• In all previous examples, we can achieve good to excellent predictions
• But we do not merely want to predict systems, but also change them!

• Causal inference goes beyond prediction by modeling the outcome of interventions
  • Ask not what \(Y\) is likely to be if \(X\) happened to be \(x\)
  • Ask what \(Y\) is likely to be if \(X\) were set to \(x\)

• It provides tools that allow us to draw causal conclusions from observational data
  • Because randomized experiments are often infeasible, unethical, or impossible

• In all previous examples, we can achieve good to excellent predictions
• But we do not merely want to predict systems, but also change them!

• Causal inference goes beyond prediction and models the outcome of interventions
• It provides tools that allow us to draw causal conclusions from observational data

• In all previous examples, we can achieve good to excellent predictions
• But we do not merely want to predict systems, but also change them!

• Causal inference goes beyond prediction and models the outcome of interventions
• It provides tools that allow us to draw causal conclusions from observational data

• In all previous examples, we can achieve good to excellent predictions
• But we do not merely want to predict systems, but also change them!

• Causal inference goes beyond prediction and models the outcome of interventions
• It provides tools that allow us to draw causal conclusions from observational data

• 1) Core concepts
  • Causality versus causal inference
  • Relating causal to probabilistic statements (DAGs)
  • Formalizing interventions, causal effects, and confounding
  • Simpson’s paradox
  • Structural Causal Models

• 2) Exercises I

• 3) Causal discovery
  • PC Algorithm
  • Invariant causal prediction
  • Restricted SCMs
  • Synthetic control
  • Interrupted Time-series

• 4) Exercises II

• David Hume defined a cause “[…] to be an object, followed by another, and where all the objects similar to the first, are followed by objects similar to the second.” (Hume, 1748, section VII)

• A key problem is that most causes are not invariably followed by their effects
  • Not everybody who smokes gets lung cancer
  • Instead, smoking increases the probability of getting lung cancer

• Causality is a big topic in philosophy
• We are interested here in causal inference
  • Identifying and estimating causal effects in the population
  • Causal effects = numerical quantities measuring changes in the distribution of an outcome under different interventions (e.g., Hernán & Robins, 2020)

• Reichenbach’s common cause principle: “If two random variables \(X\) and \(Y\) are statistically dependent (\(X \not \perp Y\)), then either (a) \(X\) causes \(Y\), (b) \(Y\) causes \(X\), or (c) there exists a third variable \(Z\) that causes both \(X\) and \(Y\). Further, \(X\) and \(Y\) become independent given \(Z\), i.e., \(X \perp Y \mid Z\).”

• Probabilistic independence of \(X\) and \(Y\) means that
  • \(P(Y, X) = P(Y \mid X) \, P(X) = P(Y) \, P(X)\)
• Probabilistic independence of \(X\) and \(Y\) given \(Z\) means that
  • \(P(Y, X, Z) = P(Y \mid Z, X) \, P(Z, X) = P(Y \mid Z) \, P(Z, X)\)

• The language of probability is limiting
• We cannot, for example, state that rain causes the streets to be wet
• We use Directed Acyclic Graphs (DAGs) to express such causal statements

• DAGs depict causal relations and imply certain (conditional) independencies

• DAGs depict causal relations and imply certain (conditional) independencies

• DAGs depict causal relations and imply certain (conditional) independencies

• d-separation allows us to read off (conditional) independencies from any DAG
• We need to define a few concepts first:
  • A path from \(X\) to \(Y\) is a sequence of nodes & edges such that the start & end nodes are \(X\) and \(Y\)
  • A conditioning set \(\mathcal{L}\) is the set of nodes we condition on (it can be empty)
  • Conditioning on a non-collider along a path blocks that path
  • A collider along a path blocks that path
  • Conditioning on a collider or any of its descendants unblocks a path (e.g., \(X \rightarrow W \leftarrow Y\))

• Two nodes \(X\) and \(Y\) are d-separated by \(\mathcal{L}\) if and only if members of \(\mathcal{L}\) block all paths between \(X\) and \(Y\)

