LaplacePrior

LaplacePrior is a Laplace (double-exponential) prior used in Bayesian statistics to regularize estimates and promote sparsity in parameter recovery. For a real-valued parameter θ, the prior density is p(θ) = (λ/2) exp(-λ |θ - μ|), where μ is the location (often 0) and λ > 0 is the rate parameter (equivalently, the scale is b = 1/λ). A common choice is μ = 0, yielding p(θ) = (λ/2) exp(-λ |θ|).

In many modeling contexts, a Laplace prior on coefficients in a linear model yields posterior modes that resemble L1-regularized solutions. With a Gaussian likelihood, the maximum a posteriori estimate under a Laplace prior corresponds to solving an L1-penalized least squares problem, linking the prior to classic sparsity-promoting regularization.

Hierarchical representations: the Laplace prior can be expressed as a scale mixture of Gaussians with a latent scale variable, which facilitates Bayesian computation. A typical formulation uses p(β | τ) = Normal(β | 0, τ) with a suitable prior on τ (often an exponential or gamma distribution). Marginalizing over τ yields the Laplace form for β. This representation supports Gibbs sampling and variational inference in sparse Bayesian models.

Applications and properties: Laplace priors are common in sparse Bayesian learning, Bayesian variable selection, compressed sensing, and signal processing. They promote shrinkage of many coefficients toward zero while allowing some to be large, but are generally less aggressive than spike-and-slab priors. Their tails are heavier than Gaussian but lighter than many power-law priors, offering a balance between shrinkage and flexibility.

See also: L1 regularization, shrinkage priors, and spike-and-slab priors.