# 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 12:02, 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