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).
85 questions
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OLS vs MLE when errors are not normally distributed (Laplace distributed)
We say that under assumptions of the Gauss-Markov theorem, OLS is BLUE. The Gauss-Markov theorem doesn't mention the normality of errors.
If the errors are distributed as per the Laplace distribution,...
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Quantifying prediction uncertainty using deep ensembles: How to combine Laplace distributions?
For a regression problem, I want to train an ensemble of deep neural networks to predict the labeled output as well as the uncertainty, similar to the approach presented in the paper Simple and ...
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Independence of real and imaginary part of the product of two independent normal variables
Let $X_1,X_2,Y_1,Y_2$ be iid standard normal variables $N(0,1).$ Let $X=X_1+iX_2,$ $Y=Y_1+iY_2$ and $Z=XY.$ We have : $Z=(X_1Y_1 - X_2Y_2) + i(X_1Y_2 + X_2Y_1).$ From https://en.wikipedia.org/wiki/...
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How can I fit a least absolute deviation model with random intercepts?
I have some data and a model formulation that I believe describes the data generating process. The model looks like this
$$y_{i,t} = \beta_1x_{i,t} + \beta_2x_{i,t-1}+\alpha_t+\epsilon_{i,t}$$
where $\...
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Sum of Independent Laplacian Variables
I have $N$ Independent Random variables Laplacian distributions with $\mu=0$ and positive $b=\sigma^2/2$. I also have dominant random variable $(X_s)$ with Laplacian distribution with $\mu=0$ and $b=\...
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What Python library functions can one use to determine if a sample is drawn from a Laplace distribution?
I have some data I suspect is drawn from a Laplace distribution rather than a Gaussian one. I can use the Kolmogrov-Smirnov test and the ability to create Uniform and Gaussian distributions of the ...
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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|+...
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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:
$$...
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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 ...
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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'...
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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$ ...
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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)$....
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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 ...
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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-\...
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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 ...
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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 ...
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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 ...
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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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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^{*}]$$.
I know that for $r*>0$ it satisfies ...
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Is the average of n independent Laplace random variables a Gaussian distribution?
Does the average $\frac{\sum^n_i X_i}{n}$ converge to a normal when $n \to \infty $. Here $X_i$ are independently distributed Laplace samples, with zero mean, and different standard deviation $\...
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Laplace and Normal Distribution Cross Entropy
I need the following integral and struggle with calculating it or finding a citable source.
$$\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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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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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$$
Using the Bonferroni method, construct a $100(1-\alpha)\%$ ...
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PDF for length of Laplace-distributed vectors
I am interested in finding an analytic expression for the length of a 3-vector whose components are distributed according to a Laplace distribution with zero mean and the same scale parameter.
I ...
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Cumulative distribution function of a squared laplace random variable
I am trying to calculate $F_Y(x)$ (CDF) of $Y=X^2$ where $X$ is a random variable of Laplace Distribution $f_X(x) = \frac{1}{2}e^{-|x|}$ (let's take a simple case when parameters $\mu=0$ and $b=1$).
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How can I get confidence interval from Laplace distribution in python?
I have a dataset and I checked that fits a Laplace distribution. I want to get different confidence intervals from it.
I know that in a normal distribution, the confidence interval of 68% is mean + ...
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linear model with laplace-distributed residuals, where scale (not location) varies
I have a dataset where I suspect the residuals are approximately Laplace-distributed.
There are three continuous predictors. When I split up the data into many bins, based on the values of these ...
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Efficient random generation from truncated Laplace distribution
We have several ways of drawing random samples from Laplace distribution. Is there any efficient way of sampling from left truncated Laplace distribution? Inverse transform sampling is an obvious ...
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Complete and sufficient statistics of Laplace Distribution [duplicate]
Let $X_{1}, X_{2},...,X_{n}$ be i.i.d from the Laplace distribution or Double exponential distribution $DE(\mu, \sigma)$ with the following pdf,
$$f(x) = \frac{1}{2\sigma} e^{\dfrac{-|x-\mu|}{\sigma}}...
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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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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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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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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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