# Questions tagged [gaussian]

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### Identity for Hilbert-valued Gaussian random vectors

Let $X$ be a zero-mean Gaussian random element in a separable Hilbert space $\mathcal{H}$ with covariance operator $\Sigma$. Let $f:\mathcal{H} \to \mathbb{R}$ be a real-valued function. Can we show ...
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### 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 $\frac{dZ}{dK}=\text{??}$ ...
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### 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 ...
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### 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 ...
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### 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 $X,Y$....
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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