Questions tagged [gaussian]
The gaussian tag has no usage guidance.
243
questions
1
vote
1answer
50 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. ...
1
vote
1answer
40 views
Partial derivative of expectation and Stein's lemma
Currently, I am reading a paper about the Gaussian Process in Neural Network [1]. In the solution of the main result in this paper, the author applied Stein's lemma and claimed an equation about the ...
0
votes
0answers
32 views
What are the boundaries of the entropy of a gaussian random variable? What is their meaning? [migrated]
Given a gaussian random variable $X\sim \mathcal N(\mu,\sigma^2)$ with p.d.f.:
$$
f(x)=\frac{1}{\sqrt{2\pi\sigma^2}}\cdot e^{-\frac{(x-\mu)^2}{2\sigma^2}}
$$
Then we can calculate its entropy (...
3
votes
1answer
87 views
Derivative of an integral of a Gaussian
I'd like to compute the derivative of an expected value w.r.t one of the parameters that define the mean of a Gaussian:
$ Z=\int \mathcal{N}(x;\mu,\Sigma)f(x) \, dx $, then $ \frac{dZ}{dK}=\text{??}$ ...
0
votes
0answers
60 views
How to calculate $\int_{\mathbb{R}^n_{\geq0}} d\mathbf{w} e^{-\frac{1}{2}\|\mathbf{w}\|^2} [\mathbf{w\cdot x}]_+ $
I am trying to evaluate
$$\int_{\mathbb{R}^n_{\geq0}} d\mathbf{w} e^{-\frac{1}{2}\|\mathbf{w}\|^2} [\mathbf{w\cdot x}]_+ $$
where $\mathbf{w} \in \mathbb{R}^n_{\geq0}$ (a real vector in the positive ...
0
votes
1answer
68 views
Why is squared exponential kernel often used in Gaussian Process regression when the most standard case is time-like X?
I might be confused about something.
Consider doing inference on $Y'\mid X',Y,X$ using standard Gaussian Process Regression with 1d $Y$ and 1d $X$. Suppose $X$ is time-like (target is stationary or ...
1
vote
1answer
72 views
Estimating the average of two gaussians' mean with minimal squared error
This is a follow-up to my previous question.
Assume that $X\sim \mathcal N(\mu_1,\sigma_1^2)$ and $Y\sim \mathcal N(\mu_2,\sigma_2^2)$.
I want to estimate $\frac{\mu_1+\mu_2}{2}$ after observing $X,Y$....
1
vote
1answer
54 views
Estimating the average of two gaussians' mean
Assume that $X\sim \mathcal N(\sigma_1,\mu_1)$ and $Y\sim \mathcal N(\sigma_2,\mu_2)$.
I want to estimate $\frac{\mu_1+\mu_2}{2}$ after observing $X,Y$.
In my setting, $\sigma_1,\sigma_2$ are known ...
1
vote
0answers
37 views
Generalization of a Gaussian measure continuity result from Hilbert to Banach space
Da Prato/Zabczyk "Second Order Partial Differential Equations in Hilbert Spaces" states the following lemma (this is a reformulation of proposition 1.3.11 in their book):
Let $\mu = \mathcal ...
1
vote
0answers
25 views
What is the distribution of determinant of multi multiplication of some Gaussian matrices?
I have a square matric $H = (ABC)(ABC)^H$ where $A$ and $C$ are complex Gaussian matrices with some correlation matrices and $B$ is a diagonal matrix with entries $e^{j \theta}$ on the diagonal such ...
0
votes
0answers
50 views
Regularity of Gaussian process sample paths
Consider a Gaussian process on $[0,1]$ given by a kernel function $K: [0,1]^2\to\mathbb{R}$. Under what conditions can we conclude that the sample paths are $C^k$ with probability 1?
This question is ...
1
vote
1answer
65 views
Tail bound on the RKHS norm of a zero-mean Gaussian process
Let $f \sim \mathcal{GP}(0, K)$ be a zero-mean Gaussian process defined on a compact set $\mathcal{D} \subset \mathbb{R}^d$, where $K \colon \mathcal{D} \times \mathcal{D} \rightarrow \mathbb{R} $ is ...
