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How do you show a function is convex?
A function f : Rn → R is convex if and only if the function g : R → R given by g(t) = f(x + ty) is convex (as a univariate function) for all x in domain of f and all y ∈ Rn. (The domain of g here is all t for which x + ty is in the domain of f.) Proof: This is straightforward from the definition.
Is a line convex?
A set is convex if it includes all convex combinations of points in the set. Or in other words, if it contains all line segment joining any two points in the set. Thus, a line is a convex set.
How do you use Jensen’s inequality?
To use Jensen’s inequality, we need to determine if a function g is convex. A useful method is the second derivative. A twice-differentiable function g:I→R is convex if and only if g″(x)≥0 for all x∈I.
What is a log-concave distribution?
If a multivariate density is log-concave, so is the marginal density over any subset of variables. 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.
How to use Jensen’s inequality in the real world?
To use Jensen’s inequality, we need to determine if a function g is convex. A useful method is the second derivative. A twice-differentiable function g: I → R is convex if and only if g ″ ( x) ≥ 0 for all x ∈ I . For example, if g ( x) = x 2, then g ″ ( x) = 2 ≥ 0, thus g ( x) = x 2 is convex over R.
What does Jensen’s inequality say about a convex function?
Jensen’s inequality states that, for any convex function g, we have E[g(X)] ≥ g(E[X]). So what is a convex function? Figure 6.2 depicts a convex function. A function is convex if, when you pick any two points on the graph of the function and draw a line segment between the two points, the entire segment lies above the graph.
Can you prove Jensen’s inequality with limiting arguments?
So we can prove the Jensen’s inequality in this case. Using limiting arguments, this result can be extended to other types of random variables. E [ g ( X)] ≥ g ( E [ X]). To use Jensen’s inequality, we need to determine if a function g is convex.
Where does the secant line lie in Jensen’s inequality?
Jensen’s inequality generalizes the statement that the secant line of a convex function lies above the graph of the function, which is Jensen’s inequality for two points: the secant line consists of weighted means of the convex function (for t ∈ [0,1]), while the graph of the function is the convex function of the weighted means,