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The propensity score is the probability of treatment given covariates. Propensity score methods balance treated and control groups on observed characteristics through matching, stratification, weighting, or regression adjustment.
From matching methods, you know the fundamental problem of causal inference: treated and control units may differ systematically in ways that affect both who receives treatment and what outcomes they'd achieve. Direct matching on observed covariates works when you have a small number of variables, but it breaks down fast. If units differ on five or ten variables, finding a "close match" in that high-dimensional space becomes nearly impossible — the curse of dimensionality. Propensity score methods solve this by collapsing all those covariates into a single number.
The propensity score p(X) = P(D=1 | X) is the conditional probability that a unit receives treatment given its observed characteristics. The key theoretical result (Rosenbaum and Rubin, 1983) is the balancing property: conditional on the propensity score, treated and control units have the same distribution of observed covariates. In other words, if two units have the same propensity score, they are comparable — even if they differ on individual covariates. This reduces a high-dimensional matching problem to a one-dimensional one.
In practice, you estimate the propensity score using logit or probit — your prerequisite from binary choice models. You regress treatment status D on all observed covariates X, and the fitted probabilities are your estimated propensity scores. Once you have scores, you can apply them in four ways: propensity score matching pairs each treated unit to the control unit with the closest score; stratification (subclassification) divides the score distribution into bins and compares averages within each bin; inverse probability of treatment weighting (IPTW) reweights the sample so treated and control groups look like they came from the same population; regression adjustment includes the score as a control variable in an outcome regression.
The critical assumption underlying all propensity score methods is unconfoundedness (also called conditional independence or selection on observables): conditional on observed covariates, treatment assignment is independent of potential outcomes. This assumption is untestable — if there are unobserved variables that affect both treatment and outcomes, propensity scores cannot remove that bias, no matter how carefully estimated. This is why checking covariate balance *after* applying the method is essential: good balance means treated and control groups look similar on observed characteristics. It doesn't guarantee good balance on unobserved ones, but poor observed balance is a definitive sign the method has failed. Sensitivity analysis tools (like Rosenbaum bounds) help assess how robust conclusions are to potential hidden confounders.