Bayesian causal inference: potential outcomes, modeling assumptions, and sensitivity

What you should be able to do after this chapter

distinguish predictive and causal questions formally;
define causal estimands through potential outcomes;
write a Bayesian model for treatment assignment and outcomes under ignorability;
explain why causal identification assumptions remain prior to the posterior;
perform basic sensitivity analysis conceptually.

1 Causal questions are not just prediction questions

Predicting what happens given observed covariates is not the same as estimating what would have happened under a counterfactual intervention. Bayesian analysis does not remove this distinction. It provides a coherent way to reason probabilistically once the causal estimand and identification assumptions have been stated (Rubin 1984; Gelman et al. 2013).

2 Potential outcomes

For each unit $i$, define:

$Y_i(1)$: outcome under treatment;
$Y_i(0)$: outcome under control.

The individual treatment effect is

\[ \tau_i = Y_i(1)-Y_i(0), \]

but only one of the two potential outcomes is observed.

Common causal estimands include:

\[ \text{ATE} = \mathbb{E}[Y(1)-Y(0)], \]

and

\[ \text{ATT} = \mathbb{E}[Y(1)-Y(0)\mid A=1]. \]

3 Identification assumptions

Bayesian inference begins after the causal assumptions are declared. Typical assumptions include:

consistency;
positivity;
ignorability or no unmeasured confounding: \[ (Y(1),Y(0)) \perp\!\!\!\perp A \mid X. \]

Without these or comparable structural assumptions, the causal estimand is not identified from the observed data.

4 Bayesian outcome model under ignorability

Suppose treatment $A_i \in \{0,1\}$ and covariates $X_i$ are observed. A simple outcome model is

\[ Y_i \mid A_i,X_i,\beta,\sigma^2 \sim \mathcal{N}(\beta_0 + \beta_A A_i + X_i^\top \beta_X,\sigma^2). \]

Under ignorability, the posterior over $(\beta,\sigma^2)$ induces a posterior over causal estimands because both potential outcomes can be generated conditional on covariates:

\[ Y_i(1) \mid X_i,\beta,\sigma^2 \sim \mathcal{N}(\beta_0+\beta_A + X_i^\top \beta_X,\sigma^2), \]

\[ Y_i(0) \mid X_i,\beta,\sigma^2 \sim \mathcal{N}(\beta_0 + X_i^\top \beta_X,\sigma^2). \]

Then

\[ \text{ATE} \mid \beta = \beta_A \]

in this additive linear model.

5 Propensity-style Bayesian modeling

A more comprehensive Bayesian workflow may model treatment assignment jointly:

\[ A_i \mid X_i,\gamma \sim \operatorname{Bernoulli}(\operatorname{logit}^{-1}(X_i^\top \gamma)), \]

and

\[ Y_i \mid A_i,X_i,\beta,\sigma^2 \sim p(\cdot). \]

This does not automatically solve confounding. But it allows the analyst to propagate uncertainty through both assignment and outcome components when the identification assumptions are accepted.

6 Mediation and structural extensions

Bayesian mediation models, structural equation models, and dynamic causal models extend the same logic:

define the causal estimand;
write a generative model consistent with the causal graph;
encode uncertainty over parameters and latent quantities;
propagate it through to the causal quantity of interest.

The challenge is still causal, not purely statistical: the model must correspond to a defensible causal story.

7 Sensitivity analysis

Because unmeasured confounding is often the core threat, Bayesian causal analysis should include sensitivity parameters whenever plausible.

For example, if an unmeasured confounder shifts the treatment-outcome relationship, one can introduce a sensitivity parameter $\delta$ into the outcome model and place a prior on it:

\[ Y_i \mid A_i,X_i,U_i,\beta,\delta \sim \mathcal{N}(\beta_0+\beta_A A_i + X_i^\top \beta_X + \delta U_i,\sigma^2). \]

Even when $U_i$ is not observed directly, priors on the induced bias structure can be varied to assess robustness.

This is often more honest than reporting a single causal estimate under heroic ignorability assumptions.

8 What Bayesian causal inference does and does not add

8.1 What it adds

coherent uncertainty propagation;
explicit prior modeling of weakly identified components;
posterior distributions for causal estimands;
a natural framework for sensitivity analysis.

8.2 What it does not add

automatic identification;
immunity to confounding;
causal validity from predictive fit alone.

The posterior can only be as causal as the assumptions that define the causal model.

