| counterfactual_yhat {SPARRAfairness} | R Documentation |
counterfactual_yhat
Description
Estimation of counterfactual quantities by resampling.
Usage
counterfactual_yhat(dat, X, x = NULL, G, g, gdash, excl = NULL, n = NULL)
Arguments
dat |
data frame containing variables in X, Yhat, G, excl. Variables in U are assumed to be colnames(dat)\(X U Yhat U G U excl) |
X |
set of variables and values on which to 'condition'; essentially we assume that any causal pathway from G to Yhat is through X |
x |
values of variables X on which to condition; can be null in which case we use marginal distribution of X |
G |
grouping variable, usually a sensitive attribute |
g |
conditioned value of g |
gdash |
counterfactual value of g |
excl |
variable names to exclude from U |
n |
number of samples; if NULL return all |
Details
Counterfactual fairness is with respect to the causal graph:
G <———-U | / | | / | | / | VV V X ——–> Yhat
where
G=group (usually sensitive attribute); Yhat=outcome; X=set of variables through which G can act on Yhat, U=set of background variables;
We want the counterfactual Yhat_g' <- G |X=x,G=g (or alternatively Yhat_g' <- G |G=g).
This can be interpreted as:
the distribution of values of Yhat amongst indivdiduals whose values of U are distributed as though they were in group G=g (and, optionally, had values X=x, but whose value of G is g'
Essentially, comparison of the counterfactual quantity above to the conditional Yhat|G=g isolates the difference in Yhat due to the effect of G on Yhat through X, removing any effect due to different distributions of U due to different values of G.
To estimate Y'=Yhat_g' <- G | G=g, we need to
Compute U'~(U|G=g)
Compute the distribution X' as X'~(X|U~U',G=g')
Sample Y'~(Yhat|X~X',U~U')
To estimate Y'=Yhat_g' <- G |X=x, G=g, we need to
Compute U'~(U|G=g,X=x)
Compute the distribution X' as X'~(X|U~U',G=g')
Sample Y'~(Yhat|X~X',U~U')
This function approximates this samplying procedure as follows
Look at individuals with G=g (and optionally X=x)
Find the values of U for these individuals
Find a second set of individuals with the same values of U but for whom G=g'
Return the indices of these individuals
The values of Yhat for these individuals constitute a sample from the desired counterfactual.
Value
indices representing sample(s) from counterfactual Yhat_g' <- G |X=x,G=g
Examples
set.seed(23173)
N=10000
# Background variables sampler
background_U=function(n) runif(n) # U~U(0,1)
# Structural equations
struct_G=function(u,n) rbinom(n,1,prob=u) # G|U=u ~ Bern(u)
struct_X=function(u,g,n) rbinom(n,1,prob=u*(0.5 + 0.5*g)) # X|U=u,G=g ~ Bern(u(1+g)/2)
struct_Yhat=function(u,x,n) (runif(n,0,x) + runif(n,0,u))/2 # Yhat|X,N ~ (U(0,X) + U(0,U))/2
