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Actually quite good. I've TA'd this class but it's surprising how many of these little facts can be helpful if you recall them at the right time. I was just re

Actually quite good. I've TA'd this class but it's surprising how many of these little facts can be helpful if you recall them at the right time. I was just reminded of: Var[Y] = E[Var[Y|X]] + Var[E[Y|X]] and it unstuck me from a little puzzle. More recent version at: http://statistics.zone http://statistics.zone

evandwight 4y ago
For those who are wondering how this equation is true: https://en.wikipedia.org/wiki/Law_of_total_variance https://en.wikipedia.org/wiki/Law_of_total_variance
willdearden 4y ago
There is a professor who was at Wisconsin, Charles Manski, who developed partial identification, which uses tons of these decompositions. Idea is let's say you have a binary survey question where 80% respond and 90% of them respond "yes". What can we say about population "yes" rate (assume sample size is huge for simplicity)? P(Yes) = P(Yes | response) * P(response) + P(Yes | no response) * P(no response) = 0.9 * 0.8 + P(Yes | no response) * 0.2 = 0.72 + P(Yes | no response) * 0.2 Then 0 <= P(Yes | no response) <= 1, so 0.72 <= P(Yes) <= 0.92. This example is somewhat trivial but it's a useful technique for showing exactly how your assumptions map to inferences.
jarenmf 4y ago
Similar to the law of total expectation, easier for me to think of partitioning a weighted mean :)
zmmmmm 4y ago
Confusing to me how Var[Y] can be computed only from terms that are conditional on X ... surely its marginal variance has components arising from variance outside its distribution that is conditional on X? (or is that a typo ....)
- addcninblue 4y ago
  You can consider the tower rule for intuition on why this can be true: E[X] = E[E[X|Y]] Essentially, what's happening is that, in E[X|Y], we first "hold" the randomness of Y, then consider the expected value of X for every possible value of Y. Once this is determined, we then take the expectation over all values of Y, which integrates out its effects, leaving us with the expected value of X. This rule is similar, and arises from the tower rule above.
  - zmmmmm 4y ago
    thanks! that kind of does make sense!

evandwight 4y ago

For those who are wondering how this equation is true: https://en.wikipedia.org/wiki/Law_of_total_variance https://en.wikipedia.org/wiki/Law_of_total_variance

willdearden 4y ago

There is a professor who was at Wisconsin, Charles Manski, who developed partial identification, which uses tons of these decompositions. Idea is let's say you have a binary survey question where 80% respond and 90% of them respond "yes". What can we say about population "yes" rate (assume sample size is huge for simplicity)? P(Yes) = P(Yes | response) * P(response) + P(Yes | no response) * P(no response) = 0.9 * 0.8 + P(Yes | no response) * 0.2 = 0.72 + P(Yes | no response) * 0.2 Then 0 <= P(Yes | no response) <= 1, so 0.72 <= P(Yes) <= 0.92. This example is somewhat trivial but it's a useful technique for showing exactly how your assumptions map to inferences.

jarenmf 4y ago

Similar to the law of total expectation, easier for me to think of partitioning a weighted mean :)

zmmmmm 4y ago

Confusing to me how Var[Y] can be computed only from terms that are conditional on X ... surely its marginal variance has components arising from variance outside its distribution that is conditional on X? (or is that a typo ....)

addcninblue 4y ago
You can consider the tower rule for intuition on why this can be true: E[X] = E[E[X|Y]] Essentially, what's happening is that, in E[X|Y], we first "hold" the randomness of Y, then consider the expected value of X for every possible value of Y. Once this is determined, we then take the expectation over all values of Y, which integrates out its effects, leaving us with the expected value of X. This rule is similar, and arises from the tower rule above.
- zmmmmm 4y ago
  thanks! that kind of does make sense!