The 3-Fiber Kullback-Liebler Poisson Experiment

Weaving Poisson arrivals λxyz\lambda_{xyz} with DKL(PQ)D_{KL}(P\|Q) texture maps.

7/25/2024

In the realm where Poisson's whispers λ softly sing,

A stochastic dance, each count a discrete spring.

From chaos’ cradle, where rare events arise,

Exponential shadows in P\mathbb{P}’s eyes.

Kullback-Leibler, the divergence keen,

Measures entropy between what’s seen.

Q\mathbb{Q}, a mirror of distorted gleam,

Reflects the divergence, a relative dream.

Consider two realms, with probabilities fine,

Where DKL(PQ)\mathcal{D}_{\text{KL}}(P \parallel Q) defines the line.

An integral feast with lnP(x)Q(x)\ln \frac{P(x)}{Q(x)} in sight,

Entropy unfolds in the measure of light.

λxeλ/x!\lambda^{x} e^{-\lambda} / x!, the Poisson’s crown,

Counts converge as dusk wears down.

Kullback’s divergence, with integrals bold,

ΩP(x)lnP(x)Q(x)dx\int_{\Omega} P(x) \ln \frac{P(x)}{Q(x)} dx, the stories told.

yippety 4o