Updated 12 September 2026

Laplace distribution

Also called the double exponential. Take two mirrored exponential tails and glue them at the mode: that’s Laplace. Independent zero-centered Laplace priors on regression coefficients produce an L1 coefficient penalty in a MAP estimate. With a Gaussian error likelihood, this gives the LASSO objective. Laplace errors instead produce absolute-residual regression; these are different uses of the same density.

What to notice

• A sharp cusp at $\mu$. Unlike the Normal’s smooth peak, the density has a kink where the two exponential tails meet. That non-differentiability is what drives L1 solutions to exact zeros.
• Heavier tails than the matched Normal. Decay is exponential in $|x-\mu|$ instead of squared, so the same standard deviation implies much more probability far from centre.
• Median is the MLE for $\mu$, not the mean. This concerns a location likelihood with Laplace errors, rather than a coefficient prior:

Why it matters

A likelihood describes observations given model parameters; a prior describes uncertainty about those parameters. Keep that distinction when connecting a density to an optimization objective. Absolute residuals from a Laplace likelihood and absolute coefficient penalties from a Laplace prior can appear in different models.

Construct the two tails

Let A and B be independent exponential waits, each with rate 1/b. Then X=μ+A−B is Laplace(μ,b). For a nonnegative difference d, integrating their joint density along A=B+d gives ∫₀∞ b⁻² exp(−(2u+d)/b) du=exp(−d/b)/(2b). Swapping A and B gives the negative half by symmetry. The table shows the actual waits used for the histogram.

For μ=1 and b=2, the example A=5 and B=1.5 produces X=4.5. Its population mean is 1 and variance is 8. If A and B were the same random wait, X would always equal 1; independence is essential to the claimed distribution.

Compare tails with the same variance

The standard deviation is √2 b. At three matched standard deviations, the Laplace outside probability is exp(−3√2)≈0.014370, compared with approximately 0.002700 for normal. But at one standard deviation the probabilities are about 0.243117 and 0.317311: a sharper center and farther tails leave less probability at some intermediate distances. “Heavier tails” describes sufficiently remote behavior, not every cutoff.

Work an absolute-error fit

For 1,2,2,3,12, a fitted location of 2 gives absolute residual sum 12 and squared residual sum 102. A location of 4 gives 16 and 82. The sample median minimizes the first, while the sample mean minimizes the second. The large observation pulls the mean farther than the median; neither optimization rule proves that its assumed likelihood is appropriate.

With known b, multiplying independent Laplace densities and taking the negative logarithm gives n log(2b) + Σ|xᵢ−μ|/b. The first term does not depend on μ, which explains the median optimizer. With an even sample size, an entire interval between the two middle observations can minimize the absolute loss.

References

Random Services: Laplace distribution gives the exponential construction, density, and moments. The likelihood calculation above follows directly by multiplying the displayed densities. Continue to normal for squared-error geometry and logistic for another smooth, exponential-tailed model.