# Questions tagged [laplace-distribution]

Use this tag when asking questions about the Laplace distribution. This probability distribution is sometimes called the double exponential distribution (not to be confused with the Gumbel distribution).

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18 views

### Laplacian noise applications

I would like to know whether Laplacian distribution can be used to model a Poisson noise. I have met this case while checking this book and here what it says (see the picture below) As far as I ...
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### ICA with a Laplace distribution

It's pretty common to set up ICA with a logistic distribution, but how would you find the loss and gradient with a Laplace distribution?
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### KL divergence between two Asymmetric Laplace distributions?

Consider two asymmetric Laplace distribution $L_1(\mu_1,\sigma_1,\tau_1)$ and $L_2(\mu_2,\sigma_2,\tau_2)$ where \begin{equation} L(x;\mu,\sigma,\tau) =\frac{\tau(1-\tau)}{\sigma} \begin{cases} ...
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### How to fit a data against a Laplace / Double Exponential Distribution (and check GoF)

I am a PhD student. I have a data set (waiting time in minutes between tweets) which looks almost symmetrically to the naked eye. I've tried a couple of distribution fits to this data and the ...
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### Why deep learning prefer the probability distribution with a sharp point?

I am reading Ian Goodfellow's book about deep learning and when it introduces exponential distribution, it says "In the context of deep learning, we often want to have a probability distribution with ...
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### Laplace Likelihood vs Gaussian Likelihood in Bayesian Regression [duplicate]

Question: What are the advantages and disadvantages of using a Laplace likelihood in regression instead of a Gaussian likelihood? Details: I know that if the unknown regression coefficients have a ...
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### 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 ...
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### What is the purpose of using a Laplacian distribution in adding noise for Differential Privacy?

I am reading up on Differential Privacy and it is mentioned that the technique relies on adding some controlled noise to the release of responses to queries towards a statistical database. This is ...
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### Conditional Density for Sigma (Bayesian Lasso)

I found that in Bayesian Lasso commonly $\beta \sim N(0,\sigma^2*diag(\tau))$ and $\sigma,\tau \sim \pi(\sigma,\tau)$ is used. Whereas $\pi(\cdot)$ is a product of Laplace distributions. Is it ...
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### Why is Laplace prior producing sparse solutions?

I was looking through the literature on regularization, and often see paragraphs that links L2 regulatization with Gaussian prior, and L1 with Laplace centered on zero. I know how these priors look ...
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### Show that a scale mixtures of normals is a power exponential

I'm trying to show that a scale mixture of normals yields a Laplace distribution. I've gotten to the point where I have $\int N(0,\tau)\times Ga(\tau\:;\:1,\frac{\lambda^{2}}{2}) \:d\tau$ should equal ...
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### Prior probability of unknown variance

Within my sampler I have to define the prior probability of the variance $\sigma$ of a random variable (drawn from $N(0,\sigma)$) Here I assume that $\sigma$ is close to zero. Its distribution is ...
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### Sampling the scale parameter of a Laplace distribution

I need to adjust the scale parameter $\lambda$ of a Laplace prior ($p(x|\lambda)=(1/2\lambda)* exp(-|x|/\lambda)$) within metropolis hastings. That means I have a couple of draws for x and now I have ...
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### Linear regression with Laplace errors

Consider a linear regression model: $$y_i = \mathbf x_i \cdot \boldsymbol \beta + \varepsilon _i, \, i=1,\ldots ,n,$$ where $\varepsilon _i \sim \mathcal L(0, b)$, that is, Laplace distribution with ...
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### A sufficient statistic for Laplace distribution

Suppose we have p dimensional vector of $X =[X_1 \dots X_n]$ where X is Laplace distributed. What will be a sufficient statistics for estimating covariance of $X$? Would it be the sample covariance,...
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### Distribution of value closest to 0

Consider $K$ independent Laplace variables $X_i$ ($1 \leq i \leq K$) with mean 0 and scale $\lambda$. Let $X′$ be the variable taking the value of the Laplace variable whose absolute value is the ...
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### How can I compare 2 means that are Laplace distributed?

I want to compare 2 sample means for 1-minute-stock returns. I assume they are Laplace distributed (already checked) and I split the returns into 2 groups. How can I check whether they are ...
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### Identifiability of a particular Independent Component Analysis model

I am considering the model : $$\mathbf{x} = \mathbf{A}\mathbf{s}$$ where $\mathbf A \in \mathcal{M}_{n,p}(\mathbb{R})$ and $\mathbf s \in \mathbb{R}^{p}$ such that the entries of $\mathbf s$ are i....
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### Laplacian distribution in arbitrary and limited ranges

I'm writing a computer program that applies Laplacian noise to data, in which λ is unbounded, and my statistical competence is limited. If data is a generic numeric value that is ok, but if domain ...
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### Efficient method for Laplace regression

I want to calculate numerically the maximum likelihood estimators of $(\beta,\sigma)$ for the linear regression model: $$y_j = x_j^{\top}\beta + \epsilon_j,$$ where $j=1,\dots,n$, $\beta$ is $p$-...
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### Generating a Laplace prior from a normal random variable with Rayleigh standard deviation.

I read on Wikipedia Laplace distribution that the following is true: If $X|Y \sim N(\mu,\sigma=Y)$ with $Y \sim \text{Rayleigh}(b)$, then $X \sim \text{Laplace}(\mu, b)$. However, there doesn't seem ...
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### Specifying a Laplace prior using a Gaussian random variable with Gamma variance

I need to place a Laplace prior on a random variable, however, I want to use a Gaussian distribution whose variance is Gamma(1,1) distributed, i.e., \begin{align} x &\sim N(\mu,\sigma^2)\\ \...
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### Difference between two i.i.d Laplace distributions?

What is the PDF of the difference of two i.i.d Laplace distributed random variables? I know that the difference of two i.i.d Normal variables is still the Normal distribution. Since the properties of ...