See also a previous version, and the notebook on causal inference this will hopefully inform one day.

We follow Pearl’s summary (Pearl 2009a), sections 1-3.

In particular, the paper surveys the development of mathematical tools for inferring (from a combination of data and assumptions) answers to three types of causal queries: (1) queries about the effects of potential interventions, (also called “causal effects” or “policy evaluation”) (2) queries about probabilities of counterfactuals, (including assessment of “regret,” “attribution” or “causes of effects”) and (3) queries about direct and indirect effects (also known as “mediation”)

In particular, I want to get to the identification of causal effects given an existing causal graphical model from observational data with unobserved covariates via criteria such as the back-door criterion. We’ll see. The approach is casual, motivating Pearl’s pronouncements without deriving everything from axioms. Will make restrictive assumptions to reduce complexity of presentation. Some concerns (e.g. over what we take as axioms and what as theorems) will be ignored.

Not covered: UGs, PDAGs, Maximal Ancestral Graphs…

## Motivational examples

- Does chocolate cause you to get Nobel prizes? (Messerli 2012)
- Does hydroxychloroquine reduce COVID-19 death rate?
- Can we learn semantic objects from video? (Li et al. 2020)
- How do our weather-dependent models generalise to a changed climate? How does our in-house trained model generalise to messy real-world data? (Subbaswamy, Schulam, and Saria 2019)
- …

## Generally

- Learning physical systems where we cannot do experiments
- Learning physical systems where someone else is intervening
- Generalising ML to out-of-sample conditions
- Learning the costs of outcomes that we do not observe
- Identify spurious inference from uncontrolled correlation

Graphical causal inference is the best-regarded tool to solve these problems (or at least, tell us when we can hope to solve these problems, which is not always.)

## Machinery

We are interested in representing influence between variables in a non-parametric fashion.

Our main tool to do this will be conditional independence DAGs, and causal use of these.
Alternative name: “Bayesian Belief networks”.^{1}

### Structural Equation Models

We have a finite collections of random variables \(\mathbf{V}=\{X_1, X_2,\dots\}\).

For simplicity of exposition, each of the RVs will be discrete so that we may work with pmfs, and write \(p(X_i|X_j)\) for the pmf. I sometimes write \(p(x_i|x_j)\) to mean \(p(X_i=x_i|X_j=x_j)\).

Extension to continuous RVs, or arbitrary RVs, is straightforward for everything I discuss here. (A challenge is if the probabilities are not all strictly positive on joint cdfs.)

Here is an abstract SEM example.
Motivation in terms of structural models: We consider each RV \(X_i\) as a *node* and introduce a non-deterministic noise \(\varepsilon_i\) at each node.
(Figure 3 in the Pearl (2009a))

\[\begin{aligned} Z_1 &= \varepsilon_1 \\ Z_2 &= \varepsilon_2 \\ Z_3 &= f_3(Z_1, Z_2, \varepsilon_3) \\ X &= f_X(Z_1, Z_3, \varepsilon_X) \\ Y &= f_Y(X, Z_2, Z_3, \varepsilon_Y) \\ \end{aligned}\]

Without further information about the forms of \(f_i\) or \(\varepsilon_i\), our assumptions have constrained the conditional independence relations to permit a particular factorization of the mass function:

\[\begin{aligned} p(x, y, z_3, z_2, z_1) = &p(z_1) \\ &\cdot p(z_2) \\ &\cdot p(z_3|z_1, z_2) \\ &\cdot p(x|z_1,z_2) \\ &\cdot p(y|x,z_2,z_3) \\ \end{aligned}\]

We would like to know which variables are conditionally independent of others, given such a factorization.

More notation. We write

\[ X \perp Y|Z \] to mean “\(X\) is independent of \(Y\) given \(Z\)”.

We use this notation for *sets* of random variables,
and bold them to emphasis this.

\[ \mathbf{X} \perp \mathbf{Y}|\mathbf{Z} \]

We can solve these questions via a graph formalism. That’s where the DAGs come in.