9 `R` workflow

The companion lab is causal_sensitivity_grid.R. It fits a simple treatment-effect regression and then perturbs a sensitivity parameter over a grid to show how the implied treatment effect changes under hidden bias assumptions.

10 Diagnostics and pitfalls

10.1 Confusing posterior certainty with causal certainty

A narrow posterior under a wrong identification assumption is still wrong.

10.2 Treating propensity models as magic

Modeling assignment helps only within a defensible causal framework.

10.3 Ignoring overlap

Weak positivity or lack of overlap makes causal estimation unstable even in Bayesian analyses.

10.4 Omitting sensitivity analysis

If unmeasured confounding is plausible, sensitivity analysis is not optional.

11 Exercises

Define the ATE and ATT formally using potential outcomes.
Explain why ignorability is an identification assumption, not a posterior conclusion.
Show how a Bayesian linear outcome model induces a posterior for the ATE.
Describe a simple sensitivity parameter and how a prior on it would be used.
Explain why strong predictive performance does not by itself imply causal validity.

12 Summary

Bayesian causal inference is Bayesian only after the causal question is specified:

potential outcomes define estimands;
causal assumptions define identifiability;
the Bayesian model propagates uncertainty conditional on those assumptions;
sensitivity analysis remains essential.

13 Further reading

Rubin (1984) is the source for the potential-outcomes framework and for the view of causal inference as a missing-data problem, which is the framing this chapter uses throughout.

Gelman et al. (2013) chapter 8 covers the Bayesian treatment of ignorability and of designs where the assignment mechanism is known, and is explicit that the causal content of a model lives in assumptions the data cannot check. Gelman and Hill (2007) chapters 9 and 10 give the applied version with worked examples.

Berger (1985) is relevant for the sensitivity analysis discussed here: when a conclusion depends on an unverifiable assumption, the honest response is to report how fast it changes as the assumption is relaxed, which is a decision-theoretic question as much as an inferential one.

References

Berger, James O. 1985. Statistical Decision Theory and Bayesian Analysis. 2nd ed. Springer.

Gelman, Andrew, John B. Carlin, Hal S. Stern, David B. Dunson, Aki Vehtari, and Donald B. Rubin. 2013. Bayesian Data Analysis. 3rd ed. CRC Press.

Gelman, Andrew, and Jennifer Hill. 2007. Data Analysis Using Regression and Multilevel/Hierarchical Models. Cambridge University Press.

Rubin, Donald B. 1984. “Bayesianly Justifiable and Relevant Frequency Calculations for the Applied Statistician.” The Annals of Statistics 12 (4): 1151–72.