• d-separation gives us an indendence model \(\perp_{\mathcal{G}}\) that is defined on graphs
• Probability theory gives us an independence model \(\perp_{\mathcal{P}}\) that is defined on random variables

• The causal Markov condition relates the two:

\[ X \perp_{\mathcal{G}} Y \mid Z \Rightarrow X \perp_{\mathcal{P}} Y \mid Z \enspace . \]

• If nodes \(X\) and \(Y\) are \(d\)-separated by \(Z\), then they are probabilistically independent given \(Z\)
• This condition implies (and is implied by) the following factorization:

\[ p(X_1, \ldots, X_n) = \prod_{i=1}^n p(X_i \mid \text{pa}^{\mathcal{G}}(X_i)) \enspace , \]

• where \(n\) is the number of nodes and \(\text{pa}^{\mathcal{G}}(X_i)\) are the parents of node \(X_i\)
• In other words, a node is independent of its non-descendants given its parents

• \(p(Y \mid X = x)\) describes what values of \(Y\) are likely if \(X\) happened to be x
  • We call this the observational distribution
• \(p(Y \mid do(X = x))\) describes what values of \(Y\) are likely if \(X\) is set to x
  • We call this the interventional distribution

• \(p(Y \mid X = x)\) describes what values of \(Y\) are likely if \(X\) happened to be x
  • We call this the observational distribution
• \(p(Y \mid do(X = x))\) describes what values of \(Y\) are likely if \(X\) is set to x
  • We call this the interventional distribution

• An intervention \(do(X = x)\) implies cutting all incoming edges to \(X\)

• \(p(Y \mid do(X = x))\) is observational distribution \(p_m(Y \mid X = x)\) in manipulated DAG
• For the left-most and right-most DAG, we have that \(p(Y \mid do(X = x)) = p(Y \mid X = x)\)
• For the two middle DAGs, we have to do some work:

\[ \begin{aligned} p(Y = y \mid do(X = x)) &= p_m(Y = y \mid X = x) \\[0.50em] &= \sum_{z} p_m(Y = y, Z = z \mid X = x) \hspace{5.25em} \text{... Sum rule} \\[0.50em] &= \sum_{z} p_m(Y = y \mid X = x, Z = z) \, p_m(Z = z) \hspace{1.45em} \text{... Product rule} \\[0.50em] &= \sum_{z} p(Y = y \mid X = x, Z = z) \, p(Z = z) \hspace{2.5em} \text{... Assumptions} \end{aligned} \]

• Assumption I: Mechanism is independent of whether \(X = x\) or \(do(X = x)\) (invariance)
• Assumption II: Interventions only change a single node (no ‘fat hand’)
• Assumption III: There is no unobserved confounding (causal sufficiency)

• The causal effect of \(X\) on \(Y\) is confounded if \(p(Y \mid do(X = x)) \neq p(Y \mid X = x)\)
• In observational data, there will always be confounding factors
  • What variables should we adjust for?
  • This requires knowledge about the underlying DAG
  • (Don’t just adjust for all variables — can induce bias!)

• One valid adjustment sets are the parents of \(X\)
• Another one is given by the backdoor criterion (e.g., Pearl, Glymour, & Jewell, 2016, p. 61)

• Rationale:
  • We block all spurious paths between \(X\) and \(Y\)
  • We leave all directed paths from \(X\) to \(Y\) unperturbed
  • We create no new spurious paths

• Causal inference is distinct from causality and requires a language that goes beyond association
• Directed Acyclic Graphs (DAGs) provide us with a way to state causal relations

• Reichenbach’s common cause principle links associations to causation
• It is implied by the causal Markov condition (e.g., Peters et al., 2017, p. 104), which
  • implies the factorization \(p(X_1, \ldots, X_n) = \prod_{i=1}^n p(X_i \mid \text{pa}^{\mathcal{G}}(X_i))\)
  • relates causal relationships to probabilistic relationships \(X \perp_{\mathcal{G}} Y \mid Z \Rightarrow X \perp_{\mathcal{P}} Y \mid Z\)

