因果推断中平均处理效应为何用τ而非τᵢ表示？

Great question—this notation quirk is a super common trip-up when starting out with causal inference, so let’s break it down clearly:

First, clarify the two core effects we’re talking about #

• Individual Treatment Effect (ITE): This is the causal effect for a single unit $i$, defined as $\tau_i = Y_i(1) - Y_i(0)$. Here, $Y_i(1)$ is the potential outcome if unit $i$ gets the treatment, and $Y_i(0)$ is the outcome if they don’t. The key note: $\tau_i$ is a fixed (but unobservable) value for each individual—since we can never observe both potential outcomes for the same person, we can’t calculate $\tau_i$ directly.
• Average Treatment Effect (ATE): This is the average of those individual effects across the entire population we care about. That’s exactly what $\tau = E[Y_i(1) - Y_i(0)]$ represents.

Why no subscript $i$ on $\tau$? #

The expectation operator $E[\cdot]$ here is taking the average over all individuals $i$ in the population. Think of it this way:

1. For every individual $i$, compute their $\tau_i = Y_i(1) - Y_i(0)$
2. Take the mean of all those $\tau_i$ values across the population

The notation $\tau = E[Y_i(1) - Y_i(0)]$ is just a compact way of writing $\tau = E[\tau_i]$—it’s a population-level summary statistic, not an individual-level quantity. Adding a subscript $i$ to $\tau$ would incorrectly imply we’re talking about an individual’s effect, not the average across all individuals.

Why is $\tau_i$ never written as $E[Y_i(1) - Y_i(0)]$? #

For a single individual $i$, $Y_i(1)$ and $Y_i(0)$ are fixed (even if unobservable) potential outcomes—they aren’t random variables from the perspective of that specific unit. Taking the expectation of a fixed value just gives you the value itself. So $E[Y_i(1) - Y_i(0)] = Y_i(1) - Y_i(0) = \tau_i$, which makes the expectation operator redundant here. The expectation only becomes meaningful when we’re averaging across multiple individuals (the population).

To sum up: The lack of a subscript on $\tau$ signals that we’re talking about a population average, not an individual effect. The expectation in the ATE formula is explicitly averaging over all units $i$ in the population.

内容的提问来源于stack exchange，提问作者user321627