### Directed Acyclic Graphs (DAGs)

A DAG is graph with directed edges, and no cycles.
(you cannot return to the same starting node travelling only *forward* along the arrows.)

Has an obvious interpretation in terms of causal graphs. Familiar from structural equation models.

DAGs are defined by a set of vertexes and (directed) edges.

\[ \mathcal{G}=(\mathbf{V},E) \]

We show the directs of edges by writing them as arrows.

For nodes \(X,Y\in \mathbf{V}\) we write \(X \rightarrow Y\) to mean there is a directed edged joining them.

We need some terminology.

- Parents
- The parents of a node \(X\) in a graph are all nodes joined to it by an incoming arrow, \(\operatorname{parents}(X)=\{Y\in \mathbf{V}:Y\rightarrow X\}.\) For convenience, we define \(X\in\operatorname{parents}(X)\)
- Children
- similarly, \(\operatorname{children}(X)=\{Y\in \mathbf{V}:X\rightarrow Y\}.\)
- Co-parent
- \(\operatorname{coparents}(X)=\{Y\in \mathbf{V}:\exists Z\in \mathbf{V} \text{ s.t. } X\rightarrow Z\text{ and }Y\rightarrow Z\}.\)

*Ancestors* and *descendants* should be clear as well.

This proceeds in 3 steps

The graphs describe

*conditional factorization relations*.We will do some work to construct from these relations some

*conditional independence relations*, which may be read off the graph.From these relations plus a causal interpretation we derive rules for

*identification of effects*

Anyway, a joint distribution \(p(\mathbf{V})\) decomposes according to a directed graph \(G\) if we may factor it

\[ p(X_1,X_2,\dots,X_v)=\prod_{X=1}^v p(X_i|\operatorname{parents}(X_i)) \]

Uniqueness?

The Pearl-style causal graphs we draw are I-graphs, which is to say &independence* graphs.
We are supposed to think of them not as the graph where every *edge* shows an *effect*, but as a graph where every *missing edge* shows *there is no effect*.

It would be tempting to suppose that a node is independent of its children given its parents or somesuch. But things are not so simple.

Questions:

- \(\text{Sprinkler}\perp \text{Rain}\)?
- \(\text{Sprinkler}\perp \text{Rain}|\text{Wet season}\)?
- \(\text{Sprinkler}\perp \text{Rain}|\text{Wet pavement}\)?
- \(\text{Sprinkler}\perp \text{Rain}|\text{Wet season}, \text{Wet pavement}\)?

We need new graph vocabulary *and* conditional independence vocabulary.

Axiomatic characterisation of conditional independence. (Pearl 2008; Steffen L. Lauritzen 1996).

**Theorem**: (Pearl 2008)
For disjoint subsets \(\mathbf{W},\mathbf{X},\mathbf{Y},\mathbf{Z}\subseteq\mathbf{V}.\)

Then the relation \(\cdot\perp\cdot|\cdot\) satisfies the following relations:

\[\begin{aligned} \mathbf{X} \perp \mathbf{Z} |\mathbf{Y} & \Leftrightarrow & \mathbf{Z}\perp \mathbf{X} | \mathbf{Y} && \text{ Symmetry }&\\ \mathbf{X} \perp \mathbf{Y}\cup \mathbf{W} |\mathbf{Z} & \Rightarrow & \mathbf{X} \perp \mathbf{Y} \text{ and } \mathbf{X} \perp \mathbf{W} && \text{ Decomposition }&\\ \mathbf{X} \perp \mathbf{Y}\cup \mathbf{W} |\mathbf{Z} & \Rightarrow & \mathbf{X} \perp \mathbf{Y}|\mathbf{Z}\cup\mathbf{W} && \text{ Weak Union }&\\ \mathbf{X} \perp \mathbf{Y} |\mathbf{Z} \text{ and } \mathbf{X} \perp \mathbf{W}|\mathbf{Z}\cup \mathbf{Y} |\mathbf{Z} & \Rightarrow & \mathbf{X} \perp \mathbf{Y}\cup \mathbf{W}|\mathbf{Z} && \text{ Contraction }&\\ \mathbf{X} \perp \mathbf{Y} |\mathbf{Z}\cup \mathbf{W} \text{ and } \mathbf{X} \perp \mathbf{W} |\mathbf{Z}\cup \mathbf{Y} & \Rightarrow & \mathbf{X}\perp \mathbf{W}\cup\mathbf{Y} | \mathbf{Z} && \text{ Intersection } & (*)\\ \end{aligned}\]

