Questions tagged [gaussian]

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

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Two definitions of a Stochastic Process?

I have two supposedly equivalent definitions of a stochastic process. A stochastic process is an indexed set of random variables. Specifically $$ y = \{y(x) \; | \; x \in \mathcal{X}\}. $$ ...
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Property $E[\mathbf{xx}^T] = \mathbf{\Sigma}-\mathbf{\mu\mu}^T$.

I'm reading through Principles of Machine Learning by Murphy for the purpose of understanding Gaussian Processes. For now I just want to understand the multivariate Gaussian distribution but I'm ...
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Bound KL divergence between two distributions by KL divergence of two Gaussian mixture models

I'm trying to bound the KL divergence between two continuous random variables with the KL divergence between two Gaussian mixture approximations motivated by the fact that the Gaussian mixture model ...
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Modeling a sequential Random Vector as a Random Processes

I'm studying random vectors. For a problem that I'm studying, I have a random vector that has a sequential nature. I know that modeling random variables as sequential processes is considered in the ...
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projection of a non-zero mean Gaussian vector into a Ball

Let $d$ denote the dimension, $\mathbf{B}_d$ denote the ball of radius one in $\mathbb{R}^d$. For $x\in \mathbb{R}^d$ let $\Pi_{\mathbf{B}_d}(x) = \frac{x}{\max\{1,\|x\|_2\}}$. Consider a fixed vector ...
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How do I prove that the distribution of $x_t$ in $x_t=x_{t-1}e^{-μt}+ θ(1-e^{-μΔt}) + \int_{t-1}^t e^{-μ(t-s)}\sigma dB_s $ is a normal distribution?

$B_s$ is brownian motion. Because $\int_{t-1}^t e^{-μ(t-s)}\sigma dB_s $ has a Brownian component, it is normally distributed according to Taylor & Karin 1998 's Introduction to Stochastic ...
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Derivation in paper Deep Neural Networks as Gaussian processes in ICLR 2018

I am trying to understand the derivation of the main equation in the seminal paper titled Deep Neural Networks as Gaussian processes (in ICLR 2018). Following is the equation number (7), which can be ...
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What is the integral of a 2D Gaussian over a disk centered at the origin

Formal Statement Let $G$ be a 2D symmetric Gaussian such that $G(x, y; x_0, y_0, \sigma) = \exp\left(-\frac{\left( (x - x_0)^2 + (y - y_0)^2 \right)}{2 \sigma^2} \right)$, for real $x, y, x_0, y_0$, ...
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Solving Backward Heat Equation with a Backward Heat Kernel?

Let $D>0$ be a constant. Imagine we have the following forward heat conduction problem: \begin{align*} \begin{cases} \partial_t u = D \partial_x^2u &, \quad (x,t) \in \mathbb{R} \times (0, \...
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How do we compute an integral over a unit simplex?

Let $ n \ge 2 $ and $ T > n $ be integers. The joint-distribution of eigenvalues in the Wishart ensemble subject to the underlying covariance matrix being equal to an identity matrix is given as ...
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Jacobian in multidimensional Gaussian integral

For the first integral shown in: reference for multidimensional gaussian integral It is mentioned that the Jacobian is 1, how is this the case? By following the calculation I found it to be $det(diag(...
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Hanson-Wright Inequality Lowering the Constant to Obtain a Factor of 2 Instead of 4

I'm reading the proof of Hanson-Wright Inequality from Rudelson & Vershynin's paper "Hanson-Wright inequality and sub-gaussian concentration". (https://arxiv.org/pdf/1306.2872.pdf) I'm ...
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Gaussian tail probability

Let $z(q)$ be the quaitle of a standard normal random variable $Z$, i.e., $z(q) = k$ when $Pr(Z\geq k) = q$. Then I would like to know why the following two results hold. (a) If we hold $\alpha$ fixed,...
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Rotation Invariance of the Distribution of $\ell_2$-norm square of Gaussian Vector

In a paper I'm reading I saw the argument below: Let $G\sim\mathcal{N}(\mathbf{0},\mathbf{I})$ and $A$ is a square matrix with SVD $A=U\Sigma V^T$ where $\Sigma=diag(\sigma_1,\dots,\sigma_n)$. Then $|...
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Finding the probability of an increment of a Brownian Bridge

I've been given a process $X_t=(1-t)B(\frac{t}{1-t}),0\leq t\leq 1$ where $B(\frac{t}{1-t})$ is a standard Brownian Motion. So far, I have proven this is a Brownian bridge. Now I need to find $P(X_{4/...
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Approximate 3D function that has certain points as peak

Given set of 3D points, how can I find a continuous function that has each point as the peak? You can imagine making a terrain editor that the inputs are 3D points and the output is a smooth mountain ...
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Frequency that a discrete brownian motion with drift process exceeds its previous maximum

This question relates to the frequency with which a discretely observed brownian motion exceeds its all time high. Assume my retirement portfolio's value can be modelled as a process with normally ...
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Computing a puzzling expectation with random variables and vectors.