2
votes
1answer
53 views
Limiting behavior of $k^{th}$ order statistics of n non-i.i.d chi square random variables
This is related to one of my previous questions here.
Let $(Z_1, Z_2, \ldots, Z_n)\sim N(0, \Omega)$, where $\Omega = (1-\mu) I_{n\times n} + \mu \boldsymbol{1}_n\boldsymbol{1}_n^\top $. Here $\...
1
vote
1answer
64 views
Norm contrained Gaussian distribution
Let $X$ be a multivariate normal $\mathcal{N}(\mu, \Sigma^2)$ and let $X$ be anisotropic, that is I am considering $\Sigma$ to be a diagonal matrix but the elements on the diagonal might be different.
...
0
votes
1answer
159 views
Obtaining the error term of binomial distribution's entropy from the differential entropy of a Gaussian distribution
It is known that the first order error term in the Shannon entropy formula for a binomial distribution is $1/n$ (for example, see the Wikipedia page Binomial distribution), where in the limit $n \to \...
0
votes
0answers
72 views
Change variables in Gaussian integral over subspace $S$
I have been thinking about a problem and I have an intuition about it but I don't seem to know how to properly address it mathematically, so I'm sharing it with you hoping to get help. Suppose I have ...
3
votes
1answer
120 views
Fast computation of convolution integral of a gaussian function
Given a convolution integral
$$
g(y) =\int_a^b\varphi(y-x)f(x)dx=\int_{-\infty}^{+\infty}\varphi(y-x)f(x)\mathbb{I}_{[a,b]}(x)dx
$$
where
$\varphi(x)= \frac{1}{\sqrt{2\pi}}\exp{\left(-\frac{x^2}{2}\...
2
votes
1answer
170 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}\...
4
votes
0answers
150 views
Approximation of integral of gaussian function over a parallelepiped
Remark: I posted this question in math stackexchange here and computer science stackexchange https://cs.stackexchange.com/ few weeks ago but obtain no answer.
Given a multi-dimensional gaussian ...
1
vote
0answers
53 views
Law of OU process with time-dependent dynamics
Fix a non-negative integer $k$ and let $M^1:\mathbb{R}^n\rightarrow \mathbb{R}^n$ and $M^2,\Sigma:\mathbb{R}^n \rightarrow \mathbb{R}^{n\times n}$ be $k$-times continuously differentiable functions, ...
3
votes
0answers
114 views
Seeking a precedent – two-stage Gaussian integration?
Sometimes, by iteration, linear algebra can be used to solve non-linear equations. For example, consider the system
$$Ax=a \qquad B(x)y=b(x), $$
where $a$ is a vector with scalar entries, $A$ is a ...
1
vote
0answers
90 views
Gaussian order statistics
Setup. Let $\alpha\in(0,1)$ fixed; and $\tau\in[0,1]$ (think of it very close to one).
Suppose $X_1,\dots,X_n$ are i.i.d. standard normal.
Let $Y_1,\dots,Y_n$ be another sequence of standard normals ...
1
vote
0answers
46 views
Explicit growth rate estimation of Gauss-Laguerre quadrature
The $n$-th Gauss-Laguerre quadrature scheme aims to approximate integral of exponentially decreassing function over $[0;+ \infty[$ by a finite sum, according to:
$ \displaystyle { \int _0 ^{+ \infty} ...
5
votes
1answer
171 views
Anti-concentration of Gaussian when conditioning on event
Let $v$ be a given vector with $\|v\|_{\Sigma^{-1}} \leq 1$, where $\Sigma$ is a positive semi-definite matrix and $\|v\|_{\Sigma^{-1}} = \sqrt{v^\top\Sigma v}$. Meanwhile, let $u$ be a random vector ...
1
vote
1answer
202 views
Ratios of Gaussian integrals with a positive semidefinite matrix
Cross-post from MSE
https://math.stackexchange.com/questions/4118128/ratios-of-gaussian-integrals-with-a-positive-semidefinite-matrix
Generally speaking, I’m wondering what the usual identities for ...
1
vote
0answers
34 views
How to characterize the variance of a linear Gaussian system with switching?
