In
convex analysis, a
non-negative function f : Rn → R+ 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
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 functionf(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:
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 fg is log-concave.
Marginals: if f(x,y) : Rn+m → R is log-concave, then
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:
If a multivariate density is log-concave, so is the
marginal density over any subset of variables.
The sum of two independent log-concave
random variables is log-concave. This follows from the fact that the convolution of two log-concave functions is log-concave.
The product of two log-concave functions is log-concave. This means that
joint densities formed by multiplying two probability densities (e.g. the
normal-gamma distribution, which always has a shape parameter ≥ 1) will be log-concave. This property is heavily used in general-purpose
Gibbs sampling programs such as
BUGS and
JAGS, which are thereby able to use
adaptive rejection sampling over a wide variety of
conditional distributions derived from the product of other distributions.
If a density is log-concave, it has a monotone
hazard rate (MHR), and is a
regular distribution since the derivative of the logarithm of the survival function is the negative hazard rate, and by concavity is monotone i.e.
which is decreasing as it is the derivative of a concave function.
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.
ISBN0-471-99545-2.
MR0489333.
Dharmadhikari, Sudhakar; Joag-Dev, Kumar (1988). Unimodality, convexity, and applications. Probability and Mathematical Statistics. Boston, MA: Academic Press, Inc. pp. xiv+278.
ISBN0-12-214690-5.
MR0954608.
Pfanzagl, Johann; with the assistance of R. Hamböker (1994). Parametric Statistical Theory. Walter de Gruyter.
ISBN3-11-013863-8.
MR1291393.
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.
ISBN0-12-549250-2.
MR1162312.
In
convex analysis, a
non-negative function f : Rn → R+ 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
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 functionf(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:
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 fg is log-concave.
Marginals: if f(x,y) : Rn+m → R is log-concave, then
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:
If a multivariate density is log-concave, so is the
marginal density over any subset of variables.
The sum of two independent log-concave
random variables is log-concave. This follows from the fact that the convolution of two log-concave functions is log-concave.
The product of two log-concave functions is log-concave. This means that
joint densities formed by multiplying two probability densities (e.g. the
normal-gamma distribution, which always has a shape parameter ≥ 1) will be log-concave. This property is heavily used in general-purpose
Gibbs sampling programs such as
BUGS and
JAGS, which are thereby able to use
adaptive rejection sampling over a wide variety of
conditional distributions derived from the product of other distributions.
If a density is log-concave, it has a monotone
hazard rate (MHR), and is a
regular distribution since the derivative of the logarithm of the survival function is the negative hazard rate, and by concavity is monotone i.e.
which is decreasing as it is the derivative of a concave function.
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.
ISBN0-471-99545-2.
MR0489333.
Dharmadhikari, Sudhakar; Joag-Dev, Kumar (1988). Unimodality, convexity, and applications. Probability and Mathematical Statistics. Boston, MA: Academic Press, Inc. pp. xiv+278.
ISBN0-12-214690-5.
MR0954608.
Pfanzagl, Johann; with the assistance of R. Hamböker (1994). Parametric Statistical Theory. Walter de Gruyter.
ISBN3-11-013863-8.
MR1291393.
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.
ISBN0-12-549250-2.
MR1162312.