Bayesian causal inference requires a model for the underlying data, Pr(X, Y (0), Y (1)), and this is where science enters. But a virtue of the framework we are presenting is that
it separates science – a model for the underlying data, from what we do to learn about science – the assignment mechanism, Pr(W|X1Y (0), Y (1)). Notice that together, these two models specify a joint distribution for all observables.
3.1. The posterior distribution of causal effects
Bayesian inference for causal effects directly confronts the explicit missing potential outcomes,Ymis = {Ymis,i}whereYmis,i =WiYi(0)+(1−Wi)Yi(1). The perspective simply takes the specifications for the assignment mechanism and the underlying data (=science), and derives the posterior predictive distribution ofYmis, that is, the distrib- ution ofYmisgiven all observed values,
(7) Pr(Ymis|X, Yobs, W ).
From this distribution and the observed values of the potential outcomes,Yobs, and co- variates, the posterior distribution of any causal effect can, in principle, be calculated.
This conclusion is immediate if we view the posterior predictive distribution in(7) as specifying how to take a random draw ofYmis. Once a value ofYmis is drawn, any causal effect can be directly calculated from the drawn values of Ymis and the ob- served values of X andYobs, e.g., the median causal effect for males: med{Yi(1)− Yi(0)|Xiindicate males}. Repeatedly drawing values ofYmisand calculating the causal effect for each draw generates the posterior distribution of the desired causal effect.
Thus, we can view causal inference completely as a missing data problem, where we multiply-impute(Rubin, 1987, 2004a)the missing potential outcomes to generate a pos- terior distribution for the causal effects. We have not yet described how to generate these imputations, however.
3.2. The posterior predictive distribution ofYmisunder ignorable treatment assignment
First consider how to create the posterior predictive distribution ofYmiswhen the treat- ment assignment mechanism is ignorable (i.e., when(6)holds). In general:
(8) Pr(Ymis|X, Yobs, W )= Pr(X, Y (0), Y (1))Pr(W|X, Y (0), Y (1))
Pr(X, Y (0), Y (1))Pr(W|X, Y (0), Y (1))dYmis. With ignorable treatment assignment, Eqs.(3),(6)becomes:
(9) Pr(Ymis|X, Yobs, W )= Pr(X, Y (0), Y (1))
Pr(X, Y (0), X(1))dYmis
.
Eq. (9) reveals that under ignorability, all that needs to be modelled is the science Pr(X, Y (0), Y (1)).
Because all information is in the underlying data, the unit labels are effectively just random numbers, and hence the array(X, Y (0), Y (1))is row exchangeable. With essen- tially no loss of generality, therefore, byde Finetti’s (1963) theorem we have that the distribution of(X, Y (0), Y (1)) may be taken to be i.i.d. (independent and identically
distributed) given some parameterθ:
(10) Pr
X, Y (0), Y (1)
=
N
i=1
f
Xi, Yi(0), Yi(1)|θ
p(θ )d(θ )
for some prior distributionp(θ ). Eq.(10)provides the bridge between fundamental the- ory and the practice of using i.i.d. models. A simple example illustrates what is required to apply Eq.(10).
3.3. Simple normal example – analytic solution
Suppose we have a completely randomized experiment with no covariates, and a scalar outcome variable. Also, assume plots were randomly sampled from a field ofN plots and the causal estimand is the mean difference betweenY (1)andY (0)across allN plots, sayY1−Y0. Then
Pr(Y )= N
i=1
f
Yi(0), Yi(1)|θ p(θ )dθ
for some bivariate densityf (ã|θ )indexed by parameterθwith prior distributionp(θ ).
Supposef (ã|θ )is normal with meansà = (à1, à0), variances (σ12, σ02) and correla- tionρ. Then conditional on (a)θ, (b) the observed values ofY, Yobs, and (c) the observed value of the treatment assignment, where the number of units withWi = K is nK
(K=0,1), we have that whenn0+n1=Nthe joint distribution of (Y1, Y0) is normal with means
1 2
¯
y1+à1+ρσ1
σ0(y¯0−à0)
, 1
2
¯
y0+à0+ρσ0
σ1(y¯1−à1)
,
variancesσ12(1−ρ2)/4n0,σ02(1−ρ2)/4n1, and zero correlation, wherey¯1andy¯0are the observed sample means ofY in the two treatment groups. To simplify comparison with standard answers, now assume largeN and a relatively diffuse prior distribution for (à1, à0, σ12, σ02) givenρ. Then the conditional posterior distribution ofY1−Y0
givenρis normal with mean
(11) E
Y1−Y0|Yobs, W, ρ
= ¯y1− ¯y0 and variance
(12) V
Y1−Y0|Yobs, W, ρ
= s12 n1 + s02
n0− 1
N σ(12−0),
whereσ(12−0)is the prior variance of the differencesYi(1)−Yi(0),σ12+σ02−2σ1σ0ρ.
Section 2.5 inRubin (1987, 2004a)provides details of this derivation. The answer given by(11)and(12)is remarkably similar to the one derived byNeyman (1923)from the randomization-based perspective, as pointed out in the discussion byRubin (1990).
There is no information in the observed data aboutρ, the correlation between the potential outcomes, because they are never jointly observed. A conservative inference forY1−Y0is obtained by takingσ(12−0)=0.
