Jensen's inequality
From Wikimization
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and <math>\,f\,</math> is constant on each <math>\,X_j\,</math> . | and <math>\,f\,</math> is constant on each <math>\,X_j\,</math> . | ||
- | Say <math>\,t_j=\mu(X_j)\,</math> and <math>\,a_j\,</math> is the value of <math>\,f\,</math> on <math>\,X_j\,</math>. | + | Say <math>\,t_j=\mu(X_j)\,</math> and <math>\,a_j\,</math> is the value of <math>\,f\,</math> on <math>\,X_j\,</math> . |
+ | |||
Then (1) and (2) say exactly the same thing. QED. | Then (1) and (2) say exactly the same thing. QED. | ||
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Let <math>\,m\,</math> be the right derivative of <math>\,\phi\,</math> | Let <math>\,m\,</math> be the right derivative of <math>\,\phi\,</math> | ||
- | at <math>\,a\,</math>, and let | + | at <math>\,a\,</math> , and let |
<math>\,L(t) = \phi(a) + m(t-a)\,</math> | <math>\,L(t) = \phi(a) + m(t-a)\,</math> | ||
The bullets above say <math>\,\phi(t)\geq L(t)\,</math> for | The bullets above say <math>\,\phi(t)\geq L(t)\,</math> for | ||
- | all <math>\,t\,</math> in the domain of <math>\,\phi\,</math> . So | + | all <math>\,t\,</math> in the domain of <math>\,\phi\,</math> . So |
<math>\begin{array}{rl}\int \phi \circ f &\geq \int L \circ f\\ | <math>\begin{array}{rl}\int \phi \circ f &\geq \int L \circ f\\ |
Revision as of 13:02, 26 July 2008
By definition is convex if and only if
whenever and are in the domain of .
It follows by induction on that if for then
(1)
Jensen's inequality says this:
If is a probability
measure on ,
is a real-valued function on ,
is integrable, and
is convex on the range
of then
(2)
Proof 1: By some limiting argument we can assume
that is simple. (This limiting argument is a missing detail to this proof...)
That is, is the disjoint union of
and is constant on each .
Say and is the value of on .
Then (1) and (2) say exactly the same thing. QED.
Proof 2:
Lemma. If and then
The lemma shows:
- has a right-hand derivative at every point, and
- the graph of lies above the "tangent" line through any point on the graph with slope equal to the right derivative.
Say
Let be the right derivative of at , and let
The bullets above say for all in the domain of . So
D. Ullrich