• Three fundamental DAG structures: chain, fork, collider
• For larger DAGs, \(d\)-separation helps us to find conditional independencies

• The \(do\)-calculus formalizes interventions
• Confounding occurs if \(p(Y \mid do(X = x)) \neq p(Y \mid X = x)\)
• DAGs help us find adjustment sets that unconfound the causal effect, in case they exist

\[ p(Y = y \mid do(X = x)) = \sum_{z} p(Y = y \mid X = x, Z = z) \, p(Z = z) \]

• 350 patients chose to take a drug and 350 chose not to
• Given these data, is the drug helpful or harmful?
• Should a doctor prescribe the drug to a patient?

• 350 patients chose to take a drug and 350 chose not to
• Given these data, is the drug helpful or harmful?
• Should a doctor prescribe the drug to a patient?

• The data are exactly the same in both cases
• Statistics alone cannot provide an answer
• We need to have an understanding of the causal story behind these data
• We can visualize the causal relations using DAGs

• Suppose we know that estrogen has a negative effect on recovery
• Note also that more women choose the drug than men
• Therefore, being a woman has an effect on drug taking as well as recovery
  • Should condition on gender!
  • This blocks the backdoor path \(D \leftarrow G \rightarrow R\)
  • And therefore unconfounds the effect \(D \rightarrow R\)

• Blood pressure is measured after taking the drug
  • It cannot cause choosing the drug
  • Instead, it is a mechanism of how the drug works
  • Should not condition on blood pressure!

• Structural Causal Models (SCM) as the fundamental building blocks of causal inference
  • We understand relations between variables in a SCM to be causal
  • Here, we will assume acyclic, linear SCMs with Gaussian error terms
  • The relations in such SCMs can be visualized in DAGs

• Example:

\[ \begin{aligned} X &:= \epsilon_X \\[.5em] Y &:= X + \epsilon_Y \\[.5em] Z &:= Y + \epsilon_Z \enspace , \end{aligned} \]

• with \(\epsilon_X, \epsilon_Y \stackrel{\text{iid}}{\sim} \mathcal{N}(0, 1)\), \(\epsilon_Z \stackrel{\text{iid}}{\sim} \mathcal{N}(0, 0.1)\), and \(\epsilon_X \perp \epsilon_Y \perp \epsilon_Z\)

• Note the factorization of the joint distribution:

\[ p(X_1, \ldots, X_n) = \prod_{i=1}^n p(X_i \mid \text{pa}^{\mathcal{G}}(X_i)) \hspace{6em} p(X, Y, Z) = p(Z \mid Y) \, p(Y \mid X) \, p(X) \]

set.seed(1)

n <- 100
x <- rnorm(n, 0, 1)
y <- x + rnorm(n, 0, 1)
z <- y + rnorm(n, 0, 0.1)

• We define the average causal effect (ACE) as

\[ ACE(Z \rightarrow Y) = \mathbb{E}\left[Y \mid do(Z = z + 1) \right] - \mathbb{E}\left[Y \mid do(Z = z) \right] \enspace . \]

• For linear Gaussian SCMs, the expectations are easy to evaluate
• In our example, the average causal effect \(Z \rightarrow Y\) is given by:

\[ ACE(Z \rightarrow Y) = \mathbb{E}[X + \epsilon_Y] - \mathbb{E}[X + \epsilon_Y] = 0 \enspace . \]

• On the other hand, the average causal effect \(X \rightarrow Y\) is given by:

\[ \begin{aligned} ACE(X \rightarrow Y) &= \mathbb{E}[X + 1 + \epsilon_Y] - \mathbb{E}[X + \epsilon_Y] \\[0.50em] &= 1 + \mathbb{E}[X + \epsilon_Y] - \mathbb{E}[X + \epsilon_Y] \\[0.50em] &= 1 \enspace . \end{aligned} \]