--- title: "Bayesian causal inference: potential outcomes, modeling assumptions, and sensitivity" --- ::: {.callout-note} ## What you should be able to do after this chapter - distinguish predictive and causal questions formally; - define causal estimands through potential outcomes; - write a Bayesian model for treatment assignment and outcomes under ignorability; - explain why causal identification assumptions remain prior to the posterior; - perform basic sensitivity analysis conceptually. ::: ## Causal questions are not just prediction questions Predicting what happens given observed covariates is not the same as estimating what would have happened under a counterfactual intervention. Bayesian analysis does not remove this distinction. It provides a coherent way to reason probabilistically once the causal estimand and identification assumptions have been stated [@rubin1984; @gelman2013]. ## Potential outcomes For each unit $i$, define: - $Y_i(1)$: outcome under treatment; - $Y_i(0)$: outcome under control. The individual treatment effect is $$ \tau_i = Y_i(1)-Y_i(0), $$ but only one of the two potential outcomes is observed. Common causal estimands include: $$ \text{ATE} = \mathbb{E}[Y(1)-Y(0)], $$ and $$ \text{ATT} = \mathbb{E}[Y(1)-Y(0)\mid A=1]. $$ ## Identification assumptions Bayesian inference begins after the causal assumptions are declared. Typical assumptions include: - consistency; - positivity; - ignorability or no unmeasured confounding: $$ (Y(1),Y(0)) \perp\!\!\!\perp A \mid X. $$ Without these or comparable structural assumptions, the causal estimand is not identified from the observed data. ## Bayesian outcome model under ignorability Suppose treatment $A_i \in \{0,1\}$ and covariates $X_i$ are observed. A simple outcome model is $$ Y_i \mid A_i,X_i,\beta,\sigma^2 \sim \mathcal{N}(\beta_0 + \beta_A A_i + X_i^\top \beta_X,\sigma^2). $$ Under ignorability, the posterior over $(\beta,\sigma^2)$ induces a posterior over causal estimands because both potential outcomes can be generated conditional on covariates: $$ Y_i(1) \mid X_i,\beta,\sigma^2 \sim \mathcal{N}(\beta_0+\beta_A + X_i^\top \beta_X,\sigma^2), $$ $$ Y_i(0) \mid X_i,\beta,\sigma^2 \sim \mathcal{N}(\beta_0 + X_i^\top \beta_X,\sigma^2). $$ Then $$ \text{ATE} \mid \beta = \beta_A $$ in this additive linear model. ## Propensity-style Bayesian modeling A more comprehensive Bayesian workflow may model treatment assignment jointly: $$ A_i \mid X_i,\gamma \sim \operatorname{Bernoulli}(\operatorname{logit}^{-1}(X_i^\top \gamma)), $$ and $$ Y_i \mid A_i,X_i,\beta,\sigma^2 \sim p(\cdot). $$ This does not automatically solve confounding. But it allows the analyst to propagate uncertainty through both assignment and outcome components when the identification assumptions are accepted. ## Mediation and structural extensions Bayesian mediation models, structural equation models, and dynamic causal models extend the same logic: - define the causal estimand; - write a generative model consistent with the causal graph; - encode uncertainty over parameters and latent quantities; - propagate it through to the causal quantity of interest. The challenge is still causal, not purely statistical: the model must correspond to a defensible causal story. ## Sensitivity analysis Because unmeasured confounding is often the core threat, Bayesian causal analysis should include sensitivity parameters whenever plausible. For example, if an unmeasured confounder shifts the treatment-outcome relationship, one can introduce a sensitivity parameter $\delta$ into the outcome model and place a prior on it: $$ Y_i \mid A_i,X_i,U_i,\beta,\delta \sim \mathcal{N}(\beta_0+\beta_A A_i + X_i^\top \beta_X + \delta U_i,\sigma^2). $$ Even when $U_i$ is not observed directly, priors on the induced bias structure can be varied to assess robustness. This is often more honest than reporting a single causal estimate under heroic ignorability assumptions. ## What Bayesian causal inference does and does not add ### What it adds - coherent uncertainty propagation; - explicit prior modeling of weakly identified components; - posterior distributions for causal estimands; - a natural framework for sensitivity analysis. ### What it does not add - automatic identification; - immunity to confounding; - causal validity from predictive fit alone. The posterior can only be as causal as the assumptions that define the causal model. ## `R` workflow The companion lab is [causal_sensitivity_grid.R](../../labs/r/causal_sensitivity_grid.R). It fits a simple treatment-effect regression and then perturbs a sensitivity parameter over a grid to show how the implied treatment effect changes under hidden bias assumptions. ## Diagnostics and pitfalls ### Confusing posterior certainty with causal certainty A narrow posterior under a wrong identification assumption is still wrong. ### Treating propensity models as magic Modeling assignment helps only within a defensible causal framework. ### Ignoring overlap Weak positivity or lack of overlap makes causal estimation unstable even in Bayesian analyses. ### Omitting sensitivity analysis If unmeasured confounding is plausible, sensitivity analysis is not optional. ## Exercises 1. Define the ATE and ATT formally using potential outcomes. 2. Explain why ignorability is an identification assumption, not a posterior conclusion. 3. Show how a Bayesian linear outcome model induces a posterior for the ATE. 4. Describe a simple sensitivity parameter and how a prior on it would be used. 5. Explain why strong predictive performance does not by itself imply causal validity. ## Summary Bayesian causal inference is Bayesian only after the causal question is specified: - potential outcomes define estimands; - causal assumptions define identifiability; - the Bayesian model propagates uncertainty conditional on those assumptions; - sensitivity analysis remains essential. ## Further reading @rubin1984 is the source for the potential-outcomes framework and for the view of causal inference as a missing-data problem, which is the framing this chapter uses throughout. @gelman2013 chapter 8 covers the Bayesian treatment of ignorability and of designs where the assignment mechanism is known, and is explicit that the causal content of a model lives in assumptions the data cannot check. @gelmanhill2007 chapters 9 and 10 give the applied version with worked examples. @berger1985 is relevant for the sensitivity analysis discussed here: when a conclusion depends on an unverifiable assumption, the honest response is to report how fast it changes as the assumption is relaxed, which is a decision-theoretic question as much as an inferential one.