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The following are among the properties of log-concave distributions: If a density is log-concave, so is its cumulative distribution function (CDF). 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 ...
Comparison of linear, concave, and convex functions when plotted using a linear scale (left) or a log scale (right). In science and engineering, a log–log graph or log–log plot is a two-dimensional graph of numerical data that uses logarithmic scales on both the horizontal and vertical axes.
The rows of Pascal's triangle are examples for logarithmically concave sequences. In mathematics, a sequence a = (a 0, a 1, ..., a n) of nonnegative real numbers is called a logarithmically concave sequence, or a log-concave sequence for short, if a i 2 ≥ a i−1 a i+1 holds for 0 < i < n.
The Brunn–Minkowski inequality asserts that the Lebesgue measure is log-concave. The restriction of the Lebesgue measure to any convex set is also log-concave.. By a theorem of Borell, [2] a probability measure on R^d is log-concave if and only if it has a density with respect to the Lebesgue measure on some affine hyperplane, and this density is a logarithmically concave function.
The sum of two concave functions is itself concave and so is the pointwise minimum of two concave functions, i.e. the set of concave functions on a given domain form a semifield. Near a strict local maximum in the interior of the domain of a function, the function must be concave; as a partial converse, if the derivative of a strictly concave ...
A sigmoid function is any mathematical function whose graph has a characteristic S-shaped or sigmoid curve. A common example of a sigmoid function is the logistic function, which is defined by the formula: [1] = + = + = ().
Log-concave may refer to: Logarithmically concave function; Logarithmically concave measure; Logarithmically concave sequence This page was last edited on 22 ...
Read's conjecture is a conjecture, first made by Ronald Read, about the unimodality of the coefficients of chromatic polynomials in the context of graph theory. [1] [2] In 1974, S. G. Hoggar tightened this to the conjecture that the coefficients must be strongly log-concave. Hoggar's version of the conjecture is called the Read–Hoggar ...