Jensen's inequality
Jensen's inequality
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DOI：
10.1002/0471667196.ess1306.pub2
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发表时间：
2005-08
期刊：
影响因子：
--
通讯作者：
K. Derpanis
中科院分区：
文献类型：
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作者：
K. Derpanis
本文回顾了凸性的概念和詹森不等式。Jensen不等式在期望最大化算法[1]的推导和极大似然估计的相合性证明中起着核心作用。定义设f(x)是定义在区间I = [a， b]上的实值函数。如果对于每个x1， x2∈[a， b]且0≥λ≥1，f(λx1 +(1−λ)x2)≥λf(x1) +(1−λ)f(x2)，则称f是凸函数，如果对x1 6= x2的等式是严格的，则称f是凸函数。如果- f(x)是凸（严格凸），则定义f(x)是凹（严格凹）的。直观地看，凸性的定义是，函数位于点（x1, f(x1)）到点（x2, f(x2)）之间的直线以下，而不高于直线（见图1）。定理0.1若f”(x)在[a， b]上存在且f”(x)在[a， b]上≥0，则f(x)在[a， b]上是凸的。定理0.2 （Jensen不等式）设f(x)是定义在区间i上的凸函数，若x1， x2，…， xN∈I, λ1, λ2，…， λN≥0,∑N i=1 λi；
In this note the concept of convexity and Jensen’s Inequality are reviewed. Jensen’s Inequality plays a central role in the derivation of the Expectation Maximization algorithm [1] and the proof of consistency of maximum likelihood estimators. Definition Let f(x) be a real valued function defined on the interval I = [a, b]. f is said to be convex if for every x1, x2 ∈ [a, b] and 0 ≥ λ ≥ 1, f(λx1 + (1− λ)x2) ≥ λf(x1) + (1− λ)f(x2) A function is said to be strictly convex if the equality is strict for x1 6= x2. Definition f(x) is said to be concave (strictly concave) if −f(x) is convex (strictly convex). Intuitively, the definition of convexity states that function falls below never above the straight line between the points (x1, f(x1)) to (x2, f(x2)) (see Fig. 1). Theorem 0.1 If f ′′(x) exists on [a, b] and f ′′(x) ≥ 0 on [a, b] then f(x) is convex on [a, b]. Theorem 0.2 (Jensen’s Inequality) Let f(x) be a convex function defined on an interval I. If x1, x2, . . . , xN ∈ I and λ1, λ2, . . . , λN ≥ 0 with ∑N i=1 λi,