[HN Gopher] Revisiting the Classics: Jensen's Inequality (2023)
       ___________________________________________________________________
        
       Revisiting the Classics: Jensen's Inequality (2023)
        
       Author : cpp_frog
       Score  : 78 points
       Date   : 2024-08-21 14:05 UTC (1 days ago)
        
 (HTM) web link (francisbach.com)
 (TXT) w3m dump (francisbach.com)
        
       | thehappyfellow wrote:
       | The proof of Young's inequality is pretty neat but has the
       | ,,magically think of taking a log of an arbitrary expression
       | which happens to work" step. But it clarifies why the reciprocals
       | of exponents have to sum up to 1: they are interpreted as
       | probabilities when calculating expected value.
       | 
       | Here's how I like to conceptualise it: bounding mixed variable
       | product by sum of single variable terms is useful. Logarithms
       | change multiplication to addition. Jensen's inequality lifts
       | addition from the argument of a convex function outside. Compose.
        
         | contravariant wrote:
         | You've got a product on one side and what looks like a convex
         | combination on the other, taking the log and applying Jensen's
         | inequality isn't as big a leap as it may sound.
        
           | thehappyfellow wrote:
           | Agreed, provided you have both sides of the inequality.
           | Coming up with that particular convex combination is a bit of
           | a leap that's not super intuitive to me.
        
             | SpaceManNabs wrote:
             | if you work with a lot of convex optimization, it comes up
             | pretty often. for example, if you learn fenchel conjugates,
             | the lead up and motivation to learning them will often
             | necessitate proving young's inequality with jensen's
             | inequality. that is why learning different maths is cool.
             | you intuit some ways to reshape the problem in order to
             | make these "not super intuitive" connections.
        
             | contravariant wrote:
             | It often happens that coming up with the right theorem is a
             | lot harder than finding its proof, but that's life. You
             | can't have _everything_ be easy, otherwise we 'd have
             | finished by now.
        
       | maxmininflect wrote:
       | A very natural explanation of "wikipedia proof 2" for
       | differentiable functions seems to be missing:
       | 
       | By linearity of expectation, both sides are linear in f, and for
       | linear f we have equality. Let's subtract the linear function
       | whose graph is the tangent hyperplane to f at E(X). By above,
       | this does not change the validity of the inequality. But now the
       | left hand side is 0, and right hand side is non-negative by
       | convexity, so we are done.
       | 
       | It's also now clear what the difference of the two sides is --
       | it's the expectation of the gap between f(X) an and the value of
       | the tangent plane at X.
       | 
       | Now in general replace tangent hyperplane with graph of a
       | subderivative, to recover what wiki says.
        
       | FabHK wrote:
       | And the extent to which the expectation of the function of the
       | random variable exceeds the function of the expectation of the
       | random variable depends on the variable's variability (or
       | variance), as can be seen eg by a Taylor expansion around the
       | expectation.
       | 
       | That's the reason why _linear (or affine)_ financial derivatives
       | (such as forwards) can be priced without using volatility as an
       | input, while products _with convexity_ (such as options) require
       | volatility as an input.
       | 
       | (Side note: I think Delta One desks should rename to Gamma
       | Zero...)
        
       ___________________________________________________________________
       (page generated 2024-08-22 17:01 UTC)