Suppose the random variables $z_i $ are i.i.d. draw from the standard normal distribution $\mathcal{N}(0, 1)$, two vector $\mathbf{v}=(v_1, 0)^\top$, $v_1>0$, $\mathbf{w}=(w_1, w_2)^\top$, the ...
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Derivation of the integral form of the numerator in the Bayesian inference equation???? (not on the denominator)

In the reference Gaussian processes: iterative sparse approximations by Csató, Lehel (Csató, Lehel. Gaussian processes: iterative sparse approximations. Diss. Aston University, 2002), on page 20, ...
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Number of estimated parameters in gaussian mixture model

Considering a gaussian mixture model with $n$ components, the model contains $n-1$ weight parameters to estimate and $2\cdot n$ parameters for mean and and variance to estimate. In total this model ...
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Covariance function of the square of a Gaussian process

Let $X_t$, $t\in\mathbb{R}$, be a Gaussian process with mean $0$. Prove that $$ Cov(X_s^2,X_t^2)=2Cov(X_s,X_t)^2 $$ I don't know how to handle the squares here.
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Farmer wants to know how wet their field is

Problem A farmer wants a better understanding of rainfall on their field. Assuming rain falls randomly and with equal likelihood over the entire field, the farmer thinks they can model the volume of ...
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Mean of Cubic and Quartic forms of Gaussians

I am trying to calculate the following means: $$ E[ (x-\mu_k)b^T(x-\mu_l)(x-\mu_l)^T ] $$ $$ E[ (x-\mu_k)(x-\mu_k)^TA(x-\mu_l)(x-\mu_l)^T ] $$ Where x is some multivariate gaussian random variable. I ...
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Distribution of Complex Gaussian random variables after multiplication

I am very new to this topic, so please excuse me if the question is too naive. I have the following equation: $y= hx+n$ where $h \sim \mathbb{CN}(0,1)$ and $n \sim \mathbb{CN}(0,N_0I)$, $h,n$ and $y$ ...
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Does a Gaussian random walk lead to a Gaussian distribution in the limit, even when the initial state is non-Gaussian?

Suppose we have an initial random variable, which is not Gaussian, but has mean $0$, std $1$. Now we add $N$ unit Gaussian variables to this initial random variable, and then renormalize to mean $0$, ...
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If $X$, $Y$, and $Z$ are Gaussian, and $E[X|Y,Z]$ is linear in $Y,Z$, does it mean that $(X,Y,Z)$ is Gaussian?

Let $X$, $Y$, and $Z$ be Gaussian random variables that are not independent. This does not imply that $(X,Y,Z)$ is a Gaussian vector. But if we add the conditions that $E[X|Y,Z]$ is a linear function ...
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How to calculate the properties of the resultant 2D Gaussian from the deconvolution of one anisotropic, 2D Gaussian with another?

Anisotropic, two-dimensional Gaussians, are described by the equation (from here): $f(x,y)=A\exp \left( -(a(x-x_0)^2 + 2b(x-x_0)(y-y_0) + c(y-y_0)^2 )\right)$ where A is the amplitude of the peak, $...
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Joint distribution of the Sum of gaussian random variables

Suppose $X_1,X_2,X_3$ are iid with distribution $\mathbb{N(\theta, \sigma^2)}$ and $Y_1 = X_1 + X_2$ and $Y_2 = X_2 + X_3$. I need to find the joint distribution of $Y_1, Y_2$. Here is my attempt: ...
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Limit of expectation of maximum of standard normal distribution

I cannot find the way to show that if $X_1,\dots,X_n$ are independent standard normal random variables, then $$ \lim_{n \rightarrow \infty} \frac {\mathbb{E} \max_{i=1,\dots,n}X_i}{\sqrt{2\log n}} = 1 ...
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high dimensional probability vershynin Exercise 6.7.5

Lemma 6.7.4(Symmetrization with Gaussians). Let $X_1,\dots,X_N$ be independent, mean zero random vectors in a normed space. Let $g_1,\dots,g_N \sim N(0,1)$ be independent Gaussian random variables, ...
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Comparing the Training Costs of Machine Learning Algorithm: A Mathematical Perspective

Recently, I was looking at the optimization functions required in training Kernel Based Methods compared to Neural Networks. 1) Kernel Methods: For instance, I was looking at the optimization in ...
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Why the scale factor of the product of two gaussian functions is the convolution of the same gaussian functions?

The product of two Gaussian functions $$ f(x)=\frac{1}{\sqrt{2 \pi} \sigma_{f}} \exp\left(-\frac{x^{2}}{2 \sigma_{f}^{2}} \right) \quad \text { and } \quad g(x-y)=\frac{1}{\sqrt{2 \pi} \sigma_{g}} \...
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Existence of non-volume preserving automorphisms of normal distribution

Let $X=\mathbb{R}^d$, let $p$ the standard normal distribution on $X$, with zero mean and identity covariance, and let $f:X \to X$ be a diffeomorphism that preserves this normal distribution. For ...
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How to do regression for this stochastic process?

