# Jensen's inequality

(Difference between revisions)
 Revision as of 12:36, 26 July 2008 (edit)← Previous diff Revision as of 12:54, 26 July 2008 (edit) (undo)Next diff → Line 3: Line 3: $\phi(ta + (1-t)b) \leq t \phi(a) + (1-t) \phi(b)$ $\phi(ta + (1-t)b) \leq t \phi(a) + (1-t) \phi(b)$ - whenever $\,0 \leq t \leq 1\,$ and $\,a, b\,$ are in the domain of $\,\phi\,$. + whenever $\,0 \leq t \leq 1\,$ and $\,a\,, b\,$ are in the domain of $\,\phi\,$. It follows by induction on It follows by induction on Line 18: Line 18:
'''Proof 1:''' By some limiting argument we can assume
'''Proof 1:''' By some limiting argument we can assume - that $\,f\,$ is simple (this limiting argument is the missing + that $\,f\,$ is simple. (This limiting argument is a missing detail to this proof...) - detail).
That is, $\,X\,$ is the disjoint union of $\,X_1 \,\ldots\, X_n\,$ +
That is, $\,X\,$ is the disjoint union of $\,X_1 \,\ldots\, X_n\,$ and $\,f\,$ is constant on each $\,X_j\,$. and $\,f\,$ is constant on each $\,X_j\,$. Line 31: Line 31: $\,(f(a) - f(b)) / (a - b) \leq (f(a') - f(b')) / (a' - b')\quad\diamond$ $\,(f(a) - f(b)) / (a - b) \leq (f(a') - f(b')) / (a' - b')\quad\diamond$ + The lemma shows: + *$\,\phi\,$ has a right-hand derivative at every point, and + *the graph of $\,\phi\,$ lies above the "tangent" line through any point on the graph with slope equal to the right derivative. - The lemma shows that $\,\phi\,$ has a right-hand + Say $\,a = \int f d \mu\,$ - derivative at every point and that the graph of $\,\phi\,$ + - lies above the "tangent" line through any point on the + - graph with slope = the right derivative. + - Say $\,a = \int f d \mu\,$, let $\,m =\,$ the right derivative of $\,\phi\,$ + Let $\,m\,$ be the right derivative of $\,\phi\,$ at $\,a\,$, and let at $\,a\,$, and let

## Revision as of 12:54, 26 July 2008

By definition $LaTeX: \,\phi\,$ is convex if and only if

$LaTeX: \phi(ta + (1-t)b) \leq t \phi(a) + (1-t) \phi(b)$

whenever $LaTeX: \,0 \leq t \leq 1\,$ and $LaTeX: \,a\,, b\,$ are in the domain of $LaTeX: \,\phi\,$.

It follows by induction on $LaTeX: \,n\,$ that if $LaTeX: \,t_j \geq 0\,$ for $LaTeX: \,j = 1, 2\ldots n\,$ then

$LaTeX: \phi(\sum t_j a_j) \leq \sum t_j \phi(a_j)$           (1)

Jensen's inequality says this:
If $LaTeX: \,\mu\,$ is a probability measure on $LaTeX: \,X\,$,
$LaTeX: \,f\,$ is a real-valued function on $LaTeX: \,X\,$,
$LaTeX: \,f\,$ is integrable, and
$LaTeX: \,\phi\,$ is convex on the range of $LaTeX: \,f\,$ then

$LaTeX: \phi(\int f d \mu) \leq \int \phi \circ f d \mu\qquad$          (2)

Proof 1: By some limiting argument we can assume that $LaTeX: \,f\,$ is simple. (This limiting argument is a missing detail to this proof...)
That is, $LaTeX: \,X\,$ is the disjoint union of $LaTeX: \,X_1 \,\ldots\, X_n\,$ and $LaTeX: \,f\,$ is constant on each $LaTeX: \,X_j\,$.

Say $LaTeX: \,t_j=\mu(X_j)\,$ and $LaTeX: \,a_j\,$ is the value of $LaTeX: \,f\,$ on $LaTeX: \,X_j\,$. Then (1) and (2) say exactly the same thing. QED.

Proof 2:

Lemma. If $LaTeX: \,a < b,\, \,a' < b',\, \,a \leq a'\,$ and $LaTeX: \,b \leq b'\,$ then

$LaTeX: \,(f(a) - f(b)) / (a - b) \leq (f(a') - f(b')) / (a' - b')\quad\diamond$

The lemma shows:

• $LaTeX: \,\phi\,$ has a right-hand derivative at every point, and
• the graph of $LaTeX: \,\phi\,$ lies above the "tangent" line through any point on the graph with slope equal to the right derivative.

Say $LaTeX: \,a = \int f d \mu\,$

Let $LaTeX: \,m\,$ be the right derivative of $LaTeX: \,\phi\,$ at $LaTeX: \,a\,$, and let

$LaTeX: \,L(t) = \phi(a) + m(t-a)\,$

The comment above says that $LaTeX: \,\phi(t) \geq L(t)\,$ for all $LaTeX: \,t\,$ in the domain of $LaTeX: \,\phi\,$. So

$LaTeX: \begin{array}{rl}\int \phi \circ f &\geq \int L \circ f\\

 &= \int (\phi(a) + m(f - a))\\ &= \phi(a) + (m \int f) - ma\\ &= \phi(a)\\ &= \phi(\int f)\end{array}$

$LaTeX: \,-\,$D. Ullrich