Estimate the parameters of the Laplacian distribution using Bayesian Distribution

Question

I have the following zero-mean Laplacian distribution, and I am trying to estimate its parameters using Bayesian Estimation.

Answer 1

Using maximum likelihood estimation with data $\{x_i\}_{i=1}^n$, parameter estimation for the Laplace distribution $\mathcal{L}(\mu,b)$ can be done by $\hat{\mu}=\text{median}\left(\{x_i\}_{i=1}^n\right)$ and $$ \hat{b} = \frac{1}{n}\sum_i |x_i - \hat{\mu}| $$ where in your case you can simply assume $\mu=0$ (and so $\hat{\mu}=0$).

However, if you want do Bayesian parameter estimation, rather than the frequentist style, you need to make some assumptions: specifically a loss function and a prior distribution for $b$.

See some of the following:

• How to choose a prior,
• Parameters as random variables in Bayesian estimation,
• Questions on continuity
• Intuition for parameter distributions
• Comparison to frequentist outlook
• Some examples/related questions: [1], [2], [3]