Mixing
Random perturbation of uniform distribution on the unit sphere
$begingroup$
Assume that $theta$ is uniformly distributed on the unit sphere in $mathbb R^d$ and let $w sim N(0, I_d)$, that is. a canonical Gaussian vector. Let $alpha in mathbb R$ be a random variable. What should be the distribution of $alpha$ such that
$$
theta + frac{sigma}{sqrt{d}} w stackrel{d}{=} theta + frac{sigma}{sqrt{d}} theta alpha
$$
where $sigma > 0$ is some fixed number (can take $sigma$ sufficiently small, and/or assume $d$ is sufficiently large). The above is an equality in distribution, i.e., we want the two sides to have the same distribution.
The idea is to recover an isotropic perturbation of $theta$ by a perturbation which is normal to the sphere at point $theta$.
probability probability-distributions
asked Jan 23 at 17:07
passerby51passerby51
2,1311018
$endgroup$
add a comment |
$begingroup$
Assume that $theta$ is uniformly distributed on the unit sphere in $mathbb R^d$ and let $w sim N(0, I_d)$, that is. a canonical Gaussian vector. Let $alpha in mathbb R$ be a random variable. What should be the distribution of $alpha$ such that
$$
theta + frac{sigma}{sqrt{d}} w stackrel{d}{=} theta + frac{sigma}{sqrt{d}} theta alpha
$$
where $sigma > 0$ is some fixed number (can take $sigma$ sufficiently small, and/or assume $d$ is sufficiently large). The above is an equality in distribution, i.e., we want the two sides to have the same distribution.
The idea is to recover an isotropic perturbation of $theta$ by a perturbation which is normal to the sphere at point $theta$.
probability probability-distributions
asked Jan 23 at 17:07
passerby51passerby51
2,1311018
$endgroup$
add a comment |
$begingroup$
Assume that $theta$ is uniformly distributed on the unit sphere in $mathbb R^d$ and let $w sim N(0, I_d)$, that is. a canonical Gaussian vector. Let $alpha in mathbb R$ be a random variable. What should be the distribution of $alpha$ such that
$$
theta + frac{sigma}{sqrt{d}} w stackrel{d}{=} theta + frac{sigma}{sqrt{d}} theta alpha
$$
where $sigma > 0$ is some fixed number (can take $sigma$ sufficiently small, and/or assume $d$ is sufficiently large). The above is an equality in distribution, i.e., we want the two sides to have the same distribution.
The idea is to recover an isotropic perturbation of $theta$ by a perturbation which is normal to the sphere at point $theta$.
probability probability-distributions
asked Jan 23 at 17:07
passerby51passerby51
2,1311018
$endgroup$
Assume that $theta$ is uniformly distributed on the unit sphere in $mathbb R^d$ and let $w sim N(0, I_d)$, that is. a canonical Gaussian vector. Let $alpha in mathbb R$ be a random variable. What should be the distribution of $alpha$ such that
$$
theta + frac{sigma}{sqrt{d}} w stackrel{d}{=} theta + frac{sigma}{sqrt{d}} theta alpha
$$
where $sigma > 0$ is some fixed number (can take $sigma$ sufficiently small, and/or assume $d$ is sufficiently large). The above is an equality in distribution, i.e., we want the two sides to have the same distribution.
The idea is to recover an isotropic perturbation of $theta$ by a perturbation which is normal to the sphere at point $theta$.
probability probability-distributions
probability probability-distributions
asked Jan 23 at 17:07
passerby51passerby51
2,1311018
asked Jan 23 at 17:07
passerby51passerby51
2,1311018
edited Jan 23 at 17:52
passerby51
asked Jan 23 at 17:07
passerby51passerby51
2,1311018
asked Jan 23 at 17:07
passerby51passerby51
2,1311018
asked Jan 23 at 17:07
passerby51passerby51
2,1311018
2,1311018
add a comment |
add a comment |
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