线性回归中E[Y]=f(X)的概念困惑及技术问询

Hey there, let's unpack this confusion step by step—this is a super common point of mix-up when first diving into ML theory, so you're not alone!

First, Let's Clarify the Symbols (Critical!) #

Your initial misunderstanding likely comes from mixing up two key details:

• What uppercase (X) and (Y) represent: In most ML textbooks, uppercase (X) and (Y) are vector-valued random variables, not just fixed sample vectors. For example:
  • (X) might be a (d)-dimensional vector (e.g., (d) features for a single data point: age, income, etc.)
  • (Y) might be a (k)-dimensional vector (e.g., in multi-output regression: house price, monthly rent; or multi-class classification: class probabilities)
• Conditional vs. unconditional expectation: You mentioned "Y的期望是单个值"—that's the unconditional expectation (E[Y]), which (if (Y) is a vector) is actually a vector too (each element is the expectation of (y_i)). But the equation we care about here is the conditional expectation (E[Y|X]), which is a function that depends directly on (X).

Why (E[Y|X] = f(X)) Makes Sense #

Let's break this down clearly:

• For any specific value of (X = x) (a fixed (d)-dimensional vector), (E[Y|X=x]) is the expected value of (Y) given that we observe (X) equals (x). This has the exact same dimension as (Y)—if (Y) is (k)-dimensional, (E[Y|X=x]) is also a (k)-dimensional vector, where each element is (E[y_i|X=x]).
• The function (f(X)) is defined exactly as this conditional expectation: (f(x) = E[Y|X=x]) for every possible (x). So when we write (E[Y|X] = f(X)), we're saying the conditional expectation of (Y) given the random variable (X) is exactly the function (f) applied to (X).

What This Conclusion Represents #

This is the theoretical backbone of supervised learning:

• (f(X)) is the optimal predictor of (Y) given (X) when using mean squared error (MSE) as our loss metric. In other words, no other function can produce a lower average squared error when predicting (Y) from (X).
• It captures the average relationship between (X) and (Y). For example, if (X) is a house's square footage and (Y) is its price, (f(x)) tells you the average price of a house with (x) square footage—this is the most reliable prediction you can make, since it's rooted in the data's average behavior.

Addressing Your Specific Confusion #

You thought "Y的每个元素y_i对应X中索引i处元素x_i的f(x_i)"—this is a misinterpretation. (f(X)) doesn't map individual elements of (X) to individual elements of (Y) (unless (X) and (Y) are 1-dimensional scalars):

• If (X) is a (d)-dimensional vector, (f) takes the entire vector (X) as input.
• If (Y) is a (k)-dimensional vector, (f(X)) outputs a (k)-dimensional vector, where each element is the conditional expectation of the corresponding (y_i) given the full (X).

Example to Make It Concrete #

Suppose:

• (X = (x_1, x_2)) = (house square footage, number of bedrooms) (2-dimensional)
• (Y = (y_1, y_2)) = (house price, monthly rent) (2-dimensional)
  Then (f(X)) is a function that takes ((x_1, x_2)) and outputs:
  [
  f(x_1, x_2) = \left( E[y_1 | x_1, x_2], E[y_2 | x_1, x_2] \right)
  ]
  This is a 2-dimensional vector, matching (Y)'s dimension—so (E[Y|X] = f(X)) is a valid equality of vectors, not a single value equated to a vector.

内容的提问来源于stack exchange，提问作者demalegabi