The analytic solution in(11)and(12)could have been obtained by simulation, as de- scribed in general in Section3.2. Simulation is a much more generally applicable tool than closed-form analysis because it can be applied in much more complicated situa- tions. In fact, the real advantage of Bayesian inference for causal effects is only revealed in situations with complications. In standard situations, the Bayesian answer often looks remarkably similar to the standard frequentist answer, as it does in the simple example of this section:
(y¯1− ¯y0)±2 s12
n1+ s02 n0
1/2
is a conservative 95% interval forY1−Y0, at least in relatively large samples.
3.4. Simple normal example – simulation approach
The intuition for simulation is especially direct in this example of Section3.3if we assumeρ =0; suppose we do so. The units withWi =1 haveYi(1)observed and are missingYi(0), and so theirYi(0)values need to be imputed. To imputeYi(0)values for them, we need to find units withYi(0)observed who are exchangeable with the Wi = 1 units, but these units are the units withWi = 0. Therefore, we estimate (in a Bayesian way) the distribution ofYi(0)from the units withWi = 0, and use this estimated distribution to imputeYi(0)for the units missingYi(0).
Since then0observed values ofYi(0)are a simple random sample of theN values ofY (0), and are normally distributed with meanà0and varianceσ02, with the standard independent noninformative prior distributions on (à0, σ02), we have for the posterior ofσ02:
σ02/s02∼ invertedXn2
0−1/(n0−1); and for the posterior distribution ofà0givenσ0:
à0∼N
¯
y0, s02/n0
;
and for the missingYi(0)givenà0andσ0: Yi(0)Wi =0i.i.d.∼ N
à0, s02 .
The missing values ofYi(1)are analogously imputed using the observed values ofYi(1).
When there are covariates observed, these are used to help predict the missing poten- tial outcomes using one regression model for the observedYi(1)given the covariates, and another regression model for the observedYi(0)given the covariates.
3.5. Simple normal example with covariate – numerical example
For a specific example with a covariate, suppose we have a large population of peo- ple with a covariate Xi indicating baseline cholesterol. Suppose the observed Xi is
dichotomous, HI versus LO, split at the median in the population. Suppose that a ran- dom sample of 100 withX0=HI is taken, and 90 are randomly assigned to the active treatment, a statin, and 10 are randomly assigned to the control treatment, a placebo.
Further suppose that a random sample of 100 withXi =LO is taken, and 10 are ran- domly assigned to the statin and 90 are assigned to the placebo. The outcome Y is cholesterol a year after baseline, withYi,obs andXi observed for all 200 units;Xi is effectively observed in the population because we know the proportion ofXi that are HI and LO.
Suppose the hypothetical observed data are as displayed inTable 1.
Table 1
Final cholesterol in artificial example Baseline y¯1 n1 y¯0 n0 s1=s0
HI 200 90 300 10 60
LO 100 10 200 90 60
Then the inferences based on the normal-model are as follows:
Table 2
Inferences for example inTable 1
HI LO Population= 12HI+12LO E(Y1−Y0|X, Yobs, W ) −100 −100 −100
V (Y1−Y0|X, Yobs, W )1/2 20 20 10√ 2
Here the notation is being slightly abused because the first entry inTable 2really should be labelledE(Y1−Y0Xi =HI|X, Yobs, W )and so forth.
The obvious conclusion in this artificial example is that the statin reduces final cholesterol for both those with HI and LO baseline cholesterol, and thus for the popula- tion which is a 50%/50% mixture of these two subpopulations. In this sort of situation, the final inference is insensitive to the assumed normality ofYi(1)givenXiand ofYi(0) givenXi; seePratt (1965)orRubin (1987, 2004a, Section 2.5)for the argument.
3.6. Nonignorable treatment assignment
With nonignorable treatment assignment, the above simplifications in Sections3.2–3.5, which follow from ignoring the specification for Pr(W|X, Y (0), Y (1)), do not follow in general, and analysis typically becomes far more difficult and uncertain. As a simple illustration, take the example in Section 3.5and assume that everything is the same except that onlyYobsis recorded, so that we do not know whether baseline is HI or LO for anyone. The actually assignment mechanism is now
Pr
W|Y (0), Y (1)
=
Pr
W|X, Y (0), Y (1) dP (X)
because X itself is missing, and so treatment assignment depends explicitly on the potential outcomes, both observed and missing, which are both correlated with the miss- ingXi.
Inference for causal effects, assuming the identical model for the science, now de- pends on the implied normal mixture model for the observedY data within each treat- ment arm, because the populationY values are a 50%/50% a mixture of those with LO and HI baseline cholesterol, and these subpopulations have different probabilities of treatment assignment. Here the inference for causal effects is sensitive to the propriety of the assumed normality and/or the assumption of a 50%/50% mixture, as well as to the prior distributions onà1,à0,σ1andσ0.
If we mistakenly ignore the nonignorable treatment assignment and simply compare the sample means of all treated with all controls, we havey¯1=.9(200)+.1(100)=190 versusy¯0 = .1(300)+.9(200) = 210; doing so, we reach the incorrect conclusion that the statin is bad for final cholesterol in the population. This sort of example is known as “Simpson’s Paradox” (Simpson, 1951)and can easily arise with incorrect analyzes of nonignorable treatment assignment mechanisms, and thus indicates why such assignment mechanisms are to be avoided whenever possible.
Randomized experiments are the most direct way of avoiding nonignorable treatment assignments. Other alternatives are ignorable designs with nonprobabilistic features so that all units with some specific value of covariates are assigned the same treatment.
With such assignment mechanisms, randomization-based inference is impossible for those units since their treatment does not change over the various possible assignments.