Questions tagged [multivariate-normal-distribution]
The multivariate normal distribution, is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. (Also called, multivariate Gaussian)
278
questions with no upvoted or accepted answers
7
votes
1
answer
957
views
Conditional expectation in the multivariate normal distribution
Suppose $(X_1, X_2, X_3)^T$ is multivariate normal.
What is the conditional expectation $E(X_1 \mid X_2, I(X_3 > 0))$?
Here, $I(X_3>0)$ is the random variable that takes the value one when $...
6
votes
0
answers
579
views
Maximum likelihood estimate for multivariate sum of normal distributions
For each $j = 1,\dots,N$, let $\mu_j \in \mathbb{R}^N$ denote a known column vector, $\Sigma_j \in \mathbb{R}^{N\times N}$ a known covariance matrix, and $\theta_j \in \mathbb{R}$ an unknown parameter,...
6
votes
0
answers
814
views
Can we apply a constraint on the distribution of the layer output?
As far I understood, the hidden layer outputs can be anything based on the learning algorithm or optimization rules. I was wondering if it possible to some constraints on the layer output. For ...
6
votes
0
answers
2k
views
Faster computation of high-dimensional multivariate normal probabilities
My goal is to find a faster way to calculate something like
mvtnorm::pmvnorm(upper = rep(1,100))
that is, the tail probability of multivariate normal distribution ...
5
votes
0
answers
106
views
Covariance as a function of explanatory variable
Suppose we have a collection of bivariate random variables $X_{1i}$ and $X_{2i}$ indexed by a continuous variable $t$ such that, for the vector ${\bf{X}} = (X_{1i} \ X_{2i})^T$ we can assume
\begin{...
5
votes
0
answers
201
views
Bayesian Multivariate Normal-Normal Model when covariance depends on the mean
Let $y$ denote a D-dimensional multivariate normal random variable with density
$$y \sim N(\mu, \Sigma(\mu)) $$
such that the covariance $\Sigma$ is a deterministic non-linear function of $\mu$. ...
4
votes
0
answers
51
views
Resulting univariate marginal distributions are not $t$ distributed - why? (S Wood Core Statistics)
In the Core Statistics by Simon Wood it says:
"If we replace the random variables $Z_i\sim_\text{i.i.d.} N(0,1)$ with random variables $T_i \sim_\text{i.i.d.} t_k$ in the definition of a ...
4
votes
1
answer
114
views
What is the expected value of $X_i/\|X\|^2$ when $X \sim \mathcal{N}(\mu, \sigma^2I)$
Let $X$ be an N-dimensional normal random vector with non-zero mean $\mu$ and diagonal covariance matrix $\sigma^2I$. I would like to understand if it is possible to derive the expected value of the ...
4
votes
0
answers
82
views
On the distribution of sample intraclass correlation for normal data
Consider i.i.d random vectors $\boldsymbol X_1,\boldsymbol X_2,\ldots,\boldsymbol X_n$ having a $p$-variate normal distribution $N_p(\boldsymbol \mu,\Sigma)$ where $\Sigma$ has compound symmetry of ...
4
votes
0
answers
117
views
truncation of bivariate normal under quadratic condition
Consider a complex normal variable $Z \sim \mathcal{CN}(\mu,2\sigma^2)$ with real component $X \sim \mathcal{N}(\mu,\sigma^2)$ and imaginary component $Y \sim \mathcal{N}(0,\sigma^2)$. We can write ...
4
votes
0
answers
85
views
How to build a probability distribution from locations with accuracies?
I have a set of $n$ GPS locations $l_i$ with latitude, longitude in degrees and accuracy in meters, corresponding to $3 σ$, i.e. the probability ≈ 0.997) $(lat_i, lng_i, acc_i)$ or $(lat_i, lng_i, σ_i ...
4
votes
0
answers
153
views
Mean and covariance estimators for IDD normal
Suppose $X_1,\cdots, X_n$ are sampled iid from a multivariate Gaussian $\mathcal N(\mu, \Sigma)$. We denote the sample mean and covariance as follows
\begin{align}
\bar X &= \frac{1}{n} \sum_{i=1}^...
4
votes
1
answer
650
views
Numerical computation of the means and covariance in a truncated bivariate normal distribution
How can I compute the means and covariance of a truncated bivariate normal distribution? I am particularly worried about the case when the truncation occurs very far from the mean. Is there a robust ...
4
votes
0
answers
118
views
Swiss Cheese Distributions
I am curious about a normal distribution with no probability mass in certain regions, sort of like the complement of the truncated normal. In particular, it will have zero mass in a circular region.
...
4
votes
0
answers
142
views
Prove that the joint density of independent multivariate normal variables is a matrix-normal
Let $X_1,...,X_n \sim N_p(\mu_i,\Sigma_i)$ be Multivariate Normal a.v. independent.
Show that $W = (X_1,...,X_n) \sim MN(M,\mathbb{I},\Sigma)$ where $M = [\mu_1 \mu_2...\mu_n]$ and $\mathbb{I}$ ...
4
votes
0
answers
195
views
Select ellipsoid data cloud "equivalent" to a spherical data cloud
What will be your considerations when you've got to realize the intuitively plain, but meditatively manifold idea to create a data cloud which "the same as" that data cloud, only that it is not ...
3
votes
0