The 3-Fiber Kullback-Liebler Poisson Experiment

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 $\mathbb{P}$’s eyes.

Kullback-Leibler, the divergence keen,

Measures entropy between what’s seen.

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

Reflects the divergence, a relative dream.

Consider two realms, with probabilities fine,

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

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

Entropy unfolds in the measure of light.

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

Counts converge as dusk wears down.

Kullback’s divergence, with integrals bold,

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

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