# Practical applications of the Laplace and Cauchy distributions

I want to know if there are any examples of real-life applications of the Laplace and Cauchy density functions. How do they differ in their applications?

This related post, however, does not answer my question.

One example is using them as robust priors for regression parameters, where Laplace prior corresponds to LASSO (Tibshirani, 1996) , but $t$-distribution, or Cauchy are other alternatives (Gelman et al, 2008).

Moreover, you can have L1 regularized regression with Laplace errors (i.e. minimizing absolute error).

Another example: Laplace noise is used in currently trendy field of .

Tibshirani, R. (1996). Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society. Series B (Methodological), 267-288.

Gelman, A., Jakulin, A., Pittau, G.M., and Su, Y.-S. (2008). A weakly informative default prior distribution for logistic and other regression models. The Annals of Applied Statistics, 2(4), 1360-1383.

• Does the Cauchy distribution have practical applications in the study of Brownian motion? – StubbornAtom Aug 12 '16 at 8:04
• @StubbornAtom en.wikipedia.org/wiki/Cauchy_process , my answer does not provide a closed list of all applications since there is much more of them. – Tim Aug 12 '16 at 8:10

The Laplace distribution is also related to median linear regression models. For a model:

$$y_i=x_i^T\beta + \epsilon_i,$$ where $\epsilon_i$ are iid Laplace with location $0$ and scale $\sigma$, the maximum likelihood estimators of $\beta$ coincide with the median regression estimators $$\hat{\beta} =\text{argmin} \sum_i\vert y_i-x_i^t\beta\vert.$$

The half-Cauchy prior is very popular in Bayesian hierarchical models:

Nicholas G. Polson and James G. Scott (2012). On the Half-Cauchy Prior for a Global Scale Parameter. Bayesian Analysis.