# 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 $$y = \{y(x) \; | \; x \in \mathcal{X}\}.$$ ...
27 views

### 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 ...
1 vote
16 views

### 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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16 views

### 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 ...
36 views

### 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 ...
30 views

### 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 ...
22 views

### 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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1 vote
48 views

### 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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1 vote
34 views

### 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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1 vote
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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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### 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 ...
• 23
1 vote