Questions tagged [gaussian]
Gaussian functions / distributions / processes...
278
questions
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33
views
Constructing a Gaussian process on $[0, 1]$ such that the sample paths are $1$-Lipschitz continuous with high probability?
In the paper [1] the authors demonstrate that for a centered Gaussian process $\{X_t\}_{t \in [0, 1]}$, if there is a constant $C > 0$ such that
$$
\mathbb{E}[(X_t - X_s)^2] \leq C~(t- s)^2,
$$
...
- 253
0
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0
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106
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When is the image of $T \colon \ell^2 \to \ell^2$ a Gaussian random variable?
In finite dimensions, if $T$ is a linear operator and $x$ is a (centered) Gaussian random variable, then $Tx$ is again a (centered) Gaussian random variable.
Now suppose that $x$ is a (say, centered) ...
- 253
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votes
1
answer
53
views
Order of orthant probabilities in a prolate multinormal distribution
This is inspired by the negative answer to the conjecture in Which orthant probabilities are the largest? (For a multivariate normal distribution).
Suppose $X$ has the $k$-dimensional multivariate ...
- 3,092
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0
answers
19
views
Distribution of the second bigger value among m Gaussian draws
$\DeclareMathOperator\erf{erf}$
Let's consider the set $\Omega_m$ of $m$ i.i.d. Gaussian variables $\{X_1, X_2, \dots, X_m \}$, with $X_i \sim \mathcal{N} \left( 0, \sigma^2 \right) \forall i$.
$\...
- 109
2
votes
1
answer
104
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Probability distribution of vectors obtained from Gram-Schmidt process on i.i.d. Gaussian vectors
Given $N$ vectors in $K$ dimensions that are independently and identically distributed according to a Gaussian distribution with mean $0$ and standard deviation equal to an identity matrix, what is ...
- 29
6
votes
1
answer
103
views
Upper-bound on the Fisher-Rao distance between multivariate Gaussian measures by the KL-divergence
Let $\mu$ and $\nu$ be two multivariate Gaussian measures on $\mathbb{R}^d$ with non-singular covariance matrices. Can the Fisher-Rao distance $d(\mu,\nu)$ computed on the information manifold of non-...
4
votes
2
answers
166
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Integral of a product between two normal distributions and a monomial
The integral of the product of two normal distribution densities can be exactly solved, as shown here for example.
I'm interested in compute the following integral (for a generic $n \in \mathbb{N}$):
$...
- 109
3
votes
1
answer
67
views
Convolution between normal distribution and the maximum over $m$ Gaussian draws
$\DeclareMathOperator\erf{erf}$
Let's consider the Gaussian distribution $P_X(x)= \frac{1}{\sqrt{2 \pi \sigma^2}} e^{- \frac{x^2}{2 \sigma^2}}$. Now consider the random variable $W \equiv \max \{ X_1, ...
- 109
1
vote
1
answer
68
views
Conditional differential entropy of sum of Gaussians
Is it possible to give an expression for the conditional differential entropy $h(A+B\mid C+D),$ where $A,B,C,D$ are normally distributed with known standard deviations $σ_A,\ldots,σ_D$ and where all ...
1
vote
0
answers
87
views
Expectation of inverse of complex Gaussian variables
If we consider a complex Gaussian random variable as $h\sim\mathcal{CN}(0,\gamma)$, where $\gamma$ is the variance. Is there any closed-form solution with $\gamma$ for $\mathbf{E}\left[\frac{1}{\lVert ...
- 11
8
votes
1
answer
439
views
Concentration bounds for martingales with adaptive Gaussian steps
Consider the following martingale: $X_1 \sim \mathcal{N}(0, 1)$, and for any $n > 1$, $X_n \sim \mathcal{N}(X_{n-1}, X_{n-1}^2)$ (notice, this is a conditional distribution given $X_{n-1}$).
I am ...
- 83
5
votes
3
answers
485
views
The relative error of approximating a binomial
Are there any good approximations for a binomial CDF that work well in terms of the relative error, as opposed to absolute? For the usual normal approximation, the absolute error is very well-studied ...
- 3,838
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votes
1
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121
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Comparison of Rademacher and Gaussian expected values under linear transformations
As per suggestion, I have decided to post the following as a new question, but it is a follow-up to this one: Comparison of Rademacher and Gaussian moments under linear transformations
Let $X$ be an $...
2
votes
1
answer
130
views
$\Psi$ in finite Wiener–Itô Chaos implies existence of continuous representative on neighborhood of Cameron–Martin space?
Let $(\Theta, H, \mu)$ be an abstract Wiener space, i.e. let $(\Theta, \lVert \cdot \rVert_{\Theta})$ be a separable Banach space, let $(H, \langle \cdot, \cdot \rangle_{H})$ be a separable Hilbert ...