(*) The Intersection axiom only holds for strictly positive distributions.

(Pearl 2008):

How can we relate this to the topology of the graph?

**Definition**:
A set \(\mathbf{S}\) *blocks* a path \(\pi\) from X to Y in a DAG \(\mathcal{G}\) if either

There a node \(a\in\pi\) which

*is not*a collider on \(\pi\) such that \(a\in\mathbf{S}\)There a node \(b\in\pi\) which

*is*a collider on \(\pi\) and \(\operatorname{descendants}(b)\cap\mathbf{S}=\emptyset\)

If a path is not blocked, it is *active*.

**Definition**:
A set \(\mathbf{S}\) *d-separates* two subsets of nodes
\(\mathbf{X},\mathbf{X}\subseteq\mathcal{G}\)
if it blocks *every* path between any every pair of nodes \((A,B)\)
such that \(A\in\mathbf{X},\, B\in\mathbf{Y}.\)

This looks ghastly and unintuitive, but we have to live with it because it is the shortest path to making simple statements about conditional independence DAGs without horrible circumlocutions, or starting from undirected graphs, which is tedious.

**Theorem**: (Pearl 2008; Steffen L. Lauritzen 1996).
If the joint distribution of \(\mathbf{V}\) factorises according to the DAG
\(\mathbf{G}\) then for two subsets of variables
\(\mathbf{X}\perp\mathbf{Y}|\mathbf{S}\) iff \(\mathbf{S}\) *d*-separates \(\mathbf{X}\) and \(\mathbf{Y}\).

This puts us in a position to make more intuitive statements about the conditional independence relationships that we may read off the DAG.

**Corollary**:
The DAG Markov property.

\[ X \perp \operatorname{descendants}(X)^C|\operatorname{parents}(X) \]

**Corollary**:
The DAG Markov blanket.

Define

\[ \operatorname{blanket}(X):= \operatorname{parents}(X)\cup \operatorname{children}(X)\cup \operatorname{coparents}(X) \]

Then

\[ X\perp \operatorname{blanket}(X)^C|\operatorname{blanket}(X) \]

## Causal interpretation

We have a DAG \(\mathcal{G}\) and a set of variables \(\mathbf{V}\) to which we wish to give a causal interpretation.

Assume

- The \(\mathbf{V}\) factors according to \(\mathcal{G}\)
- \(X\rightarrow Y\) means “causes” (The Causal Markov property)
- We additionally assume
*faithfulness*, that is, that \(X\leftrightsquigarrow Y\) iff there is a path connecting them.

So, are we done?

We add the additional condition that

- all the relevant variables are included in the graph.

The BBC raised one possible confounding variable:

[…] Eric Cornell, who won the Nobel Prize in Physics in 2001, told Reuters “I attribute essentially all my success to the very large amount of chocolate that I consume. Personally I feel that milk chocolate makes you stupid… dark chocolate is the way to go. It’s one thing if you want a medicine or chemistry Nobel Prize but if you want a physics Nobel Prize it pretty much has got to be dark chocolate.”

## do-calculus

Now we can discuss the relationship between conditional dependence and causal effect. This is the difference between, say,

\[ P(\text{Wet pavement}|\text{Sprinkler}=on) \]

and

\[ P(\text{Wet pavement}|\operatorname{do}(\text{Sprinkler}=on)). \]

This is called “truncated factorization” in the paper.
\(\operatorname{do}\)-calculus and graph surgery give us rules that allow us to calculate dependencies *under marginalisation* for *intervention distributions*.

