How common are thermostats in your field?

Resistance to formalization and some of its reasons

We know from experience that pulling a shopping cart over rough ground feels easier than pushing it. It's simple enough to explain why: when you're pushing, the tiny wheels of the cart get stuck in holes, and you end up pressing the cart even deeper into them. But if you're pulling, you're naturally lifting the wheels out. Yet turning this common-sense idea into a formal explanation using basic physics is tricky. The formula F = ma doesn't distinguish between pushing and pulling: the mass of the cart remains the same, yet the perceived muscular exertion we intuitively associate with force clearly feels greater when pushing.

To explain this difference, you would need to model the interaction between the cart wheels and the uneven surface, which is far from straightforward. Coefficients of friction, for instance, are usually measured empirically rather than derived from first principles. At the common sense level we could get away with a natural language sentence and gesturing towards "well known" facts, but a formal explanation forces us to deal with a level of detail that makes us uncomfortable. The formalism itself may even ultimately fail to fully capture our intuitive insight, leaving us unable to express our original meaning. Much like Ariel in The Little Mermaid, who traded her voice for legs, in our attempt to gain formal clarity we find ourselves having traded away our intuition.

In his 2020 paper, Potential outcome and directed acyclic graph approaches to causality: relevance for empirical practice in economics Guido Imbens writes about the paradigmatic causal questions posed in The book of why (TBOW; Pearl & Mackenzie 2018):

These types of questions are obviously all of great importance. Does the book deliver on this, or more precisely, does the methodology described in the book allow us to answer them? The answer essentially is an indirect one: if you tell me how the world works (by giving me the full causal graph), I can tell you the answers. Whether this is satisfactory really revolves around how much the researcher is willing to assume about how the world works. Do I feel after reading the book that I understand better how to answer these questions? That is not really very clear.

The rhetorical device of giving these specific examples at the beginning of the book is very helpful, but the book does not really provide context for them. Questions of this type have come up many times before, but there is little discussion of the previous approaches to answer them. The reader is never given the opportunity to compare previously proposed answers.

The focus of TBOW, [Pearl, 2000] and [Peters, Janzing, and Sch¨olkopf, 2017], is on developing machinery for answering questions of this type given two inputs. First, knowledge of the joint distribution of all the observed variables in the model, and, second, a causal model describing the phenomena under consideration. Little is said about what comes before the identification question, namely the development of the model, and what comes after the identification question, namely estimation and inference given a finite sample. The position appears to be that the specification of a causal model and the statistical analyses are problems separate from those of identification, with the integration of those problems with questions of identification less important than the benefits of the specialization associated with keeping the identification questions separate.

These are valid objections, but they could be leveled against any theory. Ask "What will the tension of this rope be if we use it to pull our cart?" instead of asking "What is the health-care cost attributable to obesity?" and Imbens' words can be used to criticize Newtonian mechanics:

These types of questions are obviously all of great importance. Does the book deliver on this, or more precisely, does the methodology described in the book allow us to answer them? The answer essentially is an indirect one: if you tell me how the world works (by giving me the full free-body diagram), I can tell you the answers. Whether this is satisfactory really revolves around how much the researcher is willing to assume about how the world works. Etc.

This is not to say that arguing against adopting a new theoretical framework in this fashion is always pointless: it depends on the value of the theory in question. Is the prince beautiful enough (or powerful enough) to justify trading our voice for a pair of legs in the hope of seducing him? Stepping out of the metaphor, what can formal causality theory do for us that our intuition around an imagined randomized experiment cannot do?

In full generality, this question is pre-theoretical, so do not expect to find it asked in a theory book, though it may be implicitly answered there. Pearl answers it here, in the comment section, but mostly starting from a situation where we have a directed acyclic graph (DAG) we can trust to correctly represent our problem.

If you have a DAG and want to find out the causal effect of a treatment variable on an outcome variable, you can use d-separation to find a minimal set of controls to include in your regression, if it exists. There are software tools to do that automatically, e.g. DAGitty. You can in fact do more than that once you have a DAG, but the elephant in the room is whether we will be able to come up with useful DAGs for the problems that matter to us. It is not clear to me that the answer must be the same in every discipline.

A thermostat. Either this is a sauna or the temperature is in Fahrenheit.

To illustrate one of the possible obstacles on the path to coming up with a good DAG, consider a thermostat controlling a house's heating system. If the thermostat works, we will observe a negative correlation between the amount of gas burned in the heater, G, and the outside temperature, O. However, there should be no correlation between G and the inside temperature T, nor between O and T. In the absence of correlation, a naive causal discovery algorithm would conclude that there is no causal relation either, so the DAG would look like either O -> G or G -> O, with T on the side. We know this conclusion to be wrong, and the real DAG looks like G -> T <- O instead, yet observational data alone won't reveal this causal structure to us¹. This is because the thermostat system lacks faithfulness: faithfulness is the property that ensures all observed conditional independencies precisely reflect the true causal structure. A thermostat is deliberately designed to break this property by dynamically adjusting gas consumption to maintain the inside temperature constant, thus effectively concealing the causal relationship in observational data².

The prevalence of thermostats -feedback loops, essentially- is likely going to be an important factor in determining whether a certain field of study is going to benefit from causal modeling in the absence of strong theoretical reasons allowing researchers to deduce a DAG from first principles. In particular, the kind of causal discovery work I am carrying out in astronomy is, at the moment, relying on the faithfulness assumption.

1. Unless we look for subtle clues in the time dependence of the data for T, showing that this variable is being dynamically adjusted. After all no thermostat is perfect, but to see this we need to be sampling T frequently enough in time to observe its relaxation towards the equilibrium value.

2. The thermostat analogy was introduced by the economist Milton Friedman to represent the behavior of a central bank that targets inflation or similar metrics. People (and their institutions) seem eager to break faithfulness whenever it suits them: whether this is going to be a major stumbling block for causality in the social sciences remains to be seen. It likely is a smaller issue in certain contexts than in others, but we won't find out unless we give it a try.