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1
vote
1
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70
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Comparison of Rademacher and Gaussian moments under linear transformations
Let $X$ be an $n$ dimensional standard Gaussian and let $U$ be an $n \times n$ orthogonal matrix. Then, the random vector $Z = U^\top X$ is also distributed as a standard Gaussian in $R^n$ and we have ...
3
votes
2
answers
123
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Maximizing expectation of gaussian process over covariance matrix with fixed trace
Let $\mathcal{A} = \{\Sigma \in PSD_{n\times n}(\mathbb{R}), \wedge \forall i,\Sigma_{ii}=1\}$. Then $\mathcal{A} \subset M_{n\times n}(\mathbb{R})$ is convex, closed, and bounded.
For each $\Sigma \...
- 143
1
vote
1
answer
115
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Cameron-Martin space of product space
Suppose you have Banach spaces $\mathcal B_\alpha$ where $\alpha$ is in some index set $I$. Let $\mu_\alpha$ be Gaussian measures on $\mathcal B_\alpha$ with Cameron-Martin spaces $\mathcal H_{\mu_\...
- 360
1
vote
1
answer
68
views
Distance between empirical measures and thickened version
Let $\mathcal{H}$ be a separable Hilbert space and let $x_1,...,x_n$ be points in $\mathcal{H}$. Let $\varepsilon >0 $ be given and consider the measures
$$
\mu := \frac1{n}\,\sum_{i=1}^n\, \...
- 4,831
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0
answers
26
views
k-means errors for a block Gaussian vector
Consider a standard centered Gaussian vector $(X_1,...,X_n)$ with an approximate block structure, i.e. there is $q$ and a partition of $\{1,...,n\}$ in $q$ classes such that if $i,j$ are in the same ...
- 1,230
6
votes
3
answers
598
views
Integral of product of gaussian CDF and PDF
Looking for an analytic solution to the integral below:
$$
\int_{-\infty}^\infty \Phi\left(\frac{x - a}{\tau}\right) \phi\left(\frac{x - b}{\sigma}\right)dx
$$
where $\Phi(\cdot)$ and $\phi(\cdot)$ ...
- 63
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votes
1
answer
130
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Given correlated Gaussian random variables, how to bound the probability that the first is the largest?
Let $Z\sim \mathcal{N}(\mu, \Sigma)$ be a Gaussian random vector in $\mathbb{R}^d$.
What are some nontrivial bounds on
$p=\mathbb{P}(S)$, where $S$ is the event $Z_1=\max_i Z_i$?
This is motivated by ...
1
vote
1
answer
70
views
Singular values of a Gaussian random times deterministic diagonal matrix
Suppose $S$ is a tall-and-skinny $m \times n$ matrix with iid Gaussian entries and $D$ is a $m \times m$ deterministic diagonal matrix. What can be said about the bounds on the largest and smallest ...
- 11
1
vote
1
answer
71
views
Multiplying a log-concave function to a Gaussian probability density reduces its variance
Let $X$ be a random Gaussian vector with probability density $p_X(x)$. Let $Y$ be the random variable with density proportional to $p_X(x)e^{-g(x)}$ for some convex function $g$. Does it hold that
$$
...
- 11
0
votes
0
answers
24
views
Maximum likelihood estimation for covariance parameters of infinite dimensional Gaussian measures
Assume that $\mu_\theta$ is an infinite dimensional Gaussian measure on a Banach or Hilbert space with a covariance operator $C_\theta$ where $\theta$ denotes a finite dimensional vector of real-...
- 1
4
votes
1
answer
160
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Reference request: path integral approach to Gaussian processes
Are there any good, rigorous and preferably modern books or papers on path integral approach to Gaussian processes? I am interested in both introductory level and deeper monographs on the subject.
I ...
- 141
3
votes
2
answers
239
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Local nondeterminism
I'm trying to understand Berman's classic paper on the subject ("Local Nondeterminism and Local Times of Gaussian Processes"). In order to define local nondeterminism, he considers the ratio
...
- 173
2
votes
0
answers
121
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Applying 1D integral to matrix integral
In the proof for finding an analytic solution to the propagation of a Hermite-Gaussian beam though a paraxial system given in the paper "The elliptical Hermite–Gaussian beam and its propagation ...
- 63
2
votes
1
answer
266
views
Evaluation of Gaussian multivariable integral
In the context of evaluating the propagation of a flattened Gaussian beam, the following integral appears:
\begin{equation}
\int (\mathbf x^T \mathbf F \mathbf x)^n \exp \left [ - \mathbf x^T \mathbf ...
