Skip to main content

All Questions

Filter by
Sorted by
Tagged with
23 votes
7 answers
5k views

What makes Gaussian distributions special?

I'm looking for as many different arguments or derivations as possible that support the informal claim that Gaussian/Normal distributions are "the most fundamental" among all distributions. ...
13 votes
1 answer
10k views

KL divergence and mixture of Gaussians

Do we have an exact formula to compute the KL divergence between 2 mixtures of Gaussians (i.e convex combinations of a finite number of Gaussian distributions)? If not exactly known, are there good ...
gradstudent's user avatar
  • 2,246
6 votes
2 answers
2k views

Are Gaussian Processes more important than other stochastic processes?

I am doing a course at university and it deals with Gaussian Processes mainly. We use them for fitting data and prediction, machine learning, regression, classification. Is there any particular reason ...
s5s's user avatar
  • 87
5 votes
1 answer
392 views

comparing Gaussian to order statistic of Gaussian

I would like to compute the probability of $$\mathbb{P}[Y > \max(X_i)], Y\sim N(0, 1), X_i \sim N(0, \sigma_i)$$ All the random variables have zero mean, but the variances are different. My ...
lhk's user avatar
  • 151
4 votes
2 answers
1k views

Reducing system of equations involving Erf, Error Function

I have a system of equations: $$1/2 + {\rm Erf}(x) - {\rm Erf}(\frac{x+y}{2})=0$$ $$-1/2 + {\rm Erf}(y) - {\rm Erf}(\frac{x+y}{2})=0,$$ Where $x \le y$ and ${\rm Erf}$ is the Error Function. By ...
Johan Ugander's user avatar
4 votes
1 answer
347 views

Concentration of functional of Gaussian random variable

Suppose I have two Gaussian distributions $p(x) = \frac{1}{(2\pi)^{d/2}|\Sigma_p|^{1/2}}\exp(-\frac{1}{2}x^\top \Sigma_p^{-1} x)$ and $q(x) = \frac{1}{(2\pi)^{d/2}|\Sigma_q|^{1/2}}\exp(-\frac{1}{2}x^\...
Wuchen's user avatar
  • 515
4 votes
0 answers
76 views

How well does an estimator perform on another dataset?

Suppose $X \sim N(0, \Sigma)$ is a $d$-dimensional Gaussian random vector. And we have $2n$ $i.i.d$ sample $X_1, \ldots, X_{n}, \ldots, X_{2n}$. Let $\hat{\Sigma}_1 = \frac{1}{n}\sum_{i=1}^nX_i X_i^\...
Wuchen's user avatar
  • 515
3 votes
1 answer
157 views

Bound for expectation of function of 3 normal distributions

Let $X,Y,Z$ be three standard normal distribution. Let $\rho_{XY},\rho_{YZ},\rho_{XZ}$ be the correlation between those random variables. Let $f()$ be a monotone, odd, bounded, and differentiable ...
clj's user avatar
  • 31
3 votes
0 answers
131 views

Matrix-Gaussian distributions

The point of this question is to ask for references on matrix-variate Gaussian distributions. But I will explain what I mean by a matrix-variate Gaussian with an example (the notion I have in mind is ...
user3826143's user avatar
3 votes
0 answers
353 views

Moments of normalized multivariate Gaussians (and Wick's/Isserlis theorems)

Suppose $x = \begin{bmatrix}x_1 \\ x_2\end{bmatrix}$ is distributed according to the real two-dimensional Gaussian with mean-$0$ and covariance matrix $\Sigma$. I am interested in a closed form for ...
user3826143's user avatar
3 votes
0 answers
75 views

Covariance of censored/clipped Gaussians

I am interested in the covariance of two clipped (or censored) Gaussian variables. More precisely, let $g_1 \sim N(0,\sigma_1^2)$ and $g_2 \sim N(0,\sigma_2^2)$ be two (dependent) Gaussians with $\...
EmmGee's user avatar
  • 53
2 votes
2 answers
690 views

Concentration and anti-concentration of gap between largest and second largest value in Gaussian iid sample

Let $n \ge 3$ be an integer and let $X=(X_1,\ldots,X_n)$ be random vector with iid coordinates from $N(0,1)$. For $1 \le k \le n$, let $X_{(k)}$ be the value of the $k$th largest coordinate of $X$. ...
dohmatob's user avatar
  • 6,853
2 votes
1 answer
256 views

About a mixture

Consider the following mixture model for a univariate density function $$ (1) \quad f(x)=\int_{(m, \sigma^2)\in D} g(x; m, \sigma^2) \mu(d(m, \sigma^2)) $$ where $D$ is a compact subset of $\mathbb{R}\...
Star's user avatar
  • 108
2 votes
0 answers
56 views

Sum of independent Wisharts

Suppose random vectors $y_1,y_2,\ldots,y_m$ are independent and the distribution of each $y_i$ is a $d$-dimensional complex Gaussian with mean $0$ and covariance $\Gamma_i$, that is $y_i \sim \mathcal{...
user3826143's user avatar
2 votes
0 answers
61 views

Approximate logarithmic bound on expected maximum via central limit theorem

If $Z_i$ are standard normal, possibly dependent, one can show that $$E\left[\max_{i=1,...,M} Z_i^2\right]\leq 3\ln M + 1.$$ I'm looking for a similar (asymptotic) bound for asymptotically normal ...
Dasherman's user avatar
  • 203
2 votes
0 answers
386 views

What is the concentration of measure for Gaussian random variables which are independent, but are transformed?

