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).
81
questions
7
votes
1
answer
197
views
Expected absolute deviation greater than standard Laplace
Could there exist a distribution, other than standard Laplace (probability density of the form $1/2e^{-|x|}$), on $\mathbb{R}$ such that $E[x]=0,E[|x|]=1$ and that
\begin{equation*}
E[|x-a|] \geq |a|+...
- 145
0
votes
0
answers
22
views
Calculation of an optimal variational distribution for covariance parameters in a Bayesian graphical lasso model
Context:
I am considering here a variational Bayesian framework where I need to calculate the optimal variational distribution for some covariance parameters.
Formally the model can be expressed as:
$$...
- 513
0
votes
0
answers
46
views
Product of a Laplace and Gaussian distribution
I have to derive some statistical distribution for a precision parameter combining one Laplace and one Gaussian distribution. However, from the absolute value arising in the Laplace density I am bit ...
- 513
0
votes
1
answer
44
views
Not fully understanding gaussian-laplace table
I have this exercise where is the answer. Using this annex(text at the top means tenths of the x), the value becomes . What I don't understand is how it becomes 2,58. Our first answer is 0,495, so I'...
4
votes
2
answers
357
views
Is it Possible to Differentiate Two Integers after Adding Random Variables from Laplace Distribution?
Suppose we have 2 reports. These reports' original values are either all 0 or all a (a>0). Let $x_i$ be independent Laplace(0, b) random variables. For each report, we generate a random noise $x_i$ ...
- 41
0
votes
0
answers
24
views
Count data analysis using GLMM in R
I need some help to find a model that fits to my data, I am new to R, I have been consulting some bibliographic resources but i could not resolve the best way to fit my model using lme4 package.
I ...
- 1
0
votes
1
answer
61
views
Writing the likelihood function for a linear combination of Normal and Laplacian random variables [duplicate]
I am working on a probability problem to solidify my knowledge and running into some difficulty.
I have the following setup:
Let $W_{i}, W_{j}$ represent 2 i.i.d random variables, drawn from $N(0,1)$....
0
votes
0
answers
59
views
Choosing between Gaussian/Laplacian prior distributions for MCMC regression
When doing a linear regression using MCMC, you have to specify prior distributions for the values of the regression coefficients of the independent variables. If all of the priors are Gaussian ...
- 111
1
vote
0
answers
91
views
Contour lines for multivariate Laplace distribution
In the p-variate normal distribution ($\mathbf{N_p(\mu,\Sigma)}$), the solid ellipsoid of x values satisfying
$(\mathbf{x-\mu)'\Sigma^{-1}(\mathbf{x}-\mu)}\leq \chi^2_p(\alpha)$
has probability $1-\...
0
votes
0
answers
12
views
Cannot obtain empirical Laplace distribution for increments of a laplace motion
Consider the Laplace motion (a special type of Levy process where the stationary and indepedent increments are Laplace distributed). One representation of the Laplace Motion is through Brownian ...
- 101
1
vote
0
answers
18
views
Connection Between Bayesian Prior and Variable Selection in Lasso [duplicate]
I am interested in learning more about the Bayesian interpretation of the Lasso model. The Lasso model assumes a Laplace distribution of coefficients and the optimal coefficients maximize the ...
- 367
1
vote
0
answers
27
views
How can I derive the distribution of the L2,1 norm if the ditribution of L1 norm is given?
I understand that the L1 norm promotes sparsity and is a Laplace prior in the LASSO regression framework. I am interested in how this prior changes when we apply L2,1 regularisation instead? Is it ...
- 11
0
votes
0
answers
152
views
How to use the Laplace distribution in a GLM model
Good morning everyone!
I am a PhD student and I am performing some statistical analysis on a behavioral dataset related to aggression of pied flycatchers males against other male intruders. The ...
4
votes
2
answers
1k
views
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/...
- 53
0
votes
0
answers
103
views
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 ...
1
vote
0
answers
69
views
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 ...
- 165
0
votes
0
answers
82
views
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 ...
