# Tag Info

### A generalization of the Law of Iterated Expectations

The way I understand conditional expectation and teach my students is the following: conditional expectation $E[Y|\sigma(X)]$ is a picture taken by a camera with resolution $\sigma(X)$ As mentioned ...
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### Intuition for Conditional Expectation of $\sigma$-algebra

One way to think about conditional expectation is as a projection onto the $\sigma$-algebra $\mathscr{G}$. (from Wikimedia commons) This is actually rigorously true when talking about square-...
• 6,379
Accepted

### Why do we care more about test error than expected test error in Machine Learning?

Why do we care more about $\operatorname{Err}_{\mathcal{T}}$ than Err? I can only guess, but I think it is a reasonable guess. The former concerns the error for the training set we have right now. ...
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Accepted

### Regular conditional distribution vs conditional distribution

Why do we need these two different concepts? Regular conditional distributions are useful because they allow us to generalize the elementary notions of conditional distribution where we consider ...
• 2,527
Accepted

### Conditioning a variable on itself and some other variable

What may be tripping you up here is a common imprecision in notation, where people (myself included) will use the same symbol to denote both a random variable, and a particular assignment or ...
• 7,702
Accepted

### In some sense, is linear regression an estimate of an estimate of an estimate?

To some extent, you had some very good point. The biggest problem in your interpretation is that you confused the concepts of approximation and estimation. By probability theory, there exists a Borel ...
• 20.5k
Accepted

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### What‘s wrong with my proof of the Law of Total Variance?

The transition from the second to the third line does not follow. Since $\mathbb{E}(X) \neq \mathbb{E}(X|Y)$ you have: \mathbb{E}[ (X - \mathbb{E}(X))^2 | Y ] \neq \mathbb{E}[ (X - \mathbb{E}(X|Y)...
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To be explicit, write $f(x_1,x_2,\rho,\sigma_1,\sigma_2)$ for the bivariate normal density. From the analysis at https://stats.stackexchange.com/a/71303/919 it is clear that the normalizing integral ...