Questions tagged [gaussian]

For questions about the Gaussian probability distribution, its definition, properties and use.

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Gaussian width after some linear transformation

The Gaussian width $w(T)$ of a set $T\in \mathbb{R}^n$ is defined as follows: $$ w(T) = \mathbb{E}\sup_{x\in T} \langle g,x\rangle $$ where $g$ is a random normal vector in $\mathbb{R}^n$. The ...
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20 views

Gaussian subdivision surfaces paper, typo?

I am reading a CG paper that came on 2019 and I am stuck at one section. After thinking about my issue I am starting to believe the author made a typo, but the likelihood of me being wrong is much ...
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27 views

Any probability density can be written as a sum of Gaussian

In Gaussian sum mixture model, any probability density function (pdf) can be written as a sum of Gaussian. Lets consider here any $n$-dimensional vector $x$ follows Gaussian distribution with mean $\...
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7 views

invert sum of Gaussian CDFS

Let $\Phi(x)$ be a the Standard Normal Univariate Gaussian CDF function, $\beta \in(0,1)$, and the standard deviations $\sigma_i$ be positive. Does anybody have intuition into solving the following ...
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9 views

reduce the dimensionality of multidimentional Gaussian process

Say my training data is $(\mathbf{x_i},y_i), i=1..n$ where $\mathbf{x_i}$ are the inputs and $y_i$ are the labels. Also, each $\mathbf{x_i}$ is a vector of two numbers $\mathbf{x_i}=(x_{i,1},x_{i,2})$....
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1answer
38 views

Maximum of integration of function over multivariate gaussian w. r. t. mean parameter

Suppose a function $f : \mathbb{R}^n \to \mathbb{R}$ that takes an unique maximum at $x_0$ (and is sufficiently well-behaved, e. g. Continuos) Suppose that for a fixed $\Sigma$, for all $\mu$, $ g(\mu)...
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16 views

Graphical models for regression problem

I am studying about the Gaussian graphical model (GGM). I have a $N\times D$ matrix X of my observations. The structure of the network has been found by using the graphical lasso method. It means I ...
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1answer
24 views

regarding the concept of infinite long vector and function

When learning the Gaussian process, there is a concept stating that " infinite long vector is similar to function", which is shown in the attached image. What does it mean, I am not very clear about ...
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12 views

Can properties for (circular symmetric) complex random matrices automatically work for real random matrices?

I am dealing with a theorem which relates to circularly symmetric complex Gaussian random matrices (CSGRM). It seems quite tempting to assume that the theorem also extends to real-valued Gaussian ...
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1answer
22 views

Is there an easy way to find the new mean of a product of gaussians?

Let's assume we have 2 gaussian functions Allowing some notation abuse they would look like: $$G_i(x) = Ce^{\frac{-1}{2}[(x-\mu_i) / \sigma_i]^2}$$ $$G_j(x) = Ce^{\frac{-1}{2}[(x-\mu_j) / \sigma_j]^...
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29 views

$\mathbf{P}\left(2<\left|X\right|+\left|Y\right|<3\right)$ where the random variables are independent $N(0,1)$?

If $X$ and $Y$ are independent $N(0,1)$ random variables, then how can I calculate the following probability: $$\mathbf{P}\left(2<\left|X\right|+\left|Y\right|<3\right)?$$ I calculated the ...
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9 views

Average error of quantization [closed]

X is normal random variable with mean 0 and variance 1. Quantizer is Q(x) = [Round(8x)](round to its nearest integer) Then what is the error of quantization? E[(Q(x)-x)^2]
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14 views

Fitting a Second Gaussian Mixture Model

I have a Gaussian mixture, $\sum_{i=1}^{K_1} v_{i} \mathcal{N}(x; a_{i}, A_{i})$. I am given another set of $K_2$ Gaussian centers, $z_{j}$, and I want to find the optimal parameters $\Sigma_{j}$ to ...
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12 views

Gaussian distribution distance function

I would be happy to get guidance for the following questions: I have a dice with 3 sides. side A: have a 50% chance of falling on it. side B: have a 30% chance of falling on it. Side C: have a 20% ...
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23 views

How to show that $\exp(X)$ has a lognormal pdf if $X$ has a normal pdf

Given: $X \sim \mathcal{N}(\mu, \sigma^2)$ Show that $\exp(X)$ has a lognormal distribution. Attempt: \begin{align} \text{Let } Y = e^X \ \ \text{and} \ \ Y \sim p_Y(y) \\ P(Y \leq t) = \...
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15 views

What is the Mathematical Terminology for Gaussian Paths

I am looking for the terminology for the path object that is Gaussian in nature, in other-words the path has a mean along it direction and variance in perpendicular direction as illustrated by the ...
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12 views

