How to prove $H(H^{-1}(v)*p)$ is convex in $v$ for every $pin [0,1]$












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In the proof of Mrs Gerber's lemma, there is a key step to prove $H(H^{-1}(v)*p)$ is convex in $v$ for every $pin [0,1]$.I don't konw how to prove it.






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  • $begingroup$
    What is Mrs. Gerber's lemma?
    $endgroup$
    – copper.hat
    Jul 9 '18 at 4:18










  • $begingroup$
    see ita.ucsd.edu/wiki/index.php?title=Mrs._Gerbers_Lemma for a statement of Mrs Gerber's lemma.
    $endgroup$
    – kimchi lover
    Jul 9 '18 at 7:25
















0












$begingroup$


In the proof of Mrs Gerber's lemma, there is a key step to prove $H(H^{-1}(v)*p)$ is convex in $v$ for every $pin [0,1]$.I don't konw how to prove it.






information-theory







share|cite|improve this question









$endgroup$












  • $begingroup$
    What is Mrs. Gerber's lemma?
    $endgroup$
    – copper.hat
    Jul 9 '18 at 4:18










  • $begingroup$
    see ita.ucsd.edu/wiki/index.php?title=Mrs._Gerbers_Lemma for a statement of Mrs Gerber's lemma.
    $endgroup$
    – kimchi lover
    Jul 9 '18 at 7:25














0












0








0





$begingroup$


In the proof of Mrs Gerber's lemma, there is a key step to prove $H(H^{-1}(v)*p)$ is convex in $v$ for every $pin [0,1]$.I don't konw how to prove it.






information-theory







share|cite|improve this question









$endgroup$




In the proof of Mrs Gerber's lemma, there is a key step to prove $H(H^{-1}(v)*p)$ is convex in $v$ for every $pin [0,1]$.I don't konw how to prove it.






information-theory




information-theory






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asked Jul 9 '18 at 3:16









DuWeiminDuWeimin

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  • $begingroup$
    What is Mrs. Gerber's lemma?
    $endgroup$
    – copper.hat
    Jul 9 '18 at 4:18










  • $begingroup$
    see ita.ucsd.edu/wiki/index.php?title=Mrs._Gerbers_Lemma for a statement of Mrs Gerber's lemma.
    $endgroup$
    – kimchi lover
    Jul 9 '18 at 7:25


















  • $begingroup$
    What is Mrs. Gerber's lemma?
    $endgroup$
    – copper.hat
    Jul 9 '18 at 4:18










  • $begingroup$
    see ita.ucsd.edu/wiki/index.php?title=Mrs._Gerbers_Lemma for a statement of Mrs Gerber's lemma.
    $endgroup$
    – kimchi lover
    Jul 9 '18 at 7:25
















$begingroup$
What is Mrs. Gerber's lemma?
$endgroup$
– copper.hat
Jul 9 '18 at 4:18




$begingroup$
What is Mrs. Gerber's lemma?
$endgroup$
– copper.hat
Jul 9 '18 at 4:18












$begingroup$
see ita.ucsd.edu/wiki/index.php?title=Mrs._Gerbers_Lemma for a statement of Mrs Gerber's lemma.
$endgroup$
– kimchi lover
Jul 9 '18 at 7:25




$begingroup$
see ita.ucsd.edu/wiki/index.php?title=Mrs._Gerbers_Lemma for a statement of Mrs Gerber's lemma.
$endgroup$
– kimchi lover
Jul 9 '18 at 7:25










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$$
H^{-1}(rtimesnu_1+(1-r)timesnu_2) leq r times H^{-1}(nu_1)+(1-r) times H^{-1}(nu_2) (1) \
H^{-1}(r times nu_1+(1-r) times nu_2)*p geq (r times H^{-1}(nu_1)+(1-r) times H^{-1}(nu_2)) * p (2) \
H^{-1}(r times nu_1+(1-r) times nu_2)*p geq r times H^{-1}(nu_1) * p+(1-r) times H^{-1}(nu_2)) * p (3) \
H(H^{-1}(r times nu_1+(1-r) times nu_2)*p) leq H(r times H^{-1}(nu_1) * p+(1-r) times H^{-1}(nu_2)) * p) (4) \
H(H^{-1}(r times nu_1+(1-r) times nu_2)*p) leq r times H(H^{-1}(nu_1) * p)+(1-r) times H(H^{-1}(nu_2)* p) (5)
$$



assume that $p$ is smaller than $1/2$. Similar derivations can be obtained for $p>1/2$.