# To see that the counterfactual 'isolates' the difference in Yhat due to the
# causal pathway from G to Yhat through X, change the definition of struct_G to
#
# struct_G=function(u,n) rbinom(n,1,prob=1/2) # G|U=u ~ Bern(1/2)
#
# so the posterior of U|G=g does not depend on g. Note that, with this definition, the
# counterfactual Yhat_{G<01}|G=1 coincides with the conditional Yhat|G=0, since
# the counterfactual G<-1 is equivalent to just conditioning on G=1.
#
# By contrast, if we change struct_G back to its original definition, but
# change the definition of struct_Yhat to
#
# struct_Yhat=function(u,x,n) (runif(n,0,1) + runif(n,0,u))/2 # Yhat|X,N ~ (U(0,1) + U(0,U))/2
#
# so Yhat depends on G only through the change in posterior of U from changing g,
# the counterfactual Yhat_{G<01}|G=1 coincides with the conditional Yhat|G=1.
# Sample from complete causal model
U=background_U(N)
G=struct_G(U,N)
X=struct_X(U,G,N)
Yhat=struct_Yhat(U,X,N)
dat=data.frame(U,G,X,Yhat)
# True counterfactual Yhat_{G <- 0}|G=1
w1=which(dat$G==1)
n1=length(w1)
UG1=dat$U[w1] # This is U|G=1
XG1=struct_X(UG1,rep(0,n1),n1)
YhatG1=struct_Yhat(UG1,XG1,n1)
# Estimated counterfactual Yhat_{G <- 0}|G=1
ind_G1=counterfactual_yhat(dat,X="X",G="G",g = 1, gdash = 0)
YhatG1_resample=dat$Yhat[ind_G1]
# True counterfactual Yhat_{G <- 0}|G=1,X=1
w11=which(dat$G==1 & dat$X==1)
n11=length(w11)
UG1X1=dat$U[w11] # This is U|G=1,X=1
XG1X1=struct_X(UG1X1,rep(0,n11),n11)
YhatG1X1=struct_Yhat(UG1X1,XG1X1,n11)
# Estimated counterfactual Yhat_{G <- 0}|G=1
ind_G1X1=counterfactual_yhat(dat,X="X",G="G",g = 1, gdash = 0,x=1)
YhatG1X1_resample=dat$Yhat[ind_G1X1]
# Compare CDFs
x=seq(0,1,length=1000)
oldpar = par(mfrow=c(1,2))
plot(0,type="n",xlim=c(0,1),ylim=c(0,1),xlab="Value",
ylab=expression(paste("Prop. ",hat('Y')," < x")))
lines(x,ecdf(dat$Yhat)(x),col="black") # Unconditional CDF of Yhat
lines(x,ecdf(dat$Yhat[which(dat$G==1)])(x),col="red") # Yhat|G=1
lines(x,ecdf(dat$Yhat[which(dat$G==0)])(x),col="blue") # Yhat|G=0
# True counterfactual Yhat_{G <- 0}|G=1
lines(x,ecdf(YhatG1)(x),col="blue",lty=2)
# Estimated counterfactual Yhat_{G <- 0}|G=1
lines(x,ecdf(YhatG1_resample)(x),col="blue",lty=3)
legend("bottomright",
c(expression(paste(hat('Y'))),
expression(paste(hat('Y'),"|G=1")),
expression(paste(hat('Y'),"|G=0")),
expression(paste(hat(Y)[G %<-% 0],"|G=1 (true)")),
expression(paste(hat(Y)[G %<-% 0],"|G=1 (est.)"))),
col=c("black","red","blue","blue","blue"),
lty=c(1,1,1,2,3),
cex=0.5)
plot(0,type="n",xlim=c(0,1),ylim=c(0,1),xlab="Value",
ylab=expression(paste("Prop. ",hat('Y')," < x")))
lines(x,ecdf(dat$Yhat[which(dat$X==1)])(x),col="black") # CDF of Yhat|X=1
lines(x,ecdf(dat$Yhat[which(dat$G==1 & dat$X==1)])(x),col="red") # Yhat|G=1,X=1
lines(x,ecdf(dat$Yhat[which(dat$G==0 & dat$X==1)])(x),col="blue") # Yhat|G=0,X=1
# True counterfactual Yhat_{G <- 0}|G=1,X=1
lines(x,ecdf(YhatG1X1)(x),col="blue",lty=2)
# Estimated counterfactual Yhat_{G <- 0}|G=1,X=1
lines(x,ecdf(YhatG1X1_resample)(x),col="blue",lty=3)
legend("bottomright",
c(expression(paste(hat('Y|X=1'))),
expression(paste(hat('Y'),"|G=1,X=1")),
expression(paste(hat('Y'),"|G=0,X=1")),
expression(paste(hat(Y)[G %<-% 0],"|G=1,X=1 (true)")),
expression(paste(hat(Y)[G %<-% 0],"|G=1,X=1 (est.)"))),
col=c("black","red","blue","blue","blue"),
lty=c(1,1,1,2,3),
cex=0.5)
# In both plots, the estimated counterfactual CDF closely matches the CDF of the
# true counterfactual.
# Restore parameters
par(oldpar)