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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7 views

What is the difference between data perturbation and differential privacy?

I cannot distinguish the terms "data perturbation" and "differential privacy". If the data perturbation is the process that adds some small value sampled from specific distributions such as Laplacian ...
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Global sensitivity of mean and variance in differential privacy?

Please explain me why global sensitivity of a mean or variance queries will be (b-a)/n and (b-a)^2/n where b is the upper ...
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39 views

Bonferroni confidence region for shifted Laplace parameters

Consider the shifted Laplace distribution with the density: $$f(y)=\frac{\theta}{2}e^{-\theta|y-\mu|}\quad, \quad y\in \mathbb R$$ Using the Bonferroni method, construct a $100(1-\alpha)\%$ ...
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41 views

PDF for length of Laplace-distributed vectors

I am interested in finding an analytic expression for the length of a 3-vector whose components are distributed according to a Laplace distribution with zero mean and the same scale parameter. I ...
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102 views

Cumulative distribution function of a squared laplace random variable

I am trying to calculate $F_Y(x)$ (CDF) of $Y=X^2$ where $X$ is a random variable of Laplace Distribution $f_X(x) = \frac{1}{2}e^{-|x|}$ (let's take a simple case when parameters $\mu=0$ and $b=1$). ...
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159 views

How can I get confidence interval from Laplace distribution in python?

I have a dataset and I checked that fits a Laplace distribution. I want to get different confidence intervals from it. I know that in a normal distribution, the confidence interval of 68% is mean + ...
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0answers
50 views

linear model with laplace-distributed residuals, where scale (not location) varies

I have a dataset where I suspect the residuals are approximately Laplace-distributed. There are three continuous predictors. When I split up the data into many bins, based on the values of these ...
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2answers
301 views

Efficient random generation from truncated Laplace distribution

We have several ways of drawing random samples from Laplace distribution. Is there any efficient way of sampling from left truncated Laplace distribution? Inverse transform sampling is an obvious ...
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0answers
182 views

Complete and sufficient statistics of Laplace Distribution [duplicate]

Let $X_{1}, X_{2},...,X_{n}$ be i.i.d from the Laplace distribution or Double exponential distribution $DE(\mu, \sigma)$ with the following pdf, $$f(x) = \frac{1}{2\sigma} e^{\dfrac{-|x-\mu|}{\sigma}}...
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2answers
417 views

Posterior computation for Laplace distribution

I am dealing with being Bayesian and looking for a closed form for a posterior for the scale parameter $\tau$ of a Laplace distribution, such that I can derive a full conditional in my Gibbs sampler. ...
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1answer
450 views

Standard Error of the MLE for Laplace Distribution

Given the Laplace distribution parametrized by $\mu$ and $b$, $f(x\mid \mu ,b)={\frac {1}{2b}}\exp \left(-{\frac {|x-\mu |}{b}}\right)\,\!$ , I know that $\hat \mu$, the maximum likelihood ...
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1answer
478 views

Fisher's Information for Laplace distribution

Say we have $f(x , \theta) = \frac{1}{2}e^{-|x-\theta|}$ Lets assume for simplicity, we only have 1 sample. We find that the log-likelihood for this distribution is: $$ l(\theta , x) = -log(2) + (\...
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23 views

Inferrence for peaked likelihoods

Suppose I have the likelihood $f(X|\theta)$ of some rich model, where $\theta\in\mathbb{R}^n$, and I have been able to find its maximum, $\hat\theta$. Suppose further that for some $i$, the plot of $...
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1answer
2k views

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 ...
6
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1answer
911 views

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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132 views

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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70 views

Normalization constant for many to one mapping (Laplace distribution)

Suppose $\alpha=U\beta$ where $U$ is $N\times K$ with $N > K$. What is the probability density function (PDF) of $\beta$, $p(\beta)$, given that we know that it is proportional to $q$, the PDF of $\...
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2answers
492 views

Is the CDF of a Laplace distribution well-defined?

