How do you show a function is log-concave?

How do you show a function is log-concave?

A function f(x) is concave if –f(x) is convex. Equivalently, the line connecting two points on the graph lies below the graph. In symbols, reverse the direction of the inequality for convexity. A function f(x) is log concave if log( f(x) ) is concave.

What is a log-concave distribution?

If a density is log-concave, so is its cumulative distribution function (CDF). 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.

Is the normal distribution log-concave?

Therefore the normal distribution has log-concave density. The exponential distribution has support, [0, ∞], and density function, f(x) = λe−λx. The c.d.f. is F(x) = 1 − e−λx. The density function is log-linear and hence log-concave, with (ln f(x)) = 0, so f(x) i.

Is log convex or concave?

Logarithm is Strictly Concave – ProofWiki.

What is the base of a logarithmic function?

The logarithm base 10 (that is b = 10) is called the decimal or common logarithm and is commonly used in science and engineering. The natural logarithm has the number e (that is b ≈ 2.718) as its base; its use is widespread in mathematics and physics, because of its simpler integral and derivative.

Is sigmoid log concave?

Properties. In general, a sigmoid function is monotonic, and has a first derivative which is bell shaped. A sigmoid function is convex for values less than a particular point, and it is concave for values greater than that point: in many of the examples here, that point is 0.

Is log of a convex function convex?

, the composition of the logarithm with f, is itself a convex function.

What is concavity in math?

What is concavity? Concavity relates to the rate of change of a function’s derivative. A function f is concave up (or upwards) where the derivative f′ is increasing. Graphically, a graph that’s concave up has a cup shape, ∪, and a graph that’s concave down has a cap shape, ∩.

Is log determinant convex?

In the following example, we apply the fact that “a function is convex if and only if its restriction to any line is convex” to prove that log determinant function is a concave function (i.e., negative log determinant function is convex). Thus, −g(t) is convex, so is −f(X).

How do you do log functions?

In this definition y=logbx y = log b x is called the logarithm form and by=x b y = x is called the exponential form. Note that the requirement that x>0 is really a result of the fact that we are also requiring b>0 ….Section 6-2 : Logarithm Functions.

Why is the second derivative important in concavity calculus?

Concavity calculus highlights the importance of the function’s second derivative in confirming whether its resulting curve concaves upward, downward, or is an inflection point at its critical points. Our discussion will focus on the following concepts and techniques: Identifying concavity and points of inflections given a function’s graph.

How to calculate the concavity of a function?

Equate the second derivative to zero. Use the solutions to divide the function’s domain into smaller intervals then find a test value to determine the function’s concavity at the intervals. Let’s say we have $f (x) = x^4 – 4x^2$. To predict its curve’s concavity, we differentiate $f (x)$ twice in a row as shown below.

Is the demand function lower case or upper case?

This function is typically called either the demand function or the price function. Be careful to not confuse the demand function, p(x) p ( x) – lower case p p, and the profit function, P (x) P ( x) – upper case P P. Bad notation maybe, but there it is.

Are there any real business applications for calculus?

There are some very real applications to calculus that are in the business world and at some level that is the point of this section. Note that to really learn these applications and all of their intricacies you’ll need to take a business course or two or three.