Variation of Jensen's inequality Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Jensen's inequalityTotal variation distance for probability measuresI need help understanding proofs of Gibbs' Inequality using Jensen's Inequality.Is a convex combination of conditional probabilities the conditional probability of a convex combination of unconditional probabilities?probability measure on $sigma$-algebraUsing Jensen's inequality to prove another inequality?Question about convex set of probabilitiesJensen's inequality applied to Liapunov's CLT conditionFrom additivity of the measure, couldn't we prove the $sigma -$additivity?Generalize Jensen's Integral Inequality to the product of two functions
When to stop saving and start investing?
Right-skewed distribution with mean equals to mode?
Is there a Spanish version of "dot your i's and cross your t's" that includes the letter 'ñ'?
ListPlot join points by nearest neighbor rather than order
When -s is used with third person singular. What's its use in this context?
How does a Death Domain cleric's Touch of Death feature work with Touch-range spells delivered by familiars?
If a contract sometimes uses the wrong name, is it still valid?
How discoverable are IPv6 addresses and AAAA names by potential attackers?
Disable hyphenation for an entire paragraph
How to motivate offshore teams and trust them to deliver?
What is a Meta algorithm?
Can inflation occur in a positive-sum game currency system such as the Stack Exchange reputation system?
What LEGO pieces have "real-world" functionality?
Is there a documented rationale why the House Ways and Means chairman can demand tax info?
What do you call a plan that's an alternative plan in case your initial plan fails?
Doubts about chords
Is 1 ppb equal to 1 μg/kg?
How does cp -a work
Is a manifold-with-boundary with given interior and non-empty boundary essentially unique?
What is the correct way to use the pinch test for dehydration?
The logistics of corpse disposal
What is the longest distance a 13th-level monk can jump while attacking on the same turn?
How much radiation do nuclear physics experiments expose researchers to nowadays?
Is the Standard Deduction better than Itemized when both are the same amount?
Variation of Jensen's inequality
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Jensen's inequalityTotal variation distance for probability measuresI need help understanding proofs of Gibbs' Inequality using Jensen's Inequality.Is a convex combination of conditional probabilities the conditional probability of a convex combination of unconditional probabilities?probability measure on $sigma$-algebraUsing Jensen's inequality to prove another inequality?Question about convex set of probabilitiesJensen's inequality applied to Liapunov's CLT conditionFrom additivity of the measure, couldn't we prove the $sigma -$additivity?Generalize Jensen's Integral Inequality to the product of two functions
$begingroup$
I have a more general question: can we always say that, given a convex function $g$ and some $h$ s.t. $sum_xinmathcalXh(x)=1$, $AsubsetmathcalX$ $$sum_xin A g(x)h(x)geq gleft(sum_xin A xh(x)right)$$
even if we do not sum on all the $x$? Why does it hold? I have seen it done a few times (practical question), like here:
$$sum_xin A left(fracP(x)Q(x)right)^ccdotfrac1P(A) P(x) geq sum_xin A left(fracP(x)Q(x)cdotfrac1P(A) P(x)right)^c$$
where $cgeq 0$ and $P,Q$ are probability measures on $mathcalX$. Is it correct? Can you give me some hints on the proof? Sorry if it is silly but I just can't see it!
probability measure-theory jensen-inequality
asked Mar 26 at 8:54
user1868607user1868607
3211210
$endgroup$
add a comment |
$begingroup$
I have a more general question: can we always say that, given a convex function $g$ and some $h$ s.t. $sum_xinmathcalXh(x)=1$, $AsubsetmathcalX$ $$sum_xin A g(x)h(x)geq gleft(sum_xin A xh(x)right)$$
even if we do not sum on all the $x$? Why does it hold? I have seen it done a few times (practical question), like here:
$$sum_xin A left(fracP(x)Q(x)right)^ccdotfrac1P(A) P(x) geq sum_xin A left(fracP(x)Q(x)cdotfrac1P(A) P(x)right)^c$$
where $cgeq 0$ and $P,Q$ are probability measures on $mathcalX$. Is it correct? Can you give me some hints on the proof? Sorry if it is silly but I just can't see it!
probability measure-theory jensen-inequality
asked Mar 26 at 8:54
user1868607user1868607
3211210
$endgroup$
$begingroup$
True if $g(0)=0$ but not otherwise.
$endgroup$
– Kavi Rama Murthy
Mar 26 at 9:03
$begingroup$
@KaviRamaMurthy thanks! Why not otherwise? How does it work with $log$?
$endgroup$
– user1868607
Mar 26 at 9:15
$begingroup$
@user1868607 If $h(x)geq0$, it's just Jensen, otherwise, It's wrong, I think.