Consider a random process described by the following linear dynamics:
$$
x_{k+1} = a x_k + n_k,
$$
where $|a|<1$ and $n_k$s are i.i.d. standard normal distributed.
It is quite easy to prove that $...
1
vote
0answers
115 views
Relation satisfied by a Gaussian random variable
I want to prove the following relation for $X\sim \mathcal{N}(0,1)$, $x\in \mathbb{R}$ and $f(x)=\mathbb{E}[\max(X,x)]$:
$$f(\frac{f(x+1)+f(x-1)}{2})\leq \frac{f(f(x)-1)+f(f(x)+1)}{2}$$
It seems that ...
3
votes
1answer
135 views
Gaussian process kernel parameter tuning
I am reading on gaussian processes and there are multiple resources that say how the parameters of the prior (kernel, mean) can be fitted based on data,specifically by choosing those that maximize the ...
0
votes
1answer
48 views
Simplification on the estimation on error of the ratio of 2 random variables
Let $Z=\dfrac{X}{Y}$ the ratio of 2 random variables.
Distribution of $Z=\dfrac{X}{Y}$
Consider the case of two independent normal variables $X$ and $Y$ with strictly positive means and variances $\...
1
vote
1answer
39 views
Asymptotics of the right singular vectors as the number of rows diverge [duplicate]
Write $X_m \in \mathbb{R}^{m \times n}$ as a Gaussian ensemble, so that $(X_m)_{ij} \sim \mathcal{N}(0, 1)$ are independent and identically distributed. Assume that $m \geq n$. Write $X_m = U_m \...
2
votes
1answer
260 views
measure of a degenerate Gaussian distribution
I want to do computations with a degenerate Gaussian measure, but I do not know how to represent it
in a close form.
After starting with a Gaussian random variable and restricting it to a condition, I ...
6
votes
1answer
206 views
Ordering preference for two zero mean Gaussian outcomes
Let $X\sim \mathcal{N}(0,1)$ be a standard Gaussian random variable. If we let $f_a(x)\triangleq\mathbb{E}[\max\{aX,x\}]$ for $a,x >0$, how to prove that $$f_a(f_b(1))<f_b(f_a(1))~~\text{for }0&...
2
votes
0answers
42 views
The optimality of Kalman filtering
It is known that the Kalman filter estimates the state of the following system recursively.
$$x_{k+1}=Ax_k+w_k, \ \ w_k \sim \mathcal{N}(0,Q)$$
$$y_k=Cx_k+v_k, \ \ v_k \sim \mathcal{N}(0,W)$$
In the ...
0
votes
0answers
79 views
A non trivial example of a Gaussian semi-Markov process?
Let $(\Omega, \mathcal A, \mathbb P)$ be a probability space and $X=(X_t)$ a real Gaussian stochastic process.
Let $\mathcal F=(\mathcal F_t)$ be the filtration generated by $(X_t)$.
$X$ is Markov ...
21
votes
7answers
4k views
What makes Gaussian distributions special?
I'm looking for as many different arguments or derivations as possible that support the informal claim that Gaussian/Normal distributions are "the most fundamental" among all distributions.
...
3
votes
1answer
113 views
Design a random variable which has the maximal correlation with another random variable
$Y$ is a Gaussian distributed random variable with zero mean and known variance: $Y\sim N(0,\sigma_y)$. We measure $Y$ with a sensor, which is corrupted by white Gaussian noise: $Z=Y+V$; $V\sim N(0,\...
3
votes
0answers
87 views
Pedestrian proof of Gaussian chaos for order-two polynomial?
Let $\ell \geqslant 1$. Let us consider $(g_n)_{n \in \mathbb{N}}$ identically distributed independent real gaussian variables and real number $(a_{n_1,\dots n_{\ell}})_{(n_1, \dots, n_{\ell}s)\in\...
2
votes
4answers
386 views
Change of variables in a Gaussian integral in matrix form
I have a problem in which I have to compute the following integral: $$\mathop{\idotsint\limits_{\mathbb{R}^k}}_{\sum_{i=1}^{k}y_i=x} e^{-N^2r(\sum_{i=1}^{k}y_i^2-\frac{1}{k}x^2)} dy_1\dots dy_{k},$$
...