• Can extent this beyond the first moment (e.g., Gische & Völkle 2020)

• Causal DAGs allow us to be explicit about our causal assumptions
  • Assuming invariance, causal sufficiency, and no ‘fat hand’, we can derive valid adjustment sets
  • This provides a path toward drawing causal conclusions from observational data

• Structural Causal Models (SCMs) are the soul of (this approach to) causal inference
  • Parameterize DAGs and allow us to make quantitative statements
  • They encode observational as well as interventional distributions

• Causal inference books freely available
  • Hernán & Robins (2020)
  • Peters, Janzing, & Schölkopf (2017)
  • Pearl, Glymour, & Jewell (2016)
  • Pearl & Mackenzie (2018)

• 1) Core concepts
  • Causality versus causal inference
  • Relating causal to probabilistic statements (DAGs)
  • Formalizing interventions, causal effects, and confounding
  • Simpson’s paradox
  • Structural Causal Models

• 2) Exercises I

• 3) Causal discovery
  • PC Algorithm
  • Invariant causal prediction
  • Restricted SCMs
  • Synthetic control
  • Interrupted Time-series

• 4) Exercises II

• We focus on purely observational data for now
• Given a \(n\) observations \(\mathbf{X} = (X_1, X_2, \ldots, X_n)\) can we learn the underlying causal graph?

• Broadly speaking, there are three approaches to learning DAGs
  • Constraint-based methods (use independence relations)
  • Score-based methods (minimize some quantity over all graphs)
  • Hybrid methods (combination of both)

• We focus on the most prominent constraint-based method, the PC-Algorithm
  • Named after Peter Spirtes & Clark Glymour (Spirtes et al., 2000)

• Given a \(n\) observations \(\mathbf{X} = (X_1, X_2, \ldots, X_n)\) can we learn the underlying causal graph?

• Given a \(n\) observations \(\mathbf{X} = (X_1, X_2, \ldots, X_n)\) can we learn the underlying causal graph?

• Uses independencies to learn about the DAG (Kalisch & Bühlmann, 2007; Kalisch et al., 2012)
• Recall the causal Markov condition: for any nodes \(X\), \(Y\), and \(Z\) we have that

\[ X \perp_{\mathcal{G}} Y \mid Z \Rightarrow X \perp_{\mathcal{P}} Y \mid Z \]

• Many causal discovery methods (including the PC-Algorithm) further assume faithfulness:

\[ X \perp_{\mathcal{P}} Y \mid Z \Rightarrow X \perp_{\mathcal{G}} Y \mid Z \]

• Faithfulness allows infering causal relations (depicted \(\mathcal{G}\)) from probabilistic associations in the data
• The PC-Algorithm further assumes no hidden or selection variables

• We learn a whole class of Markov equivalent DAGs that represent the same dependencies
• Two DAGs encode the same conditional independencies if and only if (e.g., Verma & Pearl, 1991):
  • They have the same skeleton
  • They have the same v-structures

• So-called Completed Partially Directed Acyclic Graphs (CPDAGs) encode the equivalence class
• Below are the four DAGs

• We learn a whole class of Markov equivalent DAGs that represent the same dependencies
• Two DAGs encode the same conditional independencies if and only if (e.g., Verma & Pearl, 1991):
  • They have the same skeleton
  • They have the same v-structures

• So-called Completed Partially Directed Acyclic Graphs (CPDAGs) encode the equivalence class
• Here are the skeletons of the four DAGs (simply remove arrows)

• We learn a whole class of Markov equivalent DAGs that represent the same dependencies
• Two DAGs encode the same conditional independencies if and only if (e.g., Verma & Pearl, 1991):
  • They have the same skeleton
  • They have the same v-structures

• So-called Completed Partially Directed Acyclic Graphs (CPDAGs) encode the equivalence class
• Finally, here are the CPDAGs

• The PC-Algorithm proceeds in two steps:
  • First, we estimate the skeleton of the DAG
  • Second, we orient edges if possible