- 63
0
votes
0
answers
26
views
L^2 approximation error in Gaussian Process Regression (finite data setting)
I am learning about Gaussian Process Regression. I would like to have some references or results regarding the distribution of the error between a given function, and the posterior obtained in ...
0
votes
0
answers
107
views
Converse to Cameron-Martin theorem
It is known by Cameron-Theorem that if $\mu$ is a centered Gaussian measure on Banach space $\mathcal B$, the equivalent mean-shift measures are exactly the mean-shift by the Cameron-Martin directions....
- 360
2
votes
0
answers
54
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 ...
- 211
2
votes
1
answer
140
views
Probability density of a hyperplane for a Gaussian distribution
I have a vector $\mathbf{x}$ with a multivariate Gaussian distribution
$$P[\textbf{x}\in S]
=\int_{\textbf{x}\in S}
\det(2\pi H^{-1})^{-1/2}\exp(-\frac{1}{2} \textbf{x}^T H\textbf{x}) \, d\textbf{x}$$...
- 162
3
votes
3
answers
577
views
How close are two Gaussian random variables?
Given two Gaussian random variables A and B with (mean, standard deviation) of (a,s) and (b,m) respectively, is there a scalar w in [0,1] that indicates how close A and B are?
- 159
8
votes
0
answers
265
views
Non-affine smooth transformation of Gaussian is Gaussian
Suppose $Z\sim N(0,1)$ (standard Gaussian) and $f: \mathbb{R} \to \mathbb{R}$ is a differentiable function such that $f(Z)\sim N(0,1)$. My question is whether there exists any such $f$ other than $f(x)...
- 307
3
votes
0
answers
130
views
Gaussian integral with Vandermonde determinant
I want to compute the following integral, which contains a Gaussian piece and a Vandermonde determinant:
$$
\int d^Nx \,e^{-\frac{1}{2} \sum_{k=1}^N a_k x_k^2 + \sum_{k=1}^N b_k x_k} \Delta(x),
$$
...
- 51
4
votes
0
answers
158
views
Show that $\mathbb{P}[ a V\le Z| V+Z]=\mathbb{P}[aV \ge Z| V+Z] \text{ a.s.} $ iff $V=\frac{1}{\sqrt{a}}Z'$ where $Z'$ is standard normal
Consider a pair of independent random variables $(V,Z)$ where $Z$ is standard normal. Now suppose that the following equality holds: for a given $a>0$
\begin{align}
\mathbb{P}[ a V\le Z| V+Z]=\...
- 611
6
votes
1
answer
306
views
Show that $M_X(t) = 2 E \left[ e^{tX} \Phi( aX-t) \right], \forall t \in \mathbb{R}$ iff $X$ is Gaussian
Let $M_X(t)$ denote the moment generating function of a random variable $X$. Now suppose that the following expression holds: for a given $a>0$
\begin{align}
M_X(t) = 2 E \left[ e^{tX} \Phi( aX-t) ...
- 611
4
votes
1
answer
201
views
Recovering a function from its Gaussian convolution
Let $\varphi(x)=\frac{1}{\sqrt{2\pi}}\exp(-x^2/2)$ be the Gaussian density and
$f:\mathbb{R}\to\mathbb{R}$ another measurable function.
Under what conditions can $f$ be recovered from its convolution ...
- 43
8
votes
2
answers
444
views
Ways of proving normal distribution (with a view towards Selberg's central limit theorem)
Given an random variable $Y:\Omega \to \mathbb{R}$ with finite mean $\mu$ and finite, positive variance $\sigma^2$, let $X = \frac{Y-\mu}{\sigma}$ be the renormalization with mean $0$ and variance $1$....
- 725
1
vote
1
answer
80
views
Conditional Gaussians in infinite dimensions
I asked this over on cross validated, but thought it might also get an answer here:
The law of the conditional Gaussian distribution (the mean and covariance) are frequently mentioned to extend to the ...
- 369
0
votes
0
answers
67
views
Ito formula for fractional BM + drift and supremum bound
Let $W^H$ be a fBm with Hurst parameter $H$ and let $\mathcal{H}$ be its Cameron-Martin space. Then by Girsanov theorem we know that if $\mathbb{P}$ is an fBm measure, it holds that there exists a ...
- 101
1
vote
1
answer
285
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. ...
- 11
1
vote
1
answer
122
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 ...
- 95
3
votes
1
answer
148
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{??}$ ...
- 43
0
votes
1
answer
327
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 ...
- 313
1
vote
1
answer
83
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 $X,Y$....
- 608
1
vote
1
answer
254
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 ...
- 608
1
vote
0
answers
40
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 ...
- 363
1
vote
0
answers
36
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 ...
1
vote
1
answer
172
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 ...
- 445