If we know \(P\), this is relatively easy. Marginalize out all influences to the causal variable of interest, which we show graphically as wiping out a link.

Now suppose we are not given complete knowledge of \(P\), but only *some* of the conditional distributions. (there are *unobservable variables*).
This is the setup of observational studies and epidemiology and so on.

What variables *must* we know the conditional distributions of in order to know the conditional effect?
That is, we call a set of covariates \(\mathbf{S}\) an *admissible set* (or *sufficient set*)
with respect to identifying the effect of \(X\) on \(Y\) iff

\[ p(Y=y|\operatorname{do}(X=x))=\sum_{\mathbf{s}} P(Y=y|X=x,\mathbf{S}=\mathbf{s}) P(\mathbf{S}=\mathbf{s}). \]

**Parent criterion**- The parents of a cause are an admissible set (Pearl 2009a).
**Back-door criterion**A set \(\mathbf{S}\) is sufficient if

- \(\mathbf{S}\cap\operatorname{descendants}(X)=\emptyset\)
- \(\mathbf{S}\) blocks all paths which start with an arrow
*into*\(\mathbf{X}\)

There are other criteria.

Causal properties of sufficient sets:

\[ P(Y=y|\operatorname{do}(X=x),S=s)=P(Y=y|X=x,S=s) \]

Hence

\[ P(Y=y|\operatorname{do}(X=x),S=s)=\sum_sP(Y=y|X=x,S=s)P(S=s). \]

More graph transformation rules! But this time for *interventions*.

- Rule 1 (Insertion/deletion of observations)
- \(P(Y\mid \operatorname{do}(X),Z,W)=P(Y\mid \operatorname{do}(X),W)\) if \(Y\) and \(Z\)
are
*d*-separated by \(X\cup W\) in \(\mathcal{G}^*\), the graph obtained from \(\mathcal{G}\) by removing all arrows pointing into variables in \(X\). - Rule 2 (Action/observation exchange)
- \(P(Y\mid \operatorname{do}(X),\operatorname{do}(Z),W)=P(Y\mid \operatorname{do}(X),Z,W)\) if \(Y\) and \(Z\) are
*d*-separated by \(X\cup W\) in \(\mathcal{G}^{\dagger}\), the graph obtained from \(\mathcal{G}\) by removing all arrows pointing into variables in \(X\) and all arrows pointing out of variables in \(Z.\) - Rule 3 (Insertion/deletion of actions)
- \(P(Y\mid \operatorname{do}(X),Z,W)=P(Y\mid \operatorname{do}(X),W)\) if \(Y\) and \(Z\) are
*d*-separated by \(X\cup W\) in \(\mathcal{G}^{\ddagger}\), the graph obtained from \(\mathcal{G}\) by removing all arrows pointing into variables in \(X\) (thus creating \(\mathcal{G}^*\)) and then removing all of the arrows pointing into variables in \(Z\) that are not ancestors of any variable in \(W\) in \(\mathcal{G}^*.\)

If some causal effect of a DAG is identifiable, then there exists a sequence of application of the do-calculus rules that can identify causal effects using only observational quantities (Shpitser and Pearl 2008)

## Case study: Causal GPs

Here is a cute paper (Schulam and Saria 2017) on causal Gaussian process regression. You need the supplement to the paper for it to make sense.

It uses Gaussian process regression to infer a time series model without worrying about discretization which is tricky in alternative approaches (e.g. Brodersen et al. (2015)).