• Operates according to two principles:
  • There is an edge \(X_i - X_j\) if and only if \(X_i \not \perp X_j \mid S\) for all \(S \subseteq V\setminus \{X_i, X_j\}\)
  • If \(X_i - X_j - X_k\), orient edges \(X_i \rightarrow X_j \leftarrow X_k\) iff \(X_i \not \perp X_k \mid S\) for all \(S\) where \(X_j \subset S\)
    • In other words, we only orient edges if we can identify a collider

• Algorithm works as follows:

  1. Create a fully connected undirected network \(\mathcal{G}\)
  2. For every pair of vertices \((X_i, X_j)\), test whether \(X_i \perp X_j\)
     • If independence holds, remove edge \(X_i - X_j\) from \(\mathcal{G}\)
  3. For every remaining edge with \(X_k\) being connected to \(X_i\) or \(X_j\)
     • Test whether \(X_i \perp X_j \mid X_k\)
     • If independence holds, remove edge \(X_i - X_j\) from \(\mathcal{G}\)
  4. Do this until all neighbours of both \(X_i\) and \(X_j\) are exhausted
     • For example, test \(X_i \perp X_j \mid X_k, X_l\) etc. and remove edge if independence holds
  5. Orient edges when a collider is identified
  6. If applicable, orient more edges that are logically implied using Meek’s rules (Meek, 1995)

• Suppose the true DAG is \(X \rightarrow Z \rightarrow Y\)
• The PC-Algorithm would proceed as follows

library('pcalg')
set.seed(1)

n <- 1000
X <- rnorm(n)
Z <- X + rnorm(n)
Y <- Z + rnorm(n)

suffStat <- list('C' = cor(cbind(X, Y, Z)), 'n' = n)
fit <- pc(suffStat = suffStat, indepTest = gaussCItest, p = 3, alpha = 0.01)

plot(fit, main = '')

• Suppose the true DAG is \(X \rightarrow Z \leftarrow Y\)
• The PC-Algorithm would proceed as follows

library('pcalg')
set.seed(1)

n <- 1000
X <- rnorm(n)
Y <- rnorm(n)
Z <- X + Y + rnorm(n)

suffStat <- list('C' = cor(cbind(X, Y, Z)), 'n' = n)
fit <- pc(suffStat = suffStat, indepTest = gaussCItest, p = 3, alpha = 0.01)

plot(fit, main = '')

• Assumes an oracle that can correctly tell us all possible conditional independencies
• This is usually not the case, and we rely on standard conditional independence tests from statistics
• No uncertainty quantification

• Faithfulness may not hold in contexts where we estimate population parameters (Uhler et al., 2013)

• PC algorithm cannot deal with hidden or selection variables (Kalisch et al., 2012)

• Problems in practice (see e.g., Ramsey et al., 2011, for neuroscience context)
  • Indirect measurements and measurement error (see also Westfall & Yarkoni, 2016)
  • What is the ‘correct’ granularity for causal variables? (Eberhardt, 2016; Weichwald & Peters, 2020)
    • For example, ‘bad’ versus ‘good’ cholesterol

• Given \(n\) observations \((Y, X_1, \ldots, X_n)\) can we learn the causal parents of \(Y\), \(\text{pa}^{\mathcal{G}}(Y)\)?
  • Yes, if we have data from different environments \(e \in \mathcal{E}\)
  • These can be observational and interventional, as long as we never intervene on \(Y\) directly

• ICP exploits the invariance assumption that underlies causal inference (Peters et al. 2017)

• In particular, we have that the conditional distribution \(p(Y \mid \text{pa}^{\mathcal{G}}(Y))\) is invariant across \(\mathcal{E}\)

• For linear Gaussian SCMs

\[ Y^e := \mu + \sum_{i \, \in \, \text{pa}^{\mathcal{G}}(Y)} \beta_i^e \cdot X_i^e + \varepsilon_Y^e \hspace{3em} \varepsilon_Y^e \perp (X_i, \ldots X_{|\text{pa}^{\mathcal{G}}(Y)|}) \enspace , \]