$endgroup$
– Michael Rozenberg
Mar 26 at 10:02
add a comment |
$begingroup$
I have a more general question: can we always say that, given a convex function $g$ and some $h$ s.t. $sum_xinmathcalXh(x)=1$, $AsubsetmathcalX$ $$sum_xin A g(x)h(x)geq gleft(sum_xin A xh(x)right)$$
even if we do not sum on all the $x$? Why does it hold? I have seen it done a few times (practical question), like here:
$$sum_xin A left(fracP(x)Q(x)right)^ccdotfrac1P(A) P(x) geq sum_xin A left(fracP(x)Q(x)cdotfrac1P(A) P(x)right)^c$$
where $cgeq 0$ and $P,Q$ are probability measures on $mathcalX$. Is it correct? Can you give me some hints on the proof? Sorry if it is silly but I just can't see it!
probability measure-theory jensen-inequality
asked Mar 26 at 8:54
user1868607user1868607
3211210
$endgroup$
I have a more general question: can we always say that, given a convex function $g$ and some $h$ s.t. $sum_xinmathcalXh(x)=1$, $AsubsetmathcalX$ $$sum_xin A g(x)h(x)geq gleft(sum_xin A xh(x)right)$$
even if we do not sum on all the $x$? Why does it hold? I have seen it done a few times (practical question), like here:
$$sum_xin A left(fracP(x)Q(x)right)^ccdotfrac1P(A) P(x) geq sum_xin A left(fracP(x)Q(x)cdotfrac1P(A) P(x)right)^c$$
where $cgeq 0$ and $P,Q$ are probability measures on $mathcalX$. Is it correct? Can you give me some hints on the proof? Sorry if it is silly but I just can't see it!
probability measure-theory jensen-inequality
probability measure-theory jensen-inequality
asked Mar 26 at 8:54
user1868607user1868607
3211210
asked Mar 26 at 8:54
user1868607user1868607
3211210
asked Mar 26 at 8:54
user1868607user1868607
3211210
asked Mar 26 at 8:54
user1868607user1868607
3211210
asked Mar 26 at 8:54
user1868607user1868607
3211210
3211210
$begingroup$
True if $g(0)=0$ but not otherwise.
$endgroup$
– Kavi Rama Murthy
Mar 26 at 9:03
$begingroup$
@KaviRamaMurthy thanks! Why not otherwise? How does it work with $log$?
$endgroup$
– user1868607
Mar 26 at 9:15
$begingroup$
@user1868607 If $h(x)geq0$, it's just Jensen, otherwise, It's wrong, I think.
$endgroup$
– Michael Rozenberg
Mar 26 at 10:02
add a comment |
$begingroup$
True if $g(0)=0$ but not otherwise.
$endgroup$
– Kavi Rama Murthy
Mar 26 at 9:03
$begingroup$
@KaviRamaMurthy thanks! Why not otherwise? How does it work with $log$?
$endgroup$
– user1868607
Mar 26 at 9:15
$begingroup$
@user1868607 If $h(x)geq0$, it's just Jensen, otherwise, It's wrong, I think.
$endgroup$
– Michael Rozenberg
Mar 26 at 10:02
$begingroup$
True if $g(0)=0$ but not otherwise.
$endgroup$
– Kavi Rama Murthy
Mar 26 at 9:03
$begingroup$
True if $g(0)=0$ but not otherwise.
$endgroup$
– Kavi Rama Murthy
Mar 26 at 9:03
$begingroup$
@KaviRamaMurthy thanks! Why not otherwise? How does it work with $log$?
$endgroup$
– user1868607
Mar 26 at 9:15
$begingroup$
@KaviRamaMurthy thanks! Why not otherwise? How does it work with $log$?
$endgroup$
– user1868607
Mar 26 at 9:15
$begingroup$
@user1868607 If $h(x)geq0$, it's just Jensen, otherwise, It's wrong, I think.
$endgroup$
– Michael Rozenberg
Mar 26 at 10:02
$begingroup$
@user1868607 If $h(x)geq0$, it's just Jensen, otherwise, It's wrong, I think.
$endgroup$
– Michael Rozenberg
Mar 26 at 10:02
add a comment |
0
active
oldest
votes
Your Answer
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
);
);
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3162907%2fvariation-of-jensens-inequality%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
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.
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3162907%2fvariation-of-jensens-inequality%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
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
$begingroup$
True if $g(0)=0$ but not otherwise.
$endgroup$
– Kavi Rama Murthy
Mar 26 at 9:03
$begingroup$
@KaviRamaMurthy thanks! Why not otherwise? How does it work with $log$?
$endgroup$
– user1868607
Mar 26 at 9:15
$begingroup$
@user1868607 If $h(x)geq0$, it's just Jensen, otherwise, It's wrong, I think.
$endgroup$
– Michael Rozenberg
Mar 26 at 10:02