(1) is derived because $H^{-1}(nu)$ is a convex function of $nu$,

(2) is derived because $u*p$ is a decreasing function of $u$ if $p<1/2$,

(3) is derived because $(r times H^{-1}(nu_1)+(1-r) times H^{-1}(nu_2)) * p= r times H^{-1}(nu_1)*p+(1-
r) times H^{-1}(nu_2))*p$

(4) (tricky) is derived because it can be obtained that both sides of (3) are greater than $1/2$, and $H(x)$ is a decreasing function of $x$ if $x$ is greater than $1/2$.

(5) is obtained because $H(x)$ is a concave function of $x$.






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    $begingroup$

    $$
    H^{-1}(rtimesnu_1+(1-r)timesnu_2) leq r times H^{-1}(nu_1)+(1-r) times H^{-1}(nu_2) (1) \
    H^{-1}(r times nu_1+(1-r) times nu_2)*p geq (r times H^{-1}(nu_1)+(1-r) times H^{-1}(nu_2)) * p (2) \
    H^{-1}(r times nu_1+(1-r) times nu_2)*p geq r times H^{-1}(nu_1) * p+(1-r) times H^{-1}(nu_2)) * p (3) \
    H(H^{-1}(r times nu_1+(1-r) times nu_2)*p) leq H(r times H^{-1}(nu_1) * p+(1-r) times H^{-1}(nu_2)) * p) (4) \
    H(H^{-1}(r times nu_1+(1-r) times nu_2)*p) leq r times H(H^{-1}(nu_1) * p)+(1-r) times H(H^{-1}(nu_2)* p) (5)
    $$



    assume that $p$ is smaller than $1/2$. Similar derivations can be obtained for $p>1/2$.

    (1) is derived because $H^{-1}(nu)$ is a convex function of $nu$,

    (2) is derived because $u*p$ is a decreasing function of $u$ if $p<1/2$,

    (3) is derived because $(r times H^{-1}(nu_1)+(1-r) times H^{-1}(nu_2)) * p= r times H^{-1}(nu_1)*p+(1-
    r) times H^{-1}(nu_2))*p$

    (4) (tricky) is derived because it can be obtained that both sides of (3) are greater than $1/2$, and $H(x)$ is a decreasing function of $x$ if $x$ is greater than $1/2$.

    (5) is obtained because $H(x)$ is a concave function of $x$.






    share|cite|improve this answer











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      0












      $begingroup$

      $$
      H^{-1}(rtimesnu_1+(1-r)timesnu_2) leq r times H^{-1}(nu_1)+(1-r) times H^{-1}(nu_2) (1) \
      H^{-1}(r times nu_1+(1-r) times nu_2)*p geq (r times H^{-1}(nu_1)+(1-r) times H^{-1}(nu_2)) * p (2) \
      H^{-1}(r times nu_1+(1-r) times nu_2)*p geq r times H^{-1}(nu_1) * p+(1-r) times H^{-1}(nu_2)) * p (3) \
      H(H^{-1}(r times nu_1+(1-r) times nu_2)*p) leq H(r times H^{-1}(nu_1) * p+(1-r) times H^{-1}(nu_2)) * p) (4) \
      H(H^{-1}(r times nu_1+(1-r) times nu_2)*p) leq r times H(H^{-1}(nu_1) * p)+(1-r) times H(H^{-1}(nu_2)* p) (5)
      $$



      assume that $p$ is smaller than $1/2$. Similar derivations can be obtained for $p>1/2$.