• this means that \(\beta_i^e\) and \(\sigma\left(\varepsilon^e_Y\right)\) are the same across all \(e \in \mathcal{E}\)

• Suppose we have \(n\) observations from \((X, Y, Z)\) under two different environments \(e \in \mathcal{E} = \{0, 1\}\)

• Suppose we have \(n\) observations from \((X, Y, Z)\) under two different environments \(e \in \mathcal{E} = \{0, 1\}\)

library('InvariantCausalPrediction')
set.seed(1); n <- 500

X <- c(rnorm(n), 2 + rnorm(n))
Y <- X + rnorm(n)
Z <- Y + rnorm(n)

X <- cbind(X, Z); colnames(X) <- c('X', 'Z'); indicator <- rep(1:2, each = n)
ICP(X, Y, indicator, test = 'exact', showCompletion = FALSE)

## 
##  accepted set of variables 1
##  accepted set of variables 1,2

## 
##  Invariant Linear Causal Regression at level 0.01 (including multiplicity correction for the number of variables)
##  Variable X shows a significant causal effect
##  
##     LOWER BOUND  UPPER BOUND  MAXIMIN EFFECT  P-VALUE    
## X         0.44         1.04            0.44  0.00048 ***
## Z         0.00         0.52            0.00  1.00000    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

• Does not learn the whole DAG (which is too ambitious anyway)
• Allows us to learn the causal parents of an outcome using data from different environments
• Does not require faithfulness, and comes with uncertainty quantification and error control

• Extensions include
  • Accounting for hidden variables
  • Nonlinear SCMs (Heinze-Deml, Peters, & Meinshausen 2018)
  • Time-series data (Pfister, Bühlmann, & Peters, 2018)

• For an application to neuroscience, see Weichwald & Peters (2020)

• 1) Core concepts
  • Causality versus causal inference
  • Relating causal to probabilistic statements (DAGs)
  • Formalizing interventions, causal effects, and confounding
  • Simpson’s paradox
  • Structural Causal Models

• 2) Exercises I

• 3) Causal discovery
  • PC Algorithm
  • Invariant causal prediction
  • Restricted SCMs
  • Synthetic control
  • Interrupted Time-series

• 4) Exercises II

• Conditional independence test methods require at least three variables to be applicable
• This means they cannot tell us whether \(X \rightarrow Y\) or \(Y \rightarrow X\)

• Conditional independence test methods require at least three variables to be applicable
• This means they cannot tell us whether \(X \rightarrow Y\) or \(Y \rightarrow X\)

• Suppose we have the following SCM

\[ \begin{aligned} X &:= \varepsilon_X \\ Y &:= f(X, \varepsilon_Y) \\ \end{aligned} \]

• with \(X \perp \varepsilon_Y\) and \(\varepsilon_X \perp \varepsilon_Y\)

• The problem is that there is an equivalent SCM of the form

\[ \begin{aligned} Y &:= \varepsilon_y \\ X &:= g(Y, \varepsilon_X) \\ \end{aligned} \]

• with \(Y \perp \varepsilon_X\) and \(\varepsilon_X \perp \varepsilon_Y\) (e.g., Spirtes & Zhang, 2016)

• Thus we cannot distinguish \(X \rightarrow Y\) from \(Y \rightarrow X\) when allowing these flexible SCMs
• Assumptions about \(f\) and the distribution of \(\varepsilon\) can help us uncover the direction!
• In addition to assumming no confounding and no selection bias (Mooij et al. 2016)

• If \(f\) is a linear function and \(\varepsilon\) non-gaussian, causal direction is identified (Shimizu et al., 2006)

