Mixing
Proof that the binary entropy is concave
Defining the binary entropy function as $H_{bin}(x) = - xlog(x) - (1-x)log(1-x)$, how do I show that it is concave? I can see the intuition but not the proof.
Namely, I need to prove that $H_{bin}(px_1 + (1-p)x_2) geq pH_{bin}(x_1) + (1-p)H_{bin}(x_2)$.
Following the comment posted, I take the second derivative of $H_{bin}$ with respect to $p$ and obtain $frac{partial^2 H}{partial p^2} = -(x_1 - x_2)^2 frac{1-2a}{a(1-a)}$, where $a = px_1 + (1-p)x_2$. How do I see that $frac{1-2a}{a(1-a)}$ is positive?
entropy
asked Nov 20 '18 at 16:58
user1936752
5181513
add a comment |
Defining the binary entropy function as $H_{bin}(x) = - xlog(x) - (1-x)log(1-x)$, how do I show that it is concave? I can see the intuition but not the proof.
Namely, I need to prove that $H_{bin}(px_1 + (1-p)x_2) geq pH_{bin}(x_1) + (1-p)H_{bin}(x_2)$.
Following the comment posted, I take the second derivative of $H_{bin}$ with respect to $p$ and obtain $frac{partial^2 H}{partial p^2} = -(x_1 - x_2)^2 frac{1-2a}{a(1-a)}$, where $a = px_1 + (1-p)x_2$. How do I see that $frac{1-2a}{a(1-a)}$ is positive?
entropy
asked Nov 20 '18 at 16:58
user1936752
5181513
This is best done using Calculus. Show that the second derivative is negative on this interval.
– Hans Engler
Nov 20 '18 at 17:06
@HansEngler, thanks. I think I'm close but I don't quite see the solution yet.
– user1936752
Nov 20 '18 at 18:31
add a comment |
Defining the binary entropy function as $H_{bin}(x) = - xlog(x) - (1-x)log(1-x)$, how do I show that it is concave? I can see the intuition but not the proof.
Namely, I need to prove that $H_{bin}(px_1 + (1-p)x_2) geq pH_{bin}(x_1) + (1-p)H_{bin}(x_2)$.
Following the comment posted, I take the second derivative of $H_{bin}$ with respect to $p$ and obtain $frac{partial^2 H}{partial p^2} = -(x_1 - x_2)^2 frac{1-2a}{a(1-a)}$, where $a = px_1 + (1-p)x_2$. How do I see that $frac{1-2a}{a(1-a)}$ is positive?
entropy
asked Nov 20 '18 at 16:58
user1936752
5181513
Defining the binary entropy function as $H_{bin}(x) = - xlog(x) - (1-x)log(1-x)$, how do I show that it is concave? I can see the intuition but not the proof.
Namely, I need to prove that $H_{bin}(px_1 + (1-p)x_2) geq pH_{bin}(x_1) + (1-p)H_{bin}(x_2)$.
Following the comment posted, I take the second derivative of $H_{bin}$ with respect to $p$ and obtain $frac{partial^2 H}{partial p^2} = -(x_1 - x_2)^2 frac{1-2a}{a(1-a)}$, where $a = px_1 + (1-p)x_2$. How do I see that $frac{1-2a}{a(1-a)}$ is positive?
entropy
entropy
asked Nov 20 '18 at 16:58
user1936752
5181513
asked Nov 20 '18 at 16:58
user1936752
5181513
edited Nov 20 '18 at 18:18
asked Nov 20 '18 at 16:58
user1936752
5181513
asked Nov 20 '18 at 16:58
user1936752
5181513
asked Nov 20 '18 at 16:58
user1936752
5181513
5181513
This is best done using Calculus. Show that the second derivative is negative on this interval.
– Hans Engler
Nov 20 '18 at 17:06
@HansEngler, thanks. I think I'm close but I don't quite see the solution yet.
– user1936752
Nov 20 '18 at 18:31
add a comment |
This is best done using Calculus. Show that the second derivative is negative on this interval.
– Hans Engler
Nov 20 '18 at 17:06
@HansEngler, thanks. I think I'm close but I don't quite see the solution yet.
– user1936752
Nov 20 '18 at 18:31
This is best done using Calculus. Show that the second derivative is negative on this interval.
– Hans Engler
Nov 20 '18 at 17:06
This is best done using Calculus. Show that the second derivative is negative on this interval.
– Hans Engler
Nov 20 '18 at 17:06
@HansEngler, thanks. I think I'm close but I don't quite see the solution yet.
– user1936752
Nov 20 '18 at 18:31
@HansEngler, thanks. I think I'm close but I don't quite see the solution yet.
– user1936752
Nov 20 '18 at 18:31
add a comment |
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This is best done using Calculus. Show that the second derivative is negative on this interval.
– Hans Engler
Nov 20 '18 at 17:06
@HansEngler, thanks. I think I'm close but I don't quite see the solution yet.
– user1936752
Nov 20 '18 at 18:31