Laplace Distribution

📐 Definition

The Laplace distribution centers probability mass at a location parameter with exponential decay on both sides, controlled by a scale parameter.

For $X\sim\operatorname{Laplace}(\mu,b)$, the density is

$$f(x;\mu,b) = \frac{1}{2b}\exp\!\left(-\frac{|x-\mu|}{b}\right)$$

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Domain and Codomain

Defined over real-valued outcomes; parameters set the mean and spread.

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⚙️ Key Properties

$$\mathbb{E}[X] = \mu, \qquad \operatorname{Var}(X) = 2b^2$$

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Tails decay exponentially (heavier than Gaussian, lighter than Cauchy). Inverse-CDF sampling can be expressed using $U\sim U(0,1)$ with a sign and logarithm transform.

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$(\mu,b) = (0,1)$ defines the standard Laplace density. As $b \to 0^+$, the distribution collapses to a point mass at $\mu$; as $b$ increases, variance grows linearly in $b^2$. Absolute deviations $|X-\mu|$ follow an exponential distribution.

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🎯 Special Cases and Limits

• $(\mu,b)=(0,1)$ is the standard Laplace distribution.
• $b\to 0^+$ collapses to a point mass at $\mu$.

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🔗 Related Functions

Gaussian distributions provide a lighter-tailed alternative; exponential distributions govern one-sided magnitudes; logistic and double-exponential families share similar shapes with differing normalization.

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Usage in Oakfield

Oakfield uses Laplace noise primarily via integrator stochastic hooks:

• Integrator noise option: setting integrator noise = "laplace" selects a unit-variance Laplace source (implemented via a uniform RNG and a log1p transform with clamping away from log(0)).
• Useful for heavier-tailed stochastic increments compared to Gaussian noise when running stochastic integrator steps.

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Historical Foundations

📜 Double-Exponential Laws

Laplace-type distributions arise naturally when modeling absolute-deviation penalties and exponentially decaying tails on both sides of a central location.

🌍 Modern Perspective

They are widely used in robust statistics and sparse modeling (via the $\ell_1$ penalty connection).

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📚 References

• Kotz, Kozubowski, Podgórski, The Laplace Distribution and Generalizations
• Devroye, Non-Uniform Random Variate Generation