- 113
1
vote
0
answers
51
views
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 ...
- 177
0
votes
0
answers
151
views
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 ...
- 177
1
vote
0
answers
62
views
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 ...
- 11
1
vote
0
answers
1k
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 ...
- 111
1
vote
1
answer
214
views
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 ...
- 113
0
votes
1
answer
70
views
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 ...
- 113
1
vote
1
answer
401
views
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?
- 21
0
votes
1
answer
65
views
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^{*}]$$.
I know that for $r*>0$ it satisfies ...
6
votes
2
answers
300
views
Is the average of n independent Laplace random variables a Gaussian distribution?
Does the average $\frac{\sum^n_i X_i}{n}$ converge to a normal when $n \to \infty $. Here $X_i$ are independently distributed Laplace samples, with zero mean, and different standard deviation $\...
- 341
2
votes
1
answer
455
views
Laplace and Normal Distribution Cross Entropy
I need the following integral and struggle with calculating it or finding a citable source.
$$\int_{-\infty}^{\infty}(x-\mu)^2\exp\!\left(-\frac{|x-\nu|}{\tau}\right)dx.$$
Background: I want to find ...
- 105
0
votes
0
answers
640
views
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 ...
- 105
1
vote
1
answer
227
views
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 ...
- 323
5
votes
1
answer
658
views
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 ...
- 211
1
vote
0
answers
74
views
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 ...
- 1,930
2
votes
1
answer
416
views
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}
...
- 689
1
vote
1
answer
933
views
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}+\...
2
votes
3
answers
1k
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 ...
- 203
1
vote
1
answer
2k
views
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 ...
- 111
1
vote
0
answers
84
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$$
Using the Bonferroni method, construct a $100(1-\alpha)\%$ ...
- 51
3
votes
1
answer
91
views
PDF for length of Laplace-distributed vectors
I am interested in finding an analytic expression for the length of a 3-vector whose components are distributed according to a Laplace distribution with zero mean and the same scale parameter.
I ...
- 33
1
vote
0
answers
569
views
Cumulative distribution function of a squared laplace random variable
I am trying to calculate $F_Y(x)$ (CDF) of $Y=X^2$ where $X$ is a random variable of Laplace Distribution $f_X(x) = \frac{1}{2}e^{-|x|}$ (let's take a simple case when parameters $\mu=0$ and $b=1$).
...
- 31
0
votes
0
answers
1k
views
How can I get confidence interval from Laplace distribution in python?
I have a dataset and I checked that fits a Laplace distribution. I want to get different confidence intervals from it.
I know that in a normal distribution, the confidence interval of 68% is mean + ...
- 111
1
vote
0
answers
412
views
linear model with laplace-distributed residuals, where scale (not location) varies
I have a dataset where I suspect the residuals are approximately Laplace-distributed.
There are three continuous predictors. When I split up the data into many bins, based on the values of these ...
- 438
6
votes
2
answers
757
views
Efficient random generation from truncated Laplace distribution
We have several ways of drawing random samples from Laplace distribution. Is there any efficient way of sampling from left truncated Laplace distribution? Inverse transform sampling is an obvious ...
Tim♦
- 134k
1
vote
0
answers
765
views
Complete and sufficient statistics of Laplace Distribution [duplicate]
Let $X_{1}, X_{2},...,X_{n}$ be i.i.d from the Laplace distribution or Double exponential distribution $DE(\mu, \sigma)$ with the following pdf,
$$f(x) = \frac{1}{2\sigma} e^{\dfrac{-|x-\mu|}{\sigma}}...
- 335
4
votes
2
answers
2k
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.
...
- 411
7
votes
1
answer
2k
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 ...
- 143
6
votes
1
answer
3k
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) + (\...
- 770
1
vote
0
answers
35
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 $...
- 505
5
votes
1
answer
5k
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 ...
- 512
8
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 ...
- 201
1
vote
0
answers
295
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 ...
- 103
1
vote
0
answers
302
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 $\...
- 862