# 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.

## 2 Answers

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.