I wonder if there is some good way to determine the parameters for a random process. In my research work, I have a random process of the following form: $$ X_t=\alpha_{1}G_{t-1}+\alpha_{2}G_{t-2}+...+\...
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Is the distribution of a random variable normal if it is conditionally normal and the conditioning variable is normal?

Suppose we have a variable $\xi\sim N(\mu_2,\sigma_2)$ and another conditional variable $\eta | \xi \sim N(\mu_1,\sigma_1)$. Is $\eta$ normally distributed, and if so what is its mean and variance? I ...
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Representing rectangular function with Gaussian series

Is it possible to represent the rectangular function with an infinite series of scaled Gaussian functions? https://en.wikipedia.org/wiki/Rectangular_function Thank you, Alex
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Creeping for a Levy processes with infinite total mass

I am reading Theorem 7.11 from the book on Levy processes by Kyprianoy. I cannot intuitively grasp this: In this theorem it is said that if the process has a Gaussian component then it creeps upwards. ...
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PRML - Ex. 1.20. Behavior of probability mass & density of high-dimensional Gaussian distribution

I am following PRML by Bishop. At exercise 1.20, there is a question I am not quite able to understand the significance of. Would appreciate clarification around the same. Question 1.20 - In this ...
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Distribution of multivariate gaussian under linear constraints

Let $Z_1 \ldots Z_n$ be i.i.d Gaussians and introduce a set of linear constraints 1 through k, where the $j$'s constraint is the a linear combination of the $Z$'s take a specific value, i.e. $ \sum_{i=...
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Boundedness of Gaussian Process for Borell-TIS inequality

I have a question regarding the Borell-TIS Theorem which presents a result for a Gaussian process. An assumption is that the Gaussian Process $\{f_t\}_{t\in T} $ is almost surely finite, i.e. $P(\sup_{...
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Understanding well the definition of a Gaussian vector according to Le Gall's book.

I am reading the book "Brownian Motion, Martingales, and Stochastic Calculus", Jean-François Le Gall and I have a doubt regarding the Gaussian vector definition. First, the enveiroment is $(\...
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Computation of the Laplace transform of the Gaussian heat kernel

Currently I'm interested in the Laplace transform of the Gaussian heat kernel $$ k_t(x):=\frac{1}{(4\pi t)^\frac{d}{2}}e ^{-\frac{|x|^2}{4t}}. $$ Using the Laplace transform $$ G_\lambda(x)=\int_0^\...
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How to generate random number of non-standardized t distribution

Foreword: In the sparse Bayesian learning, the signal of interest $\mathbf{x}$ is assumed to follow conditional Gaussian distribution $p(\mathbf{x}|\boldsymbol{\alpha})=\prod_{i=1}^N\mathcal{N}(x_i|0,\...
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Martingale with Gaussian marginals that is not jointly Gaussian

Let $X_0,X_1$ be random variables such that $E[X_1|X_0]=X_0$, $X_0 \sim \mathcal N (0, \sigma_0^2)$, and $X_1 \sim \mathcal N (0, \sigma_1^2)$ (notice that $\sigma_0^2 \leq \sigma_1^2$ by the ...
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Uniform-Gaussian mixture

enter image description hereWould you help me finding the probability density function of a mixture of Uniform random variable with Gaussian random variable. Z=x*y Where x is Uniform distribution and ...
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Quantiles of pairwise distance between samples from a multivariate Gaussian distribution?

Assume that I have a $D$-variate multivariate Gaussian distribution $\mathcal{N}\left(\mathbf{\mu},\mathbf{\Sigma}\right)$ with known mean $\mathbf{\mu}$ and covariance $\mathbf{\Sigma}$. I could draw ...
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There's a metric to express the disagreement of predictions mathematically?

Let be $y = \{a_1, a_2, ..., a_n\}$ the distribution of predicted values given a variable $x$ such that $\sum_{k = 1}^K a_k = 1$. Let's consider now that for a given $x$ in fact we have $\{y\}_1^T$ ...
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exponential power distribution

I see two different definitions of the exponential power distribution: http://www.math.wm.edu/~leemis/chart/UDR/PDFs/Exponentialpower.pdf $$f(x)= \exp(1-\exp(\lambda x^k)) \exp(\lambda x^k) \lambda k ...
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A dimension-free upper bound for $\| X \|_{\infty}$ when $X \sim N(0,\Sigma)$

For a $p$-dimensional random vector $X \sim N(0,\Sigma)$, it is known that with high probability $$ \| X \|_{\max} \le C \sqrt{\log p} $$ where $C$ is some constant possibly depending on $\Sigma$. I ...
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Question about PDF and probability in continuous distribution.

The integral of a pdf is the probability, and the probability of drawing a specific value from a continues distribution is zero. So why can we compute the probability of, say drawing $x=1$ from $N(0, ...
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