Logarithmically concave function

In convex analysis, a non-negative function f: R → R+ is logarithmically concave (or log-concave for short) if its domain is a convex set, and if it satisfies the inequality

0 < θ < 1. If log⁡f(θx+(1−θ)y)≥θlog⁡f(x)+(1−θ)log⁡f(y)

for all f(θx+(1−θ)y)≤f(x)θf(y)1−θ

for all f: Z → R+, it is log-concave if

f(x) = 2</sup>/2)"}},"i":0}}]}'/>exp(−x/2) which is log-concave since x. But f″(x)=e−x22(x2−1)≰0

• From above two points, concavity x satisfying &nbsp;0"}},"i":0}}]}'/>f(x) > 0,

f(x)f″(x)≤(f′(x))2

• Products: The product of log-concave functions is also log-concave. Indeed, if log f and f g is log-concave.
  • Marginals: if g(x)=∫f(x,y)dy

    is log-concave (see Prékopa–Leindler inequality).

    • This implies that convolution preserves log-concavity, since f and ddxlog⁡(1−F(x))=−f(x)1−F(x) which is decreasing as it is the derivative of a concave function.

      See also

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      • logarithmically concave sequence
      • logarithmically concave measure
      • logarithmically convex function
      • convex function

      Notes

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      Grechuk, Bogdan; Molyboha, Anton; Zabarankin, Michael (May 2009). "Maximum Entropy Principle with General Deviation Measures". Mathematics of Operations Research. 34 (2): 445–467. doi:10.1287/moor.1090.0377.
    • 1 2 See Barndorff-Nielsen, Ole (1978). Information and exponential families in statistical theory. Wiley Series in Probability and Mathematical Statistics. Chichester: John Wiley \& Sons, Ltd. pp. ix+238 pp. ISBN 0-471-99545-2. MR 0489333.
    • Pečarić, Josip E.; Proschan, Frank; Tong, Y. L. (1992). Convex functions, partial orderings, and statistical applications. Mathematics in Science and Engineering. Vol. 187. Boston, MA: Academic Press, Inc. pp. xiv+467 pp. ISBN 0-12-549250-2. MR 1162312.