Gaussian process properties and stationarity

What properties of the Gaussian Process makes it so unique? Im trying to get more intuition, or a proof for some of these theorems : If X is a Gaussian Process and iid therefore X is a SSS process (...
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10 views

Probability of observing a multivariate Gaussian given another Gaussian

I have two multivariate, $n$-dimensional Gaussians: $g_a = \mathcal{N}(x; a, A)$ and $g_b = \mathcal{N}(x; b, B)$. Three parameters are known: $a, A, b$ and I want to compute the maximum-likelihood ...
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14 views

Multivariate Gaussian density function notation

I need to come up with an explicit expression for the transition density $p(s_{k+1}|s_k)$, which is engineering notation for $P_{S_{k+1}|S_k=s_k}(s_{k+1})$. I have derived that this is equal to the ...
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8 views

Bounding expectation of a function Gaussian random vector

Let $x\sim N(0,\Sigma)$ be a normal random vector. I am interested in finding lower and upper bound for the following expectation $$\mathbb{E}||V\varphi(Wx)-\tilde{V}\varphi(\tilde{W}x)||^2$$ Where $...
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5 views

Fit Gaussian curve given a set of 2D points, expected value and variance

Hi guys I'm facing the problem wrote in the title, could you help me please? I have a set of 10 2D points with x in [1,2,3,..,10] and $$\sum_{i=1}^{10}y_{i}=1$$ It is required to fit a new ...
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Gaussian Isoperimetric Inequality Metric Restrictions

Does the Gaussian Isoperimetric Inequality extend to all metrics, or is it restricted to n-dimensional Euclidean space? I understand the inequality and its implications, but I am unclear on whether ...
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2answers
26 views

The difference between standard deviation $=0$ and no standard deviation [closed]

I would like to ask if there is the difference in the interpretation standard deviation of the Gaussian $=0$, and there is no standard deviation for the Gaussian. Is it the same? Or there is some ...
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6 views

Total variance distance between high-dimension Gaussians with different means

I am trying to compute the total variation distance, also called total variance distance, and by abuse of notation sometimes even statistical distance (at least in the field of research of lattice-...
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1answer
24 views

Sampling from a Multivariate Gaussian whose covariance form is given by Cholesky

I've read a paper "structured uncertainty prediction networks", and I don't understand how to sample from a multivariate Gaussain in the paper. Here is a sampling method used in the paper. Suppose ...
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26 views

How should one interpret the Normal/Gaussian Distribution function? [duplicate]

While studying linear regression, my text book indicates that we account for noise, which is given by the function: $$ \varepsilon_i \sim \mathcal{N}(0,\beta^{-1}).$$ I know this is a normal ...
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36 views

Cant figure out what this symbol means

Ive been reading a paper and I cant seem to understand what this cursive symbol "N" means ath the beginning. Its in the context of a staircase function and Kac-Rice method. Its meant to be a function ...
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7 views

What Does Gaussian Process Become When the Non-Linearity ReLU Is Considered?

As the simplest regression problem, $y=wx+\epsilon$, $\epsilon\sim\mathcal{N}(0,\sigma^2)$, priori is $w\sim\mathcal{N}(0,\tau^2)$, then we have a Gaussian Process $\mathbf{f}|\mathbf{x}\sim\mathcal{N}...
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1answer
27 views

A simple question for a Joint Gaussian Random variable. Beginner.

I am a beginner in probability theory. The Wikipedia definition of joint Gaussian random variables is https://en.wikipedia.org/wiki/Multivariate_normal_distribution#Notation_and_parametrization it ...
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17 views

derivation of 2d/integration of gaussian distribution/bayesian inference

I don't know how to calculate the integration of many pdf which are Gaussian distributions. Why is $\frac{1}{2\pi }\sqrt\frac{J_{vest}J_{vis}J_{obj}J{p}}{2\pi(J_{vest}+J_{vis}+J_{obj}+J{p})}e^{^{-\...
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20 views

Finding multiple peaks of gaussians?

If I have a Gaussian function $f(x)=c e^{-(x-a)^2}$. I can calculate the $x$ value where it peaks using: $$a = \frac{\int_{-\infty}^{\infty} x f(x) dx}{\int_{-\infty}^{\infty} f(x) dx} $$ But ...
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1answer
18 views

Expectation of square of Gaussian random variable.

Let $Z\sim \mathcal{N}(0,1)$. For $\lambda <1/2$, why does this hold? $$ \mathbb{E}\left[\exp\left(\lambda\left(Z^2 - E[Z^2]\right)\right)\right] = \exp\left(-\frac{1}{2}\log(1-2\lambda) - \...
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Estimate of mean and variance of a Gaussian random variable from its error function

Let's say we have a Gaussian random variable X with mean and variance to be m and s, which are unknown. What can be observed is the error function sampled at a few values of $X=x_i$. Let say $\...
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10 views

Discrete sequence and surface optimization

Imagine I define two sequences, $v_n$ and $x_n$, intuitively the speed and position of a marble over a surface. If I redefine $v_n$ as an estimation variable $\hat{v}_n + z_n$, $z_n$ a zero-mean ...
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23 views

Expectation of function of normal vector

Given a random vector $z \sim CN(0,\Sigma)$ and $g(x)$ is a positive scalar function, there is an explicit expression for $E[g(z)zz^H]$?
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Are there other distributions over functions besides the Gaussian process?

