Ratio of sums vs sum of ratioIs anyone aware of any general (or perhaps not so general) relationship (inequality for instance) relating A ( x , y ) = ∑ z f ( x , y , z ) ∑ z g ( y , z ) and B ( x , y ) = ∑ z ( f ( x , y , z ) g ( y , z ) ) ?Specific context (for what I'm dealing with, but not necessarily the question) is that ∑ x , y , x f ( x , y , z ) = 1 and ∑ y , x g ( y , z ) = 1 and f ( x , y , z ) ≥ 0 and g ( y , z ) ≥ 0 ∀ x , y , z. I.e. probabilities (or more generally, I guess, measures).It seems like it could 'vaguely' be related to log sum inequalities (when transformed) or Jensen's inequality perhaps?

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Ratio of sums vs sum of ratioIs anyone aware of any general (or perhaps not so general) relationship (inequality for instance) relating  A  (  x  ,  y  )  =                    ∑        z            f      (      x      ,      y      ,      z      )                      ∑        z            g      (      y      ,      z      )      and  B  (  x  ,  y  )  =      ∑    z        (                  f        (        x        ,        y        ,        z        )                    g        (        y        ,        z        )              )  ?Specific context (for what I&#039;m dealing with, but not necessarily the question) is that       ∑          x      ,      y      ,      x        f  (  x  ,  y  ,  z  )  =  1 and       ∑          y      ,      x        g  (  y  ,  z  )  =  1 and   f  (  x  ,  y  ,  z  )  ≥  0 and   g  (  y  ,  z  )  ≥  0    ∀  x  ,  y  ,  z. I.e. probabilities (or more generally, I guess, measures).It seems like it could &#039;vaguely&#039; be related to log sum inequalities (when transformed) or Jensen&#039;s inequality perhaps?

tradirasi · Accepted Answer

If you assume that both   f and   g are nonnegative, you have   A  (  x  ,  y  )  ≤  B  (  x  ,  y  )Proof:            f      (      x      ,      y      ,      z      )                      ∑                  z                    g      (      y      ,      z      )        ≤            f      (      x      ,      y      ,      z      )              g      (      y      ,      z      )      , since   g  (  y  ,  z  )  ≤      ∑          z        g  (  y  ,  z  )So   A  (  x  ,  y  )  =                    ∑        z            f      (      x      ,      y      ,      z      )                      ∑        z            g      (      y      ,      z      )        =      ∑    z              f      (      x      ,      y      ,      z      )                      ∑        z            g      (      y      ,      z      )        ≤      ∑    z              f      (      x      ,      y      ,      z      )              g      (      y      ,      z      )        =  B  (  x  ,  y  )

Ratio of sums vs sum of ratio Is anyone aware of any general (or perhaps not so general) relationsh

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