A normal distribution has the familiar bell curve shape. A random variable has a $${\displaystyle {\textrm {Laplace}}(\mu ,b)}$$ distribution if its probability density function is The Laplace distribution is convenient and conventional in differential privacy. When mathematical statistics and differential privacy combine, it could be convenient to “approximate” a Laplace distribution by a normal distribution [2]. There’s no need to ask whether it is realistic because Laplace noise is added deliberately; the distribution assumption is exactly correct by construction. probability density drops very rapidly as you move further from the middle, like exp(-x²). You could just set the two scale parameters to be the same, but that’s similar to the Greek letter fallacy, assuming two parameters have the same meaning just because they have the same symbol. The question then becomes how to choose the scale parameters. The latter is messier and harder to interpret. This site is not liable for any informational errors, incompleteness, or delays, or for any actions taken in reliance on information contained herein. A Laplace distribution, also known as a double exponential distribution, it pointed in the middle, like a pole holding up a circus tent. Your email address will not be published. (See this post for details.). In green, the density of a Laplace distribution. Rather than working with ε-differential privacy you have to work with (ε, δ)-differential privacy. normal-Laplace random variable and provides a useful parametric form for modelling size distributions. We look forward to exploring the opportunity to help your company too. Whether it is realistic in application depends on context, but it’s convenient and conventional. normal distribution vs Laplace Predicting Stock Market Returns—Lose the Normal and Switch to Laplace. Elles sont également appelées lois gaussiennes, lois de Gauss ou lois de Laplace-Gauss des noms de Laplace (1749-1827) et Gauss (1777-1855), deux mathématiciens, astronomes et physiciens qui l'ont étudiée. I heard the phrase “normal approximation to the Laplace distribution” recently and did a double take. It is not intended as advice to buy or sell any securities. One way to replace a Laplace distribution with a normal would be to pick the scale parameter of the normal so that both two quantiles match. Because the two distributions have different tail weights, their scale parameters serve different functions. All content on this site is provided for informational and entertainment purposes only and is not intended for trading purposes or advice. My colleagues and I have decades of consulting experience helping companies solve complex problems involving data privacy, math, statistics, and computing. A bottom-up simulation points to the Laplace distribution as a much better choice. Question: We obtain the asymptotic distributions of the test statistics and it is observed that they are independent of the unknown parameters. A normal distribution has the familiar bell curve shape. [2] You could use a Gaussian mechanism rather than a Laplace mechanism for similar reasons, but this makes the differential privacy theory more complicated. We find two quantiles of the Laplace distribution, then use the method in that post to find the corresponding normal distribution scale (standard deviation). A Laplace distribution, also known as a double exponential distribution, it pointed in the middle, like a pole holding up a circus tent. where Φ is the cumulative distribution function of the standard normal. The generalized normal-Laplace (GNL) distribution is both inﬁnitely divisible and closed under summation. There are numerous generalizations of univariate to multivariate Laplace distributions; we follow Kozubowski et al. The Laplace distribution has moderate tails, going to zero like exp(-|x|). Here in blue is the density of returns, based on 10 years of historical data of 5-minutes chart of EUR/USD. If the underlying distribution is normal the asymptotic Required fields are marked *. Adding Gaussian or Laplacian noise for privacy. Both distributions are symmetric about their means, so it’s natural to pick the means to be the same. Using the formula derived in the previously mentioned post. (Especially, it has fatter tails than normal distribution, as required). So if you wanted to replace a Laplace distribution with a normal distribution, which one would you choose? probability density drops very rapidly as you move further from the middle, like exp(-x²). “Thick tailed” and “thin tailed” are often taken to mean thicker than exponential and thinner that exponential respectively. The normal distribution does not approximate the Laplace! The Laplace distribution has moderate tails [1], going to zero like exp(-|x|). I’ve written before about how to solve for scale parameters given two quantiles. [1] The normal distribution is the canonical example of a thin-tailed distribution, while exponential tails are conventionally the boundary between thick and thin. So why would you want to replace one by the other? So normal and Laplace distributions are qualit… A Laplace distribution, also known as a double exponential distribution, it pointed in the middle, like a pole holding up a circus tent. In this paper we consider the logarithm of the ratio of the maximized likelihoods to discriminate between the two distribution functions. The normal distribution is convenient to use in mathematical statistics. Predicting Stock Market Returns—Lose the Normal and Switch to Laplace. Many of the products/companies that I mention in my posts advertise on this site and I receive revenue from those advertisements. Both normal and Laplace distributions can be used to analyze symmetric data. It is possible to construct a L´evy process whose increments follow the GNL distribution. Everyone agrees the normal distribution isn’t a great statistical model for stock market returns, but no generally accepted alternative has emerged. Everyone agrees the normal distribution isn’t a great statistical model for stock market returns, but no generally accepted alternative has emerged. Please do your own homework and accept full responsibility for any investment decisions you make. The Laplace distribution with scale s has density, If we want to solve for the quantile x such that Prob(X > x) = p, we have. November 11, 2019 March 18, 2016 by Vance Harwood.

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