1
vote
1answer
92 views
Conditions for Gaussianity of SDE
Fix $T>0$, $x \in \mathbb{R}^n$, and let $\mu$ and $\sigma_1,\dots,\sigma_m$ be (globally) Lipschitz-continuous functions from $[0,T]\times \mathbb{R}^n$ to $\mathbb{R}^n$. Thus, for every $0\leq ...
0
votes
2answers
263 views
Bounds for the sum of dependent gaussian random variables
Let $X_1,...,X_n$ be $n$ gaussian random variables $N(0,1)$ not necessarily independent or jointly correlated, $S=\sum_{i=1}^n w_i X_i$ be the weighted sum of these gaussian variables (because $(X_i)_{...
2
votes
2answers
343 views
Concentration and anti-concentration of gap between largest and second largest value in Gaussian iid sample
Let $n \ge 3$ be an integer and let $X=(X_1,\ldots,X_n)$ be random vector with iid coordinates from $N(0,1)$. For $1 \le k \le n$, let $X_{(k)}$ be the value of the $k$th largest coordinate of $X$.
...
0
votes
0answers
41 views
Hardness of generating Gaussian distribution
I would like to ask about the computational complexity of the problem of generating integers so that the obtained distribution is asymptotic to the Gaussian distribution. Any related reference is very ...
1
vote
1answer
292 views
Definite integral of 2d Gaussian
Is there some analytic expression or even an approximation of the definite 2D Gaussian integral of the form: $$E=\int_a^b Dg \int_{cg+d}^\infty Dh$$ where $Dg=\frac{dg}{\sqrt{2 \pi}} e^{-g^2/2}$ and a,...
0
votes
0answers
55 views
How to compute the following probability involving two normal random variables?
$\alpha$ and $\alpha'$ are two independent standard normal random variables. What's the conditional probability $$\mathbb{P}[\alpha >0, \alpha' >0|c_1<|\alpha - \alpha'|<c_2],$$ where $c_1$...
2
votes
3answers
86 views
On the probability of the multivariate normal with fixed pairwise correlations being coordinate-wise non-negative
This problem itself, admittedly, is not a research problem; but rather an intermediate step I've encountered in my research.
Let $(X_i:1\le i\le N)$ be a multivariate normal random vector where i) ...
5
votes
2answers
167 views
Integrability of Gaussian sums
Let $(X_1, \ldots, X_n)$ be a Gaussian vector, and $Z = \sum_{i=1}^n |X_i|$.
Since the map $x \mapsto e^{x^2}$, is convex, for any $t>0$
$$
e^{tZ^2} \, = \, e^{t \big(\sum_{i=1}^n |X_i| \big)^2}...
0
votes
1answer
65 views
Algorithm for economically sampling method for Gaussian matrix product
Let $A$ be an $n\times n$ random matrix with i.i.d. $N(0,\sigma)$ entries, for some $\sigma>0$ and let $x\in \mathbb{R}^n$. A direct computation shows that $Ax \sim N(0,\sigma x^{\top}x)$.
I would ...
0
votes
0answers
307 views
Does additive Gaussian noise preserves the Shannon entropy ordering?
Suppose that $Z$ is a Gaussian random variable independent of $X$ and $Y$. Moreover suppose that $h(X) \geq h(Y)$, where $h(\cdot)$ is the differential Shannon entropy.
Does relation $h(X+Z) \geq h(Y+...
1
vote
0answers
61 views
Bayesian inference of stochastically evolving model parameters
I have a question related to self-calibration in radio interferometry, but I will try to phrase it as generic as possible. I have a set of data points, $D = \{ d_{0, t_0}, d_{1, t_0}, ..., d_{M, t_0}, ...
1
vote
1answer
392 views
Convolution of two Gaussian mixture model
Suppose I have two independent random variables $X$, $Y$, each modeled by the Gaussian mixture model (GMM). That is,
$$
f(x)=\sum _{k=1}^K \pi _k \mathcal{N}\left(x|\mu _k,\sigma _k\right)
$$
$$
g(y)=\...