So I'm asked to argue whether the CDF of a Laplace distribution is well-defined or not. Now I don't completely understand what well-defined actually means. CDF given by: Acording to wikipedia: "A ...
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1k views

Sufficiency of Sample Mean for Laplace Distribution

I recently started reading about sufficient statistics. I have the following questions: 1) Is sample mean a sufficient statistic for Laplace Distribution (aka Double Exponential) if we already know ...
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2answers
543 views

First and second moments of truncated laplace distribution

I'm trying to estimate a distribution that looks like a truncated Laplace distribution. However, I can't find closed-form expressions of its first and second moments. I'm expecting closed-form ones as ...
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1answer
91 views

Fractional moments of the Laplacian distribution larger than of the normal

How can I show that the fractional moments of the (unit variance) Laplacian distribution are higher than of the standard normal distribution, for moments higher than 2? Formally, if $l \sim Laplace(...
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2k views

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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1answer
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Computing the mean and variance of the ratio of two Laplace variables

I know that Laplacian distribution function is defined as follow $$ f(x)=\frac{b}{2}\exp(-b|x-\mu|) $$ Also, I know that the mean and variance for the ratio between two normal variables like $$c=\...
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What is meant by “Laplace noise”?

I am currently writing algorithm for differential privacy using the Laplace mechanism. Unfortunately I have no background in statistics, therefore a lot of terms are unknown to me. So now I'm ...
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3answers
1k views

Does this distribution have a name? $f(x)\propto\exp(-|x-\mu|^p/\beta)$

It occurred to me today that the distribution $$ f(x)\propto\exp\left(-\frac{|x-\mu|^p}{\beta}\right) $$ could be viewed as a compromise between the Gaussian and Laplace distributions, for $x\in\...
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1answer
386 views

How does one find the sample median of for a group of iid random variables with Laplace distribution?

I know a median is $$\frac{1}{2} = \int_{-\infty}^{\mu} f(x)dx$$ I understand how to solve this for simple distributions. However, I am learning how to do it for iid samples, which I haven't done ...
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1answer
830 views

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 - ...
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1answer
1k views

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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0answers
77 views

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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1answer
11k views

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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2answers
827 views

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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1answer
77 views

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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79 views

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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1answer
1k views

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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1answer
824 views

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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1answer
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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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1answer
433 views

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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1answer
36 views

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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1answer
318 views

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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515 views

Laplace approximation for binomial distribution in matlab

i using bionrnd() function to generate a random vector and Laplace approximation formula to approximate the binomial distribution. but Laplace histogram dose not ...
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1answer
437 views

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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1answer
959 views

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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1answer
1k views

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 ...
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1answer
3k views

Does there exist a conjugate prior for the Laplace distribution?

Does there exist a conjugate prior for the Laplace distribution? If not, is there a known closed form expression that approximates the posterior for the parameters of the Laplace distribution? I've ...
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1answer
247 views

How to apply Laplace rule of succession for any possible value in a range?

I am looking to have People rate certain items on a scale of 0-10 based on those items perceived utility to the particular Person performing the rating. Fractional values are permitted. Giving no ...
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2answers
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Sum of two normal products is Laplace?

It is apparently the case that if $X_i \sim N(0,1)$, then $X_1 X_2 + X_3 X_4 \sim \mathrm{Laplace(0,1)}$ I've seen papers on arbitrary quadratic forms, which always results in horrible non-central ...
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1answer
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Multivariate Laplace distribution

I find it strange, but I can't find what multivariate Laplace distribution looks like. What is its pdf? I googled for a while but couldn't find a good description. I wasn't paying attention to ...
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2answers
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What processes could generate Laplace-distributed (double exponential) data or parameters?

Lots of distributions have "origin myths", or examples of physical processes that they describe well: You can get normally distributed data from sums of uncorrelated errors via the Central Limit ...
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5answers
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Random number generation using t-distribution or laplace distribution

I am a newbie in stat. I am completing my thesis in Evolutionary algorithm. I have to generate some random numbers from T-distribution or Laplace distribution. How can I do this? An easy and simple ...