Manuscript Oai119-A@1 · Oai119-A.tex · section · fdff7292

Three different scalar functions

Stage 0: objects, normalizations, and claim status › Three different scalar functions · lines 172–207

Oai119-A@1#eq:profiles, Oai119-A@1#eq:softinfo

To avoid the conflicting uses of $H$ in the source manuscripts, define

ℓ h (t) ψ (t) H (t) = ln 2, = - \frac{1 + t}{2} ln \frac{1 + t}{2} - \frac{1 - t}{2} ln \frac{1 - t}{2}, = ℓ - h (t) = \frac{1 + t}{2} ln (1 + t) + \frac{1 - t}{2} ln (1 - t), = 1 - t^{2} . - 1 \leq t \leq 1,

The convention is $0 ln 0 = 0$ . Thus $h$ is sign entropy in nats, $ψ$ is its deficit from $ℓ$ , and $H$ is the Hellinger profile. They are not interchangeable.

For a soft summary $u$ , put

I (u) = h (E u) - E h (u) = E ψ (u) - ψ (E u) .

Binary entropy in bits is

h_{2} (p) = - p lo g_{2} p - (1 - p) lo g_{2} (1 - p) .

We write $I_{2} (A; B)$ for classical mutual information in bits and $I (A; B) = ℓ I_{2} (A; B)$ for the same quantity in nats.

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Oai119-A@1#eq:profiles · line 176
$ℓ h (t) ψ (t) H (t) = ln 2, = - \frac{1 + t}{2} ln \frac{1 + t}{2} - \frac{1 - t}{2} ln \frac{1 - t}{2}, = ℓ - h (t) = \frac{1 + t}{2} ln (1 + t) + \frac{1 - t}{2} ln (1 - t), = 1 - t^{2} . - 1 \leq t \leq 1,$
Oai119-A@1#eq:softinfo · line 195
$I (u) = h (E u) - E h (u) = E ψ (u) - ψ (E u) .$
unlabeled · line 201
$h_{2} (p) = - p lo g_{2} p - (1 - p) lo g_{2} (1 - p) .$

Three different scalar functions

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