Statistical Analysis Stack Exchange
Statistical Analysis Stack Exchange

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).

The Laplace distribution is a distribution symmetric about the mean $\mu$ with tails decaying at an exponential rate. More on this density may be found in the [Wikipedia page].(https://en.wikipedia.org/wiki/Laplace_distribution)

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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 like, but I don't understand, how it translates to,…
Dmitry Smirnov
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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 $x\in\mathbb{R}, p\in[1,2]$ and $\beta>0.$ Does such a…
Sycorax
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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) $$ {\rm loss} = \| y - X \beta \|_2^2 + \lambda \| \beta \|_1 $$ is…
Grant Izmirlian
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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 Theorem You can get binomially distributed data from…
David J. Harris
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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 googled around quite a lot with no success so my…
Rasmus Bååth
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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 chi-squared expressions. The above simple…
Corvus
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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 stumbling over the term: Laplace noise. To make a…
Axolotl
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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 $0$ mean and $b$ scale parameter, are all are…
nmerci
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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 significantly different? I think I cannot treat them like a…
Rob
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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 Laplace. Now, all of a sudden when I need it, I can't…
user34790
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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 estimator of $\mu$, is the sample median. But how do we…
set_seed_1234
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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 a sharp point at x=0." I don't understand why the…
Flamingo
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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 a Laplace. It's unclear to me how to solve the…
ilanman
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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 $\sigma_i$. I know this can be applied from the general…
Phong Le
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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 solution, but maybe there is something better?
Tim
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