Causal inference on DAGs

Confounding! This scientist performed a miracle graph surgery intervention and you won’t believe what happened next

Inferring cause and effect from nature, especially from observational (as opposed to ideal experimental) data where it is hard. Graphical models and related techniques for doing it. Avoiding the danger of folk statistics. Observational studies, confounding, adjustment criteria, d-separation, identifiability, interventions, moral equivalence… Avoidance of Ecological fallacy/ Simpson’s paradox.

The gold standard, of course, is to work out if A causes B by doing an experiment where no input but A changes, then observing B, which is what a controlled trial is. In practice this is unattainable because it would usually require cloning the entire state of the universe and running multiple copies in parallel. Statistically it can be nearly as good to do the experiment where we change A and all other influences apart from are at least uncorrelated with A, which is more usually what we do — a randomised controlled trial. In many circumstances though, (budget restrictions, ethical constraints, bad experimental design…) we cannot do these ideal experiments, and a mathematical crutch is needed to get us the next-best outcome, which is to control for the things that we must (and not control for the things we must not).

Control trials are the gold standard but are not always feasible. From Girl Genius

In classic-flavoured causal inference, we use graphical models with the additional assumption that $$A\rightarrow B$$ may be read as “A causes a change in B”. This is what you end up with if you use a Structural Equation Model (a.k.a. hierarchical models) to impose a causal structure on the observations. The result is a particular type of graph, a Directed Acyclic Graph_ (DAG) which, informally put, summarises what can possibly affect what in the model. Slightly more formally, it summarises what cannot (conditionally) affect what. C&C conditional treatment effect estimation by potential outcomes.

With this tool in hand I can answer the question of when I can use my crappy observational data, collected without a good experimental design for whatever reason, to do interventional inference? There is a lot of research in this area; I should summarise the salient bits for myself.

What can go wrong?

Inferring the optimal intervention requires accounting for which arrows are independent of which.

What can I actually identify? For a start, if we are resorting to this more difficult methodology that already suggests that we might be trying to use data which was collected with no regard to our actual statistical needs, and thus we might really stretch to imagine that we can actually find actual appropriate instruments in the data. Here is an essay on that theme.

• we have at least some control over the light but choose to let it fall where it may and then proclaim that whatever it illuminates is what we were looking for all along

Teaching

Yanir Seroussi’s Causal inference resources recommends

Miguel Hernán and Jamie Robins’ causal inference book has a free draft online. See Yanir Seroussi’s review.

Jonas Peters’ notes from his teaching in 2015 (I think I took this couse).

Samantha Kleinberg write two (introductory and advanced. The latter is notable for handling for time-dependent causality.

Tutorial: David Sontag and Uri Shalit, Causal inference from observational studies. Mastering Metrics: The Path from Cause to Effect A resource list for causality in statistics, data science and physics.

Brady Neal’s Introduction to Causal Inference includes his draft textbook.

Chapter 3 of (some edition of) Pearl’s book is available as an author’s preprint: Part 1, 2, 3, 4, 5, 6.

Various classic introductions . Notably not recommended as a pedagogic experience (although as a reference text it is great and will make you smarter).

The dagitty intro is an interactive guide via visualizations. Likewise, the ggdag bias structure vignette shows of the useful explanation diagrams available in ggdag and is also a good introduction to selection bias and causal DAGs themselves.

Still confused? Overwhelmed? I am. How about a diagram?

TODO.

do-calculus

Pearl’s do calculus

In modern machine learning

Cunning modern nonparametric approaches such as Künzel et al. (2019) are covered in the causality notebook.

Continuously indexed fields

More generally that the typical framing where we have a few distinct variables joined by arrows of inference, we might be concerned with continuously indexed random fields.

Potential outcomes approach

A.k.a. Neyman-Rubin school. The two connect by, e.g. Single World Intervention Graphs . See potential outcomes.

Interactions

This point is almost obvious but worth saying. Interaction effects are already included in classic DAGs. If we are used to imagining that our DAGs are generated by linear structural equation models we might miss this. I have an apparently it is not uncommon . Some researchers propose extending the formalism to make interaction terms explicit .

Inferring a causal graph from data

Uh oh. You don’t know what causes what? Or specifically, you can’t eliminate a whole bunch of potential causal arrows a priori? Much more work. Many more philosophical difficulties also. Anyway, we can start from classic identification of graphical models and then hope that stuff is not too much harder for causal interventions specifically.

Causal time series

As with other time series methods, has its own issues.

🏗 find out how Causal impact works. (Based on Brodersen et al. (2015).)

The CausalImpact R package implements an approach to estimating the causal effect of a designed intervention on a time series. For example, how many additional daily clicks were generated by an advertising campaign? Answering a question like this can be difficult when a randomized experiment is not available. The package aims to address this difficulty using a structural Bayesian time-series model to estimate how the response metric might have evolved after the intervention if the intervention had not occurred.

More generally, we might be concerned with continuous time.

Tools

Many. See, e.g. CausalDiscoveryToolbox, ijmbarr/causalgraphicalmodels: Causal Graphical Models in Python dagR does R. Recent and backed by Microsoft, DoWhy is a python toolbox.

Incoming

I speculate that in realistic causal networks or DAGs, the number of possible correlations grows faster than the number of possible causal relationships. So confounds really are that common, and since people do not think in DAGs, the imbalance also explains overconfidence.

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