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












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












$begingroup$


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






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Jul 9 '18 at 3:16









DuWeiminDuWeimin

62




62












  • $begingroup$
    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










1 Answer
1






active

oldest

votes


















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











$endgroup$













    Your Answer





    StackExchange.ifUsing("editor", function () {
    return StackExchange.using("mathjaxEditing", function () {
    StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
    StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
    });
    });
    }, "mathjax-editing");

    StackExchange.ready(function() {
    var channelOptions = {
    tags: "".split(" "),
    id: "69"
    };
    initTagRenderer("".split(" "), "".split(" "), channelOptions);

    StackExchange.using("externalEditor", function() {
    // Have to fire editor after snippets, if snippets enabled
    if (StackExchange.settings.snippets.snippetsEnabled) {
    StackExchange.using("snippets", function() {
    createEditor();
    });
    }
    else {
    createEditor();
    }
    });

    function createEditor() {
    StackExchange.prepareEditor({
    heartbeatType: 'answer',
    autoActivateHeartbeat: false,
    convertImagesToLinks: true,
    noModals: true,
    showLowRepImageUploadWarning: true,
    reputationToPostImages: 10,
    bindNavPrevention: true,
    postfix: "",
    imageUploader: {
    brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
    contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
    allowUrls: true
    },
    noCode: true, onDemand: true,
    discardSelector: ".discard-answer"
    ,immediatelyShowMarkdownHelp:true
    });


    }
    });














    draft saved

    draft discarded


















    StackExchange.ready(
    function () {
    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2845173%2fhow-to-prove-hh-1vp-is-convex-in-v-for-every-p-in-0-1%23new-answer', 'question_page');
    }
    );

    Post as a guest















    Required, but never shown

























    1 Answer
    1






    active

    oldest

    votes








    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    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












      $begingroup$

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



      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












        0








        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














        share|cite|improve this answer



        share|cite|improve this answer








        edited Dec 19 '18 at 15:08

























        answered Dec 19 '18 at 7:30









        HaytanHaytan

        12




        12






























            draft saved

            draft discarded




















































            Thanks for contributing an answer to Mathematics Stack Exchange!


            • Please be sure to answer the question. Provide details and share your research!

            But avoid …



            • Asking for help, clarification, or responding to other answers.

            • Making statements based on opinion; back them up with references or personal experience.


            Use MathJax to format equations. MathJax reference.


            To learn more, see our tips on writing great answers.




            draft saved


            draft discarded














            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2845173%2fhow-to-prove-hh-1vp-is-convex-in-v-for-every-p-in-0-1%23new-answer', 'question_page');
            }
            );

            Post as a guest















            Required, but never shown





















































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown

































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown







            -

            Popular posts from this blog

            Bressuire

            Image
            .mw-parser-output .infobox{border:1px solid #aaa;background-color:#f9f9f9;color:black;margin:.5em 0 .5em 1em;padding:.2em;float:right;clear:right;width:22em;text-align:left;font-size:88%;line-height:1.6em}.mw-parser-output .infobox td,.mw-parser-output .infobox th{vertical-align:top;padding:0 .2em}.mw-parser-output .infobox caption{font-size:larger}.mw-parser-output .infobox.bordered{border-collapse:collapse}.mw-parser-output .infobox.bordered td,.mw-parser-output .infobox.bordered th{border:1px solid #aaa}.mw-parser-output .infobox.bordered .borderless td,.mw-parser-output .infobox.bordered .borderless th{border:0}.mw-parser-output .infobox-showbutton .mw-collapsible-text{color:inherit}.mw-parser-output .infobox.bordered .mergedtoprow td,.mw-parser-output .infobox.bordered .mergedtoprow th{border:0;border-top:1px solid #aaa;border-right:1px solid #aaa}.mw-parser-output .infobox.bordered .mergedrow td,.mw-parser-output .infobox.bordered .mergedrow th{border:0;border-right:1px solid ...
            Read more

            Cabo Verde

            Image
              Nota: Cabo-verdiano redireciona para este artigo. Para a língua falada em Cabo Verde, veja crioulo cabo-verdiano. Para outros significados, veja Cabo Verde (desambiguação). Coordenadas: 14° 55' N 23° 31' O Cabo Verde República de Cabo Verde Bandeira Brasão de Armas Lema: Unidade, Trabalho, Progresso Hino nacional: Cântico da Liberdade Gentílico: cabo-verdiano [ 1 ] Capital Praia 14° 55' N 23° 31' O Cidade mais populosa Praia Língua oficial Português Governo República unitária semipresidencialista [ 2 ]  - Presidente Jorge Carlos Fonseca  - Primeiro-ministro Ulisses Correia e Silva História    - Descoberta 1460   - Independência 00 • de Portugal 5 de Julho de 1975   - Multipartidarismo 13 de Janeiro de 1990  Área    - Total 4 033 km² (146.º)  - Água (%) 0,08 População    - Estimativa para 2017 560 899 [ 3 ] hab. (173.º) ...
            Read more

            Gyllenstierna

            Image
            .mw-parser-output .infobox{border:1px solid #aaa;background-color:#f9f9f9;color:black;margin:.5em 0 .5em 1em;padding:.2em;float:right;clear:right;width:22em;text-align:left;font-size:88%;line-height:1.6em}.mw-parser-output .infobox td,.mw-parser-output .infobox th{vertical-align:top;padding:0 .2em}.mw-parser-output .infobox caption{font-size:larger}.mw-parser-output .infobox.bordered{border-collapse:collapse}.mw-parser-output .infobox.bordered td,.mw-parser-output .infobox.bordered th{border:1px solid #aaa}.mw-parser-output .infobox.bordered .borderless td,.mw-parser-output .infobox.bordered .borderless th{border:0}.mw-parser-output .infobox-showbutton .mw-collapsible-text{color:inherit}.mw-parser-output .infobox.bordered .mergedtoprow td,.mw-parser-output .infobox.bordered .mergedtoprow th{border:0;border-top:1px solid #aaa;border-right:1px solid #aaa}.mw-parser-output .infobox.bordered .mergedrow td,.mw-parser-output .infobox.bordered .mergedrow th{border:0;border-right:1px solid ...
            Read more