      (1) is derived because $H^{-1}(nu)$ is a convex function of $nu$,

      (2) is derived because $u*p$ is a decreasing function of $u$ if $p<1/2$,

      (3) is derived because $(r times H^{-1}(nu_1)+(1-r) times H^{-1}(nu_2)) * p= r times H^{-1}(nu_1)*p+(1-
      r) times H^{-1}(nu_2))*p$

      (4) (tricky) is derived because it can be obtained that both sides of (3) are greater than $1/2$, and $H(x)$ is a decreasing function of $x$ if $x$ is greater than $1/2$.

      (5) is obtained because $H(x)$ is a concave function of $x$.






      share|cite|improve this answer











      $endgroup$
















        0












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        0





        $begingroup$

        $$
        H^{-1}(rtimesnu_1+(1-r)timesnu_2) leq r times H^{-1}(nu_1)+(1-r) times H^{-1}(nu_2) (1) \
        H^{-1}(r times nu_1+(1-r) times nu_2)*p geq (r times H^{-1}(nu_1)+(1-r) times H^{-1}(nu_2)) * p (2) \
        H^{-1}(r times nu_1+(1-r) times nu_2)*p geq r times H^{-1}(nu_1) * p+(1-r) times H^{-1}(nu_2)) * p (3) \
        H(H^{-1}(r times nu_1+(1-r) times nu_2)*p) leq H(r times H^{-1}(nu_1) * p+(1-r) times H^{-1}(nu_2)) * p) (4) \
        H(H^{-1}(r times nu_1+(1-r) times nu_2)*p) leq r times H(H^{-1}(nu_1) * p)+(1-r) times H(H^{-1}(nu_2)* p) (5)
        $$



        assume that $p$ is smaller than $1/2$. Similar derivations can be obtained for $p>1/2$.

        (1) is derived because $H^{-1}(nu)$ is a convex function of $nu$,

        (2) is derived because $u*p$ is a decreasing function of $u$ if $p<1/2$,

        (3) is derived because $(r times H^{-1}(nu_1)+(1-r) times H^{-1}(nu_2)) * p= r times H^{-1}(nu_1)*p+(1-
        r) times H^{-1}(nu_2))*p$

        (4) (tricky) is derived because it can be obtained that both sides of (3) are greater than $1/2$, and $H(x)$ is a decreasing function of $x$ if $x$ is greater than $1/2$.

        (5) is obtained because $H(x)$ is a concave function of $x$.






        share|cite|improve this answer











        $endgroup$



        $$
        H^{-1}(rtimesnu_1+(1-r)timesnu_2) leq r times H^{-1}(nu_1)+(1-r) times H^{-1}(nu_2) (1) \
        H^{-1}(r times nu_1+(1-r) times nu_2)*p geq (r times H^{-1}(nu_1)+(1-r) times H^{-1}(nu_2)) * p (2) \
        H^{-1}(r times nu_1+(1-r) times nu_2)*p geq r times H^{-1}(nu_1) * p+(1-r) times H^{-1}(nu_2)) * p (3) \
        H(H^{-1}(r times nu_1+(1-r) times nu_2)*p) leq H(r times H^{-1}(nu_1) * p+(1-r) times H^{-1}(nu_2)) * p) (4) \
        H(H^{-1}(r times nu_1+(1-r) times nu_2)*p) leq r times H(H^{-1}(nu_1) * p)+(1-r) times H(H^{-1}(nu_2)* p) (5)
        $$



        assume that $p$ is smaller than $1/2$. Similar derivations can be obtained for $p>1/2$.

        (1) is derived because $H^{-1}(nu)$ is a convex function of $nu$,

        (2) is derived because $u*p$ is a decreasing function of $u$ if $p<1/2$,

        (3) is derived because $(r times H^{-1}(nu_1)+(1-r) times H^{-1}(nu_2)) * p= r times H^{-1}(nu_1)*p+(1-
        r) times H^{-1}(nu_2))*p$

        (4) (tricky) is derived because it can be obtained that both sides of (3) are greater than $1/2$, and $H(x)$ is a decreasing function of $x$ if $x$ is greater than $1/2$.

        (5) is obtained because $H(x)$ is a concave function of $x$.







        share|cite|improve this answer














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        share|cite|improve this answer








        edited Dec 19 '18 at 15:08

























        answered Dec 19 '18 at 7:30









        HaytanHaytan

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