cs.ucla.edu/pub/stat_ser/r420.pdf

Of course, this and other difficulties will not dissuade econometricians to use graphs; that would rake a scientific revolution of Kuhnian proportions. (see http://ftp.cs.ucla.edu/pub/stat_ser/r391.pdf) Still, awareness of these complexity issues should give inquisitive students the ammunition to hasten the revolution and equip econometrics with modern tools of causal analysis.

They eventually will.

## February 12, 2016

### Winter Greeting from the UCLA Causality Blog

Friends in causality research,
This greeting from the UCLA Causality blog contains:

A. An introduction to our newly published book, Causal Inference in Statistics – A Primer, Wiley 2016 (with M. Glymour and N. Jewell)
B. Comments on two other books: (1) R. Klein’s Structural Equation Modeling and (2) L Pereira and A. Saptawijaya’s on Machine Ethics.
C. News, Journals, awards and other frills.

A.
Our publisher (Wiley) has informed us that the book “Causal Inference in Statistics – A Primer” by J. Pearl, M. Glymour and N. Jewell is already available on Kindle, and will be available in print Feb. 26, 2016.
http://www.amazon.com/Causality-A-Primer-Judea-Pearl/dp/1119186846
http://www.amazon.com/Causal-Inference-Statistics-Judea-Pearl-ebook/dp/B01B3P6NJM/ref=mt_kindle?_encoding=UTF8&me=

This book introduces core elements of causal inference into undergraduate and lower-division graduate classes in statistics and data-intensive sciences. The aim is to provide students with the understanding of how data are generated and interpreted at the earliest stage of their statistics education. To that end, the book empowers students with models and tools that answer nontrivial causal questions using vivid examples and simple mathematics. Topics include: causal models, model testing, effects of interventions, mediation and counterfactuals, in both linear and nonparametric systems.

http://bayes.cs.ucla.edu/PRIMER/
A book website providing answers to home-works and interactive computer programs for simulation and analysis (using dagitty)  is currently under construction.

B1
We are in receipt of the fourth edition of Rex Kline’s book “Principles and Practice of Structural Equation Modeling”, http://psychology.concordia.ca/fac/kline/books/nta.pdf

This book is unique in that it treats structural equation models (SEMs) as carriers of causal assumptions and tools for causal inference. Gone are the inhibitions and trepidation that characterize most SEM texts in their treatments of causation.

To the best of my knowledge, Chapter 8 in Kline’s book is the first SEM text to introduce graphical criteria for parameter identification — a long overdue tool
in a field that depends on identifiability for model “fitting”. Overall, the book elevates SEM education to new heights and promises to usher a renaissance for a field that, five decades ago, has pioneered causal analysis in the behavioral sciences.

B2
Much has been written lately on computer ethics, morality, and free will. The new book “Programming Machine Ethics” by Luis Moniz Pereira and Ari Saptawijaya formalizes these concepts in the language of logic programming. See book announcement http://www.springer.com/gp/book/9783319293530. As a novice to the literature on ethics and morality, I was happy to find a comprehensive compilation of the many philosophical works on these topics, articulated in a language that even a layman can comprehend. I was also happy to see the critical role that the logic of counterfactuals plays in moral reasoning. The book is a refreshing reminder that there is more to counterfactual reasoning than “average treatment effects”.

C. News, Journals, awards and other frills.
C1.
Nominations are Invited for the Causality in Statistics Education Award (Deadline is February 15, 2016).

The ASA Causality in Statistics Education Award is aimed at encouraging the teaching of basic causal inference in introductory statistics courses. Co-sponsored by Microsoft Research and Google, the prize is motivated by the growing importance of introducing core elements of causal inference into undergraduate and lower-division graduate classes in statistics. For more information, please see http://www.amstat.org/education/causalityprize/ .

Nominations and questions should be sent to the ASA office at educinfo@amstat.org . The nomination deadline is February 15, 2016.

C.2.
Issue 4.1 of the Journal of Causal Inference is scheduled to appear March 2016, with articles covering all aspects of causal analysis. For mission, policy, and submission information please see: http://degruyter.com/view/j/jci

C.3
Finally, enjoy new results and new insights posted on our technical report page: http://bayes.cs.ucla.edu/csl_papers.html

Judea

### UAB’s Nutrition Obesity Research Center — Causal Inference Course

Filed under: Announcement,Uncategorized — bryantc @ 1:03 am
We received the following announcement from Richard F. Sarver (UAB):

UAB’s Nutrition Obesity Research Center invite you to join them at one or both of our five-day short courses at the University of Alabama at Birmingham.