The Gaussian process allows you to sample a continuous function $f(x)$ evaluated at arbitrary points $\vec{x}$, \begin{align} f(x) &\sim GP\left(\mu(x), K(x,x')\right). \end{align} Are there any ...
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30 views

Does it make sense to take the standard deviation of a uniform distribution of values?

You can get the mean and standard deviation of any set of numbers and come up with a Gaussian fitting them. What does it mean if you do this with another distribution, such as a uniform distribution? ...
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1answer
33 views

Can anyone helps me understanding this integral solution?

I was reading a research article in which authors used Chebyshev-Gauss Quadrature on an integral which is given as in the attached image. Can anyone help me out with any citation to understand how $...
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13 views

Standard deviation ellipse vs standard deviation curve (of 2D Gaussians)

I've read this two articles about the "standard deviation" of bi (and multi) variate gaussians: newer, older So lets have the following gaussian: $$f(x,y)=A\exp\bigg[-\frac{1}{2(1-\rho^2)}\bigg(\frac{...
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28 views

Nonlinear function of two Gaussians - Stein's Lemma

Let $g,h$ be independent standard normal variables ($\cal{N}(0,1)$). Fix $\sigma>0$ and pick $f:\mathbb{R}\rightarrow \mathbb{R}$. Under what conditions on $f$, we have that $$ \mathbb{E}[f(g+\...
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1answer
34 views

Two random variables not gaussian but whose sum is gaussian

It is well known that if $X$ and $Y$ are two independent gaussian random variables, or if $(X,Y)$ is jointly bivariate gaussian, then the sum $X+Y$ is a gaussian random variable. It is also known that ...
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30 views

Predictive distribution of SPGP

Eq(8) in Sparse Gaussian Processes using Pseudo-inputs states that \begin{align*} "p(y^*|x^*,D,\bar{X})=\int{p(y^*|x^*,\bar{X},\bar{f})p(\bar{f}|D,\bar{X})d\bar{f}}" \end{align*} which can be ...
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20 views

gaussian-bernulli mixture model MLE estimate

I am looking at a Gaussian mixture model which has the following probability: $P(X) = \sum_{i=1}^{N} \sum_{j=1}^{M} \log (\sum_{k=1}^{2} N(x_{ij} \mid \mu_k, \Sigma_k) p(z_{ij} = k \mid y_i) p(y_i))...
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38 views

A uniform distributed random vector on euclidean ball is sub gaussian

Consider a random vector $X\sim Uniform(B(0,\sqrt{n}))$, that is $X$ is uniformly distributed on the Euclidean ball $B(0,\sqrt{n})$ in $R^n$ centered at the origin with radius $\sqrt{n}$. $B(0,\sqrt{n}...
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1answer
26 views

how was the gaussian distribution developed? (question of an answer already done)

I was looking for an answer for this question, I found it here on math stackexchage, but there's something in the answer I did not understand. I tried to add a comment too the answer, but I couldn't, ...
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2answers
22 views

Joint Gaussian distribution implies Gaussian + independence?

Assume that $X = (X_1, ... , X_n)$ are jointly Gaussian distributed. Can I then say that each of $X_1, ... , X_n$ is Gaussian distributed? Can I deduce that they are pairwise independent?
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18 views

Distribution of Gaussian projected on orthogonal direction

Let $X$ be a matrix-valued random variable with each entries are independent Gaussian. Let $\mathbf{u}$ be the first left singular vector of $X$. What is the distribution if we project $X$ on $\mathbf{...
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Expectation: sigmoid times mixture of Gaussians

Let $Y$ be a random variable where \begin{align} Y&=Z\theta^*+W \end{align} Here $Z$ is a Rademacher random variable, $W\sim \mathcal{N}(0,\sigma^2 I_p)$, and $\theta\in \mathbb{R}^p$ is a fixed ...
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2answers
46 views

Searching for a second order ODE whose solution is bell shape (Gaussian function)

I'm studying a nonequilibrium dynamics of a stochastic system. I found that in mean-field approximation the numerical solution resembles a bell shaped function (Gaussian function) with is zero at ...
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31 views

What is a Gaussian kernel?

I am trying to understand a paper where the term "Gaussian Kernel" is used often. Upon first reading it I thought kernel and $\sigma$ (standard deviation) were synonyms, but upon reading this document,...