Logarithmically concave function

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

for all x,y ∈ dom f and 0 < θ < 1. If f is strictly positive, this is equivalent to saying that the logarithm of the function, log ∘ f, is concave; that is,

for all x,y ∈ dom f and 0 < θ < 1.

Examples of log-concave functions are the 0-1 indicator functions of convex sets (which requires the more flexible definition), and the Gaussian function.

Similarly, a function is log-convex if it satisfies the reverse inequality

for all x,y ∈ dom f and 0 < θ < 1.

Properties[edit]

  • A log-concave function is also quasi-concave. This follows from the fact that the logarithm is monotone implying that the superlevel sets of this function are convex.[1]
  • Every concave function that is nonnegative on its domain is log-concave. However, the reverse does not necessarily hold. An example is the Gaussian function f(x) = exp(−x2/2) which is log-concave since log f(x) = x2/2 is a concave function of x. But f is not concave since the second derivative is positive for |x| > 1:
  • From above two points, concavity log-concavity quasiconcavity.
  • A twice differentiable, nonnegative function with a convex domain is log-concave if and only if for all x satisfying f(x) > 0,
,[1]
i.e.
is
negative semi-definite. For functions of one variable, this condition simplifies to

Operations preserving log-concavity[edit]

  • Products: The product of log-concave functions is also log-concave. Indeed, if f and g are log-concave functions, then log f and log g are concave by definition. Therefore
is concave, and hence also f g is log-concave.
  • Marginals: if f(x,y) : Rn+m → R is log-concave, then
is log-concave (see Prékopa–Leindler inequality).
  • This implies that convolution preserves log-concavity, since h(x,y) = f(x-yg(y) is log-concave if f and g are log-concave, and therefore
is log-concave.

Log-concave distributions[edit]

Log-concave distributions are necessary for a number of algorithms, e.g. adaptive rejection sampling. Every distribution with log-concave density is a maximum entropy probability distribution with specified mean μ and Deviation risk measure D.[2] As it happens, many common probability distributions are log-concave. Some examples:[3]

Note that all of the parameter restrictions have the same basic source: The exponent of non-negative quantity must be non-negative in order for the function to be log-concave.

The following distributions are non-log-concave for all parameters:

Note that the cumulative distribution function (CDF) of all log-concave distributions is also log-concave. However, some non-log-concave distributions also have log-concave CDF's:

The following are among the properties of log-concave distributions:

See also[edit]

Notes[edit]

  1. ^ a b Boyd, Stephen; Vandenberghe, Lieven (2004). "Log-concave and log-convex functions". Convex Optimization. Cambridge University Press. pp. 104–108. ISBN 0-521-83378-7.
  2. ^ Grechuk, B.; Molyboha, A.; Zabarankin, M. (2009). "Maximum Entropy Principle with General Deviation Measures". Mathematics of Operations Research. 34 (2): 445–467. doi:10.1287/moor.1090.0377.
  3. ^ See Bagnoli, Mark; Bergstrom, Ted (2005). "Log-Concave Probability and Its Applications". Economic Theory. 26 (2): 445–469. doi:10.1007/s00199-004-0514-4.
  4. ^ a b Prékopa, András (1971). "Logarithmic concave measures with application to stochastic programming". Acta Scientiarum Mathematicarum. 32: 301–316.

References[edit]

  • 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.
  • Dharmadhikari, Sudhakar; Joag-Dev, Kumar (1988). Unimodality, convexity, and applications. Probability and Mathematical Statistics. Boston, MA: Academic Press, Inc. pp. xiv+278. ISBN 0-12-214690-5. MR 0954608.
  • Pfanzagl, Johann; with the assistance of R. Hamböker (1994). Parametric Statistical Theory. Walter de Gruyter. ISBN 3-11-013863-8. MR 1291393.
  • Pečarić, Josip E.; Proschan, Frank; Tong, Y. L. (1992). Convex functions, partial orderings, and statistical applications. Mathematics in Science and Engineering. 187. Boston, MA: Academic Press, Inc. pp. xiv+467 pp. ISBN 0-12-549250-2. MR 1162312.