June: The Mathematical Sciences in Obesity Research The mathematical sciences including engineering, statistics, computer science, physics, econometrics, psychometrics, epidemiology, and mathematics qua mathematics are increasingly being applied to advance our understanding of the causes, consequences, and alleviation of obesity. These applications do not merely involve routine well-established approaches easily implemented in widely available commercial software. Rather, they increasingly involve computationally demanding tasks, use and in some cases development of novel analytic methods and software, new derivations, computer simulations, and unprecedented interdigitation of two or more existing techniques. Such advances at the interface of the mathematical sciences and obesity research require bilateral training and exposure for investigators in both disciplines. July: Strengthening Causal Inference in Behavioral Obesity Research Identifying causal relations among variables is fundamental to science. Obesity is a major problem for which much progress in understanding, treatment, and prevention remains to be made. Understanding which social and behavioral factors cause variations in adiposity and which other factors cause variations is vital to producing, evaluating, and selecting intervention and prevention strategies. In addition, developing a greater understanding of obesity’s causes, requires input from diverse disciplines including statistics, economics, psychology, epidemiology, mathematics, philosophy, and in some cases behavioral or statistical genetics. However, applying techniques from these disciplines does not involve routine well-known ‘cookbook’ approaches but requires an understanding of the underlying principles, so the investigator can tailor approaches to specific and varying situations. For full details of each of the courses, please refer to our websites below: Mon 6/13/2016 – Fri 6/17/2016: The Mathematical Sciences in Obesity, http://www.soph.uab.edu/energetics/shortcourse/third Mon 7/25/2016 – Fri 7/29/2016: Strengthening Causal Inference in Behavioral Obesity Research, http://www.soph.uab.edu/energetics/causal_inference_shortcourse/second Limited travel scholarships are available to young investigators. Please apply by Fri 4/1/2016 and be notified of acceptance by Fri 4/8/2016. Women, members of underrepresented minority groups and individuals with disabilities are strongly encouraged to apply. We look forward to seeing you in Birmingham this summer!

## July 19, 0001

### Where does the model come from? – A rebuttal to reviewers

Filed under: Uncategorized — judea @ 5:32 pm

Readers have written to me that they have encounter criticism from reviewers who question the reliance of
an assumed causal graphs or some of their derivatives. I have encountered this criticism myself and I would like to share my rebuttal with readers, so as to help minimize this line of criticism.

Here is a letter I send to the program committee of a conference in Machine Learning.

Dear Program Committee,
I would like to call your attention to an attitude against which I have been fighting  for the past 30 years and which evidently is still common among researchers and reviewers.

I am referring to remarks made by Reviewer #7 of our paper, which judge model-based, methodological works to be “incremental and unlikely to have much impact”, ostensibly because “we do not have the model” or “one can never tell if the model is correct”, or “where the structures that are reasoned about come from” or “we require knowledge of the process that will be unknown”.

If we examine carefully the developments of ideas in computer science, or even in statistics, we find that most innovative works germinated from model-based questions, assuming that a model is known and asking: (1) what should the world look like for our question to me well-defined, (2) what algorithms can exploit its idiosyncratic features, (3) Can any algorithm produce the desired result and (4) whether the model has any testable implications.

I think conferences and journals should encourage such analytical work to continue and, at the same time, discourage reviewers from dismissing such work as “incremental” because “we do not have the model”.

I hope you share my feelings about this issue. I do not know what we can do to enforce this understanding community wide, but I hope in your communication with reviewers you succeed in conveying this spirit.

Sincerely yours,
Judea Pearl

## May 22, 0001

### draft post

Filed under: Uncategorized — judea @ 8:36 pm

### FORMULATING ASSUMPTIONSTHREE LANGUAGES

1. English: Smoking (X), Cancer (Y), Tar (Z), Genotypes (U)
2. Counterfactuals:
 Zx(u) = Zyx(u) Xy(u) = Xzy(u) = Xz = X(u), Yz(u) = Yzx(u), Zx(u) {Yz,X}

Not too friendly:

Consistent?, complete?, redundant?, arguable?

4. Structural

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