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Causal Inference
BIDMach has several basic causal estimators. IPTW stands for Inverse Probability of Treatment Weighting and is a widely used method technique for causal inference on binary treatments. We start with some data features <math>X</math>, a response <math>Y</math>, and a "treatment" <math>Z</math>. In causal inference we are interested in the effects of directly changing <math>Z</math> on <math>Y</math>. This is different from the conditional probability of <math>Y</math> given <math>Z</math> which depends on joint probability distribution of the system "as is". The causal effect instead models a change to the system where we force <math>Z</math> to a new value. The simplest approach is to regress <math>Y</math> on <math>X,Z</math>, using linear or logistic regression. The coefficient of <math>Z</math> in the regression model captures the direct influence of <math>Z</math> on <math>Y</math>. If the regression model is exact, i.e. if <math>Y</math> really is a linear function of the inputs, then this coefficient accurately captures the influence of <math>Z</math> on <math>Y</math> (for logistic regression we need to use instead the influence of the regression coefficient, which is <math>L(1,X)-L(0,X)</math> for a particular input X, where <math>L(Z,Z)</math> is the logistic predictor).
However, a regression model often wont be exact, and a different kind of estimate is needed. The next approach is to simulate randomization of <math>Z</math>. If we had randomly assigned each user to classes <math>Z=0</math> and <math>Z=1</math> we could simply use the difference in responses as the causal effect. This is the approach taken in randomized trials. But given a dataset, we can't change the assignments to <math>Z</math> that were made. The actual assignment could depend in an arbitrary fashion on the other features <math>X</math> and then the difference in response will depend on the influence of those features as well as <math>Z</math>.
and A-IPTW is an augmented form of IPTW which guards against errors