# Solved – If the LASSO is equivalent to linear regression with a Laplace prior how can there be mass on sets with components at zero

We are all familiar with the notion, well documented in the literature, that LASSO optimization (for sake of simplicity confine attention here to the case of linear regression)
\$\$
{rm loss} = | y – X beta |_2^2 + lambda | beta |_1
\$\$
is equivalent to the linear model with Gaussian errors in which the parameters are given the Laplace prior
\$\$
exp(-lambda | beta |_1 )
\$\$
We are also aware that the higher one sets the tuning parameter, \$lambda \$, the larger the portion of parameters get set to zero. This being said, I have the following thought question:

Consider that from the Bayesian point of view we can calculate the posterior probability that, say, the non-zero parameter estimates lie in any given collection of intervals and the parameters set to zero by the LASSO are equal to zero. What has me confused is, given that the Laplace prior is continuous (in fact absolutely continuous) then how can there be any mass on any set that is a product of intervals and singletons at \${0}\$?

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