- (Consistency). Let \(Y\) be the observed outcome, \(A \in \mathcal{C}\) be the observed action, and \(Y[a]\) be the potential outcome for action \(a \in \mathcal{C},\) then: \((Y \triangleq Y[a]) \mid A=a\)
- (Continuous-Time NUC). For all times \(t\) and all histories \(\mathcal{H}_{t^{-}}\), the densities \(\lambda^{*}(t)\), \(p^{*}\left(z_{y}, z_{a} \mid t\right),\) and \(p^{*}\left(a \mid y, t, z_{a}\right)\) do not depend on \(Y_{s}[\) a \(]\) for all times \(s>t\) and all actions a.
- (Non-Informative Measurement Times). For all times t and any history \(\mathcal{H}_{t^{-}}\), the following holds: \(p^{*}\left(y \mid t, z_{y}=1\right) \mathrm{d} y=P\left(Y_{t} \in \mathrm{d} y \mid \mathcal{H}_{t^{-}}\right)\)

Notation decoder ring:

\[(Y[a]|X):= P(Y|X,\operatorname{do}(A=a)\]

What are unobserved \(u_1,u_2\)? Event and action models? Where are latent patience health states? \(u_2\) only effects outcome measurements. Why is there a point process at all? Does each patient possess a true latent class?

How does causality flow through GP?

Why is the latent mean 5-dimensional?

## Recommended reading

### Quick intros

- Mohan and Pearl’s 2012 tutorial
- Elwert’s intro (Elwert 2013)
- Shalit and Sontag’s intro
*d*-separation without tears is an interactive verson of Pearl’s original based on daggity, a graphical model digramming tool.- 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

### Textbooks

People recommend me Koller and Friedman (2009), which includes many different flavours of DAG model and many different methods, but it didn’t suit me, being somehow too detailed and too non-specific at the same time.

Spirtes, Glymour, and Scheines (2001) and Pearl (2009a) are readable. Steffen L. Lauritzen (1996) is clear but the details of the constructions are long and detailed and more general than here. (partially directed graphs.)

The shorter Steffen L. Lauritzen (2000) is good Not overwhelming, starts with a slightly more general formalism (DAGs as a special case of PDAGs, moral graphs everywhere). Murphy (2012) has a minimal introduction intermingled with some related models, with a more ML, “expert systems”-flavour and more emphasis on Bayesian inference applications.

## Questions

Propensity score matching, where does that fit in?

## References

*Statistical Methods in Medical Research*25 (5): 2294–314. https://doi.org/10.1177/0962280213520436.

*Proceedings of the National Academy of Sciences*106 (51): 21544–49. https://doi.org/10.1073/pnas.0908800106.

*Conditional Specification of Statistical Models*. Springer Science & Business Media. https://books.google.com.au/books?hl=en&lr=&id=lKeKu_HtMdQC&oi=fnd&pg=PA1&dq=arnold+castillo+sarabia+conditional+specification+of+statistical+models&ots=gxWoVEdsde&sig=p0BJlEeB5yQ052m5YhfQ_A6Kmoo.

*Proceedings of the National Academy of Sciences*113 (27): 7345–52. https://doi.org/10.1073/pnas.1510507113.

*AAAI*, 2410–16. http://ftp.cs.ucla.edu/pub/stat_ser/r425.pdf.

*The Annals of Applied Statistics*9 (1): 247–74. https://doi.org/10.1214/14-AOAS788.

*Mathematical Methods of Operations Research*77 (3): 357–70. https://doi.org/10.1007/s00186-012-0404-7.

*Annual Review of Statistics and Its Application*1 (1): 255–78. https://doi.org/10.1146/annurev-statistics-022513-115545.

*Statistical Methods in Medical Research*22 (5): 466–92. http://smm.sagepub.com/content/22/5/466.short.

*The Annals of Statistics*40 (1): 294–321. http://projecteuclid.org/euclid.aos/1333567191.

*Biometrika*, October, asr041. https://doi.org/10.1093/biomet/asr041.

*Handbook of Causal Analysis for Social Research*, edited by Stephen L. Morgan, 245–73. Handbooks of Sociology and Social Research. Dordrecht: Springer Netherlands. https://doi.org/10.1007/978-94-007-6094-3_13.

*American Journal of Sociology*117 (3): 955–66. https://doi.org/10.1086/662659.

*Proceedings of the 10th International Workshop on Artificial Intelligence and Statistics*, 128–35. Citeseer. http://www.cs.toronto.edu/~osindero/PUBLICATIONS/HintonOsinderoBao05_CLMRF.pdf.

