= Inverse probability weighting
{wiki}
<Inverse probability weighting> weights observations by reciprocals of the <probabilities> that generated their observed assignment or selection. For binary <treatment> with <propensity score> $e(X)$, the average-treatment-effect observation weight is
$$
w(A,X)=\frac A{e(X)}+\frac{1-A}{1-e(X)}.
$$
It uses the <probability> of the <treatment> actually received. Under <conditional exchangeability>, <consistency in causal inference> and <positivity>, the identification identity is
$$
\mathbb E\left[\frac{\mathbf1\{A=a\}Y}{P(A=a\mid X)}\right]=\mathbb E[Y(a)].
$$
To prove it, condition on $X$: the assignment <probability> cancels its denominator, leaving $\mathbb E(Y\mid A=a,X)$. Average over $X$ and apply the <causal identification> assumptions. The same cancellation with a function $g(X)$ instead of $Y$ proves population balance of the measured <covariates> under correctly specified assignment <probabilities>.
If a stratum has <treatment> <probability> $0.2$, each treated observation there receives weight five: it represents units with the same measured baseline characteristics who usually receive the other <treatment>. Very small assignment <probabilities> give large weights and unstable estimates. Estimated assignment models, trimming and <normalization> can change finite-sample behavior and, for trimming, the target population. <Overlap weights> deliberately define a different target rather than estimating the same full-population contrast by another name.
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