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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Relationship between laplace and l1 regularization

It is well known that an L1 regularized linear regression is equivalent to a regression with a Laplace prior on the distribution of the coefficients. This is explained here: https://bjlkeng.github.io/...
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MLE of Laplacian with linear parameters

I'm stumped on this problem and was hoping someone could give me some guidance. I'm new to this sort of thing so it's possible that I'm leaving something out of this question or that my question isn't ...
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Is a Laplace Prior the same thing or related to a Laplace Transformation?

Context: I was watching this video https://youtu.be/pOYAXv15r3A?t=796 about Facebook Prophet and the speaker mentioned they use a Laplace Prior $$\delta \sim Laplace(\lambda)$$. What I have gleaned so ...
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Is there an any advantage to using Laplace transforms over Fourier transforms in Statistics?

Do you know the advantage of the Laplace transform over the Fourier transform in statistics? I say, it's more restrictive and I couldn't find any advantage I asked this because the professor used ...
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Output Distribution of ReLU given a Laplace Distribution as its Input

If input to a ReLU function (Max(X, 0)) is a Laplace Distribution, what would be the output distribution? will it have a density function? how would it look like? assuming that mean of the Laplace is ...
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Linear Transformation of a Random Variable with a Laplace Distribution

I have read these two posts ( 1 and 2) about linear transformation of a random variable with a Gaussian distribution. I would like to find the first two moments of a linearly transformed Laplace ...
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Distribution of the dot product of a multivariate Laplace random variable and a fixed vector

This question is basically a follow-up: Distribution of the dot product of a multivariate gaussian random variable and a fixed vector But instead of a multivariate Gaussian random variable, what about ...
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Laplace distribution as an Exponential Distribution and Minimizitaion of KL Divergence

In the context of Expectation Propagation [Minka's thesis-2001], I would like to approximate an unknown distribution with a Laplace distribution. This can be solved by minimizing KL-Divergence. In ...
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345 views

Logit Laplace Loss Function

In a recent OpenAi paper, the authors propose a novel loss function for the reconstruction term of a VAE coined Logit-Laplace loss. They detail the math on page 13 of the paper but I am having trouble ...
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Why does Chebyshev's inequality yield that the probability of Laplacian noise being bigger than x is bounded like this?

I am trying to understand this proof of the bounds of Laplacian noise used in a paper on differential privacy. Given a random variable $Lap\left ( \frac{\Delta f}{\varepsilon } \right )$, apparently ...
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Can a folded LaPlace distribution (or other folded distributions) be used with Ɛ-differential privacy

I have a single value in (or over) our dataset, let's say a count of something, and we want to keep that value private within a certain range. This range is the sensitivity. The adversary can ask if a ...
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MLE of a Laplace density [closed]

How do you evaluate MLE of theta, considering a simple random sample of size n from a Laplace density?
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Laplace Inequality

I am trying to prove that if $r_i \sim Lap(0,1/\varepsilon)$ where $\varepsilon >0$ then: $$Pr[r_i \geq 1+r^*] \geq e^{-\varepsilon}Pr[r_i \geq r^{*}]$$. I know that for $r*>0$ it satisfies ...
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2 answers
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Is the average of n independent Laplace random variables a Gaussian distribution?

Does the average $\frac{\sum^n_i X_i}{n}$ converge to a normal when $n \to \infty $. Here $X_i$ are independently distributed Laplace samples, with zero mean, and different standard deviation $\...
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2 votes
1 answer
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Laplace and Normal Distribution Cross Entropy

I need the following integral and struggle with calculating it or finding a citable source. $$\int_{-\infty}^{\infty}(x-\mu)^2\exp\!\left(-\frac{|x-\nu|}{\tau}\right)dx.$$ Background: I want to find ...
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KL Divergence Normal and Laplace densities

I want to calculate the KL-Divergence between a Laplacian density g and a normal density f. I can decompose $KL(G|F)$ to $\mathbb{E}_g[\log g(X)]-\mathbb{E}_g[\log f(X)]$. I am already stuck with my ...
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sparsity assumption in Bayesian linear regression

I have a simple question. Is the assumption of sparsity only useful when p > n, that is when you have a large number of features compared to observation. When ...
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What is the PDF of a Normal convolved with a Laplace

I'd like to see if using Stan or similar I can successfully model Laplace noise added to data through the use of a convolved Normal-Laplace distribution and MCMC sampling. In the literature I can only ...
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Surprising nonlinear variance-based scale est (bias adj) for Laplace Distribution competes with MLE?

Background: Using the quantile function (inverse cumulative distribution) for the Laplace distribution supplied with uniform random deviates (per the RAND() spreadsheet function), I examined an ...
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2 votes
1 answer
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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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Simple Linear Regression With Laplace Distribution (Double Exponential)

I have a question on how it would look the linear regression model given that $\epsilon_{i}\sim Laplace(0,\lambda)$ with a reparametrization $b=\frac{1}{\lambda}$. $Y_{i}=\alpha+\beta x_{i}+\...
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3 answers
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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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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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3 votes
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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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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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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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1 vote
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257 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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6 votes
2 answers
591 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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1 vote
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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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4 votes
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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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7 votes
1 answer
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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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6 votes
1 answer
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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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1 vote
0 answers
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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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5 votes
1 answer
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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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7 votes
1 answer
2k 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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1 vote
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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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1 vote
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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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3 votes
2 answers
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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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5 votes
2 answers
3k 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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2 votes
2 answers
1k 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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2 votes
1 answer
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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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5 votes
3 answers
3k 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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-1 votes
1 answer
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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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12 votes
2 answers
9k views

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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23 votes
3 answers
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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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1 vote
1 answer
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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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22 votes
1 answer
2k 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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4 votes
1 answer
2k 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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2 votes
0 answers
108 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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