*Advances in Neural Information Processing Systems 21*, edited by D. Koller, D. Schuurmans, Y. Bengio, and L. Bottou, 689–96. Curran Associates, Inc. http://papers.nips.cc/paper/3548-nonlinear-causal-discovery-with-additive-noise-models.pdf.

*Artificial Intelligence*182-183 (May): 1–31. https://doi.org/10.1016/j.artint.2012.01.002.

*IEEE Transactions on Information Theory*56 (10): 5168–94. https://doi.org/10.1109/TIT.2010.2060095.

*The Handbook of Brain Theory and Neural Networks*, 490–96. http://www.cs.iastate.edu/ honavar/jordan2.pdf.

*Handbook of Neural Networks and Brain Theory*. http://mlg.eng.cam.ac.uk/zoubin/course03/hbtnn2e-I.pdf.

*Learning in Graphical Models*. Cambridge, Mass.: MIT Press.

*Journal of Machine Learning Research*8 (May): 613–36. http://jmlr.org/papers/v8/kalisch07a.html.

*IJCAI*, 83:190–93. San Francisco, CA, USA: Citeseer. https://www.ijcai.org/Proceedings/83-1/Papers/041.pdf.

*Probabilistic Graphical Models : Principles and Techniques*. Cambridge, MA: MIT Press.

*Journal of the Royal Statistical Society. Series B (Methodological)*50 (2): 157–224. http://intersci.ss.uci.edu/wiki/pdf/Lauritzen1988.pdf.

*Graphical Models*. Clarendon Press.

*Complex Stochastic Systems*, 63–107. CRC Press. https://books.google.ch/books?hl=en&lr=&id=gCENL6qflA8C&oi=fnd&pg=PA63&ots=vgUI_QIs0y&sig=4WEKa7ToKKqHC1fsSt5prFZSL4Q.

*Nature Methods*7 (4): 247–48. https://doi.org/10.1038/nmeth0410-247.

*The Annals of Statistics*37 (December): 3133–64. https://doi.org/10.1214/09-AOS685.

*Proceedings of the National Academy of Sciences*107 (14): 6286–91. https://doi.org/10.1073/pnas.0913357107.

*2018 IEEE Data Science Workshop (DSW)*, 6–10. https://doi.org/10.1109/DSW.2018.8439889.

*New England Journal of Medicine*367 (16): 1562–64. https://doi.org/10.1056/NEJMon1211064.

*Proceedings of the 24th International Conference on Machine Learning*, 625–32. ACM. http://dl.acm.org/citation.cfm?id=1273575.

*Journal of Machine Learning Research*17 (32): 1–102. http://jmlr.org/papers/v17/14-518.html.

*Counterfactuals and Causal Inference*. Cambridge University Press.

*Machine Learning: A Probabilistic Perspective*. 1 edition. Adaptive Computation and Machine Learning Series. Cambridge, MA: MIT Press.

*Machine Learning and Knowledge Extraction*1 (1, 1): 312–40. https://doi.org/10.3390/make1010019.

*Learning Bayesian Networks*. Vol. 38. Prentice Hal, Paperback. https://analyticsconsultores.com.mx/wp-content/uploads/2019/04/Learning-Bayesian-Networks-R.E.-Neapolitan-Northeastern-Illinois-University.pdf.

*Advances In Neural Information Processing Systems*. http://arxiv.org/abs/1911.07420.

*Social Networks*33 (3): 211–18. https://doi.org/10.1016/j.socnet.2011.05.003.

*In Proceedings of the National Conference on Artificial Intelligence*, 133–36. http://www.aaai.org/Papers/AAAI/1982/AAAI82-032.pdf.

*Artificial Intelligence*29 (3): 241–88. https://doi.org/10.1016/0004-3702(86)90072-X.

*Quantified Representation of Uncertainty and Imprecision*, edited by Philippe Smets, 367–89. Handbook of Defeasible Reasoning and Uncertainty Management Systems. Dordrecht: Springer Netherlands. https://doi.org/10.1007/978-94-017-1735-9_12.

*Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference*. Rev. 2. print., 12. [Dr.]. The Morgan Kaufmann Series in Representation and Reasoning. San Francisco, Calif: Kaufmann.

*Statistics Surveys*3: 96–146. https://doi.org/10.1214/09-SS057.

*Causality: Models, Reasoning and Inference*. Cambridge University Press.

*2011 49th Annual Allerton Conference on Communication, Control, and Computing (Allerton)*, 958–65. https://doi.org/10.1109/Allerton.2011.6120270.

*Annals of Statistics*30 (4): 962–1030. https://doi.org/10.1214/aos/1031689015.

*Latent Variable Modeling and Applications to Causality*, edited by Maia Berkane, 69–117. Lecture Notes in Statistics. New York, NY: Springer. https://doi.org/10.1007/978-1-4612-1842-5_4.

*Statistical Science*21 (2): 206–22. https://doi.org/10.1214/088342306000000259.

*Use of Directed Acyclic Graphs*. Agency for Healthcare Research and Quality (US). https://www.ncbi.nlm.nih.gov/books/NBK126189/.

*Proceedings of the 31st International Conference on Neural Information Processing Systems*, 1696–706. NIPS’17. Red Hook, NY, USA: Curran Associates Inc. http://papers.nips.cc/paper/6767-reliable-decision-support-using-counterfactual-models.pdf.

*Sociological Methods & Research*40 (2): 211–39. https://doi.org/10.1177/0049124111404820.

*The Journal of Machine Learning Research*9: 1941–79.

*BMC Medical Research Methodology*8 (1): 70. https://doi.org/10.1186/1471-2288-8-70.

*Proceedings of the Conference on Empirical Methods in Natural Language Processing*, 145–56. Association for Computational Linguistics. http://dl.acm.org/citation.cfm?id=1613737.

*BMJ*327 (7429): 1459–61. https://doi.org/10.1136/bmj.327.7429.1459.

*Causation, Prediction, and Search*. Second Edition. Adaptive Computation and Machine Learning. The MIT Press. https://www.cs.cmu.edu/afs/cs.cmu.edu/project/learn-43/lib/photoz/.g/scottd/fullbook.pdf.

*The 22nd International Conference on Artificial Intelligence and Statistics*, 3118–27. PMLR. http://proceedings.mlr.press/v89/subbaswamy19a.html.

*Proceedings of the Twenty-Seventh Conference on Uncertainty in Artificial Intelligence*, 681–88. UAI’11. Arlington, Virginia, USA: AUAI Press. http://arxiv.org/abs/1202.3764.

*Statistical Methods in Medical Research*21 (1): 7–30. https://doi.org/10.1177/0962280210387717.

*The Annals of Mathematical Statistics*5 (3): 161–215. https://doi.org/10.1214/aoms/1177732676.

*Exploring Artificial Intelligence in the New Millennium*, edited by G. Lakemeyer and B. Nebel, 239–36. Morgan Kaufmann Publishers. http://www.merl.com/publications/TR2001-22.

*Proceedings of the Thirtieth AAAI Conference on Artificial Intelligence*, 3315–21. AAAI’16. Phoenix, Arizona: AAAI Press. https://www.tcs.uni-luebeck.de/downloads/papers/2016/full-version.pdf.

*Proceedings of the UAI 2014 Conference on Causal Inference: Learning and Prediction - Volume 1274*, 11–24. CI’14. Aachen, DEU: CEUR-WS.org. https://staff.fnwi.uva.nl/j.m.mooij/articles/uai2014ci_proceedings.pdf#page=17.

*Proceedings of the 24th International Conference on Artificial Intelligence*, 3243–49. IJCAI’15. Buenos Aires, Argentina: AAAI Press. https://www.ijcai.org/Proceedings/15/Papers/457.pdf.

Terminology confusion: This is orthogonal to subjective probability.↩︎

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