[HN Gopher] Revisiting the Classics: Jensen's Inequality (2023)
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       Revisiting the Classics: Jensen's Inequality (2023)
        
       Author : cpp_frog
       Score  : 41 points
       Date   : 2024-08-21 14:05 UTC (8 hours 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.
        
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       (page generated 2024-08-21 23:00 UTC)