• If \(f\) is a nonlinear function and \(\varepsilon\) Gaussian, causal direction is identified (Hoyer et al., 2009, Peters et al., 2014)

library('pcalg')
options(scipen = 99, digits = 4)

url <- 'https://webdav.tuebingen.mpg.de/cause-effect/'
dat <- read.table(paste0(url, 'pair0064.txt'), col.names = c('drinking_water_access', 'infant_mortality'))

lingam(dat)$Bpruned

##        [,1] [,2]
## [1,]  0.000    0
## [2,] -1.817    0

library('mgcv')
library('dHSIC')
options(scipen = 99, digits = 4)

url <- 'https://webdav.tuebingen.mpg.de/cause-effect/'
dat <- read.table(paste0(url, 'pair0038.txt'), col.names = c('age', 'bmi'))

m1 <- gam(bmi ~ s(age), data = dat)
m2 <- gam(age ~ s(bmi), data = dat)

library('mgcv')
library('dHSIC')
options(scipen = 99, digits = 4)

url <- 'https://webdav.tuebingen.mpg.de/cause-effect/'
dat <- read.table(paste0(url, 'pair0038.txt'), col.names = c('age', 'bmi'))

m1 <- gam(bmi ~ s(age), data = dat)
m2 <- gam(age ~ s(bmi), data = dat)

c(
  dhsic.test(resid(m1), dat$age, method = 'gamma')$p.value, # Age \perp error
  dhsic.test(resid(m2), dat$bmi, method = 'gamma')$p.value  # BMI \perp error
)

## [1] 0.12313742 0.00002698

• Can’t say that face masks curb infections from these data
• The problem is that there are likely many confounding variables
• We could try to find a suitable control group. Maybe compare Japan to Iran?
• But this choice is very subjective (and obviously incorrect)

• The synthetic control method provides a data-driven solution (Abadie et al., 2010; Abadie et al., 2011)

• Compare a treated unit \(Y\) against a synthetic unit \(Y_s\)
  • \(Y_s\) is a weighted average of control units
  • It best approximates the most relevant characteristics of \(Y\) prior to treatment

• Observe \(j = 1, \ldots, J + 1\) units for time \(t = 1, \ldots, T\)
• Suppose that \(j = 1\) is the treated unit \(j = 2, \ldots, J + 1\) are the control units
• Some policy takes place at time point \(T_0\), so that \(t = 1, \ldots, T_0\) is pre-intervention
• Define two potential outcomes
  • \(Y_{1t}^N\) is the value of the treated unit at \(t\) where no intervention took place
  • \(Y_{1t}^I\) is the value of the treated unit at \(t\) where an intervention took place
  • The treatment effect is given by \(\alpha_t =Y_{1t}^I - Y_{1t}^N\)

• The issue is, of course, that we don’t observe \(Y_{1t}^N\) for \(t > T_0\)
• The synthetic control method aims to best approximate \(Y_{1t}^N\) for \(t > T_0\)

• Mitze et al. (2020) apply the synthetic control method to face masks
• Used a weighted combination of covariates from 409 regions to build the synthetic control

• Mitze et al. (2020) apply the synthetic control method to face masks
• Used a weighted combination of covariates from 409 regions to build the synthetic control

• Use the S&P500 as a synthetic control

• Similar to synthetic control, except that we do not use a control unit (Bernal et al., 2017, 2019)
• Leefting & Hinne (2020) apply it to Kundalini Yoga

• Causal discovery is very ambitious
• PC algorithm learns Markov equivalent DAGs from observational data
  • Assumes faithfulness, no hidden and no selection variables
  • Faithfulness is a strong assumption, conditional independence testing is hard

• ICP uses data from different environments to learn causal parents
  • Assumes invariant mechanisms (standard in causal inference)
  • Does not require faithfulness, can be extended to nonlinear case, hidden variables, time-series

• Restricted SCMs learn the causal direction (\(X \rightarrow Y\) or \(Y \rightarrow X\))
  • LiNGAMs assume linear function with non-Gaussian error
  • Nonlinear Gaussian additive noise models assume non-linear function with Gaussian error

• Synthetic control methods create a control to learn causal effect (usually after some intervention)
• Interrupted time-series similar in spirit, except no synthetic control