This might be a too easy question for Mathoverflow, but Googling led to similar questions and answers here (though not the one I was looking for). The question is split into two: I have a matrix $X \...
kloop's user avatar
  • 131
1 vote
2 answers
331 views

Anti-concentration of gaussian variable

Let $X$ be $\mathcal{N}(\mu,\sigma^2)$ gaussian. Its expectation $\mu$ is positive. Can we derive a lower bound on $$\mathbb{P}(X\geq\epsilon)\geq g(\epsilon,\mu,\sigma) \text{ where } \epsilon\leq\mu$...
tony's user avatar
  • 405
1 vote
1 answer
281 views

A uniqueness proposition involving Erf, the error function

This is a generalization of a previous MO question, "Reducing system of equations involving Erf, Error Function". Consider the system of equations: $$1/2 + {\rm Erf}(x) - \alpha {\rm Erf}(\frac{x+y}{...
Johan Ugander's user avatar
1 vote
1 answer
140 views

Reference request: Cover times, Mixing Times and DGFF applied in statistics?

I am trying to find if in active research in statistics, there is interest in mixing times, cover times of graphs, and/or the discrete Gaussian free field? I haven't found anything so far for the ...
noitseuq's user avatar
1 vote
2 answers
388 views

Average Multivariate Gaussian

Suppose we have a (possibly infinite) collection k-variate gaussian distributions $\{(\mathcal{N}(\mu_{\lambda}, \Sigma_{\lambda}))\}$ ($\lambda$ is just a label), and for each distribution $\mu \in [-...
user76775's user avatar
1 vote
1 answer
613 views

Integral of the product of a gaussian pdf and cdf

I am trying to solve the integral of a gaussian cumulative distribution function and a gaussian probability function. On this site I have seen solutions of similar, less general integrals (e.g. ...
Kurt Z.'s user avatar
  • 11
1 vote
1 answer
115 views

Supremum of centered jointly generalized chi-square random variables

Let $\zeta_n$ be a sequence of centered jointly generalized chi-square random variables, i.e. $\zeta_n = \sum_{k=1}^{m_n} a_{k,n}(\xi_{k,n}^2 - E[\xi_{k,n}^2])$, and $\xi_{k,n}$ are centered jointly ...
zhoraster's user avatar
  • 1,533
1 vote
1 answer
207 views

Computing probability that $Ax\geq0$ where $x$ is a vector of iid gaussians and $A$ is matrix of $1$s and $0$s

This question came up in my research: What is the probability that $Ax\geq0$ where $x$ is a vector of iid gaussians and $A$ is matrix of $1$s and $0$s? So far I only figured out that I can do Monte ...
nivwusquorum's user avatar
1 vote
1 answer
241 views

Expectation of top-K selection of squared Gaussian random variables

Let us have $$ Z = [z_1, z_2, \dots, z_n], $$ where $z_i \sim N(0, \sigma^2)$ and are iid. Additionally, consider $$ X_k := \{ x \in \{0, 1\}^n : e^T x = k \} $$ If $Y = \max_{X \in X_k} |Z^T X|^2,$ ...
Alireza Khayatian's user avatar
1 vote
0 answers
104 views

Efficient evaluation of multidimensional kernel density estimate

Edit I have copied this discussion to the stats community site here, since I feel it is more relevant. Please feel free to close this in due course. I've seen a reasonable amount of literature about ...
Gabriel's user avatar
  • 161
0 votes
1 answer
110 views

Positivity of linear combination of gaussian variables

Consider a collection of independent standard Gaussian variables $w_i$ for $i = 1, 2, \ldots, N$. Define its linear combination $f:=\sum_{i=1}^Na_iw_i+b_i$, where $a_i=pb_i$ ($p$ is a fixed parameter),...
happyle's user avatar
  • 49
0 votes
1 answer
99 views

Expressing a multivariate normal distribution as a mixture of uniform distributions?

Context: Given a scalar normal distribution $X\sim \mathrm{N}(\mu, \sigma^2)$, it is possible to express $X$ as a mixture of uniform distributions over intervals (compound probability distributions), ...
PiePiePie's user avatar
0 votes
1 answer
61 views

What can we say about the order of convergence of a critical point of Gaussian mixture density to its limit when the parameter $h$ goes to $0?$

Density of Gaussian mixture with $n$ components is given by: $$f(x):=C \sum_{i=1}^{n}e^{-\frac{1}{2}||\frac{x-x_i}{h}||^2}, x_i \in \mathbb{R}^d, h > 0$$ where $C$ is a normalization constant ...
Learning math's user avatar
0 votes
0 answers
29 views

k-means errors for a block Gaussian vector

Consider a standard centered Gaussian vector $(X_1,...,X_n)$ with an approximate block structure, i.e. there is $q$ and a partition of $\{1,...,n\}$ in $q$ classes such that if $i,j$ are in the same ...
kaleidoscop's user avatar
  • 1,352
-2 votes
1 answer
92 views

Existence or impossibility of Gaussian factory

Gaussian factory problem: given an iid sequence $x_i \sim \mathcal{N}(\mu,\sigma^2)$, $i=1,2,\dots$, with $\mu$ and $\sigma^2$ both unknown, construct a realization $y \sim \mathcal{N}(0,1)$.
Sebastian Nowozin's user avatar