Jensen's Inequality as an Intuition Tool (2021)
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Jensen's Inequality as an Intuition Tool (2021)
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Re: Jensen's Inequality as an Intuition Tool (2021)
#2Jensen's inequality says that the average value of a function is biased toward the value where the derivative is closer to 0. That's where moving away from a sample point has the least impact on the output.
It applies in scenarios that are "convex", which means that the derivative is monotonically increasing or decreasing, so "closer to 0" is a consistent direction.
Re: Jensen's Inequality as an Intuition Tool (2021)
#3I would have spent longer on this article but it did not define Jensen’s inequality before I got frustrated.
I skimmed several section looking for it. :(
Re: Jensen's Inequality as an Intuition Tool (2021)
#4This was interesting, especially with the DCF example at the end - it’s pertinent to business sell decisions (assuming your ownership structure allows you to make a decision) should I sell at an 8x multiple of revenue, or hold at an X% growth rate and Y% cash flow? What’s my net after 10 years?
The point of Jensen’s inequality if I understand correctly is that you’d underestimate the value of holding using a basic estimate approach, because you’ll underestimate the compounding cash flow from growth?
Re: Jensen's Inequality as an Intuition Tool (2021)
#5This was interesting, especially with the DCF example at the end - it’s pertinent to business sell decisions (assuming your ownership structure allows you to make a decision) should I sell at an 8x multiple of revenue, or hold at an X% growth rate and Y% cash flow? What’s my net after 10 years? The point of Jensen’s inequality if I understand correctly is that you’d underestimate the value of holding using a basic es…
It depends on whether future returns increase. One tends to draw the optimistic version of Jensen’s inequality (and in general, of convex curves), but it also applies to decreasing functions.
Re: Jensen's Inequality as an Intuition Tool (2021)
#6Jensen's inequality says that the average value of a function is biased toward the value where the derivative is closer to 0. That's where moving away from a sample point has the least impact on the output. It applies in scenarios that are "convex", which means that the derivative is monotonically increasing or decreasing, so "closer to 0" is a consistent direction.
Isn’t this just saying things seek a local minima (or maxima)?
Re: Jensen's Inequality as an Intuition Tool (2021)
#7I would have spent longer on this article but it did not define Jensen’s inequality before I got frustrated. I skimmed several section looking for it. :(
I had the exact same reaction and was about to rant here but I looked up the wikipedia article and to be fair to the author there is no simple easy way to explain this. The article actually does a decent job. :)
Re: Jensen's Inequality as an Intuition Tool (2021)
#8I would have spent longer on this article but it did not define Jensen’s inequality before I got frustrated. I skimmed several section looking for it. :(
There is a problem with the organization. At least one of the explanatory paragraphs in "2" of the table of contents should come before "why I find it interesting".
Definitely recommend reading a little of this first:
https://en.m.wikipedia.org/wiki/Jensen%27s_inequality
(Could just provide a link to it near the beginning of the article for reference.)
Re: Jensen's Inequality as an Intuition Tool (2021)
#9Looks like a reasonable intuition guide with a couple of caveats. This intuition only works for gaussian or may be a few other distributions, not for a general p(x). Then the EV rows in the tables are not actual EV quantities, only the total is an EV, that can confuse somebody. Overall I think, the point could have been carried better with a few nice charts showing the p(x), f(p(x)) and the E.
Re: Jensen's Inequality as an Intuition Tool (2021)
#10Looks like a reasonable intuition guide with a couple of caveats. This intuition only works for gaussian or may be a few other distributions, not for a general p(x). Then the EV rows in the tables are not actual EV quantities, only the total is an EV, that can confuse somebody. Overall I think, the point could have been carried better with a few nice charts showing the p(x), f(p(x)) and the E.
> This intuition only works for gaussian or may be a few other distributions, not for a general p(x)
I don't think that's true. It's that if X is a random variable and φ is a convex function, then
φ(E[X]) ≤ E[φ(X)].
An intuitive way of thinking about it is if φ is convex then it is cup-shaped. So if I sample two points from X and draw a line φ(x_1) to φ(x_2) then that line will clearly lie above the points in x that are between x_1 and x_2 in the cup right? Jensen's inequality just generalises that to say what if I take all the points from X, then the expectation of φ(X) is going to sit above φ(E[X]). Because E[X] is just going to sit somewhere in the middle of X so φ(E[X]) is going to be down in the middle of the cup so is going to be smaller than (φ(x_1)+φ(x_2)+...φ(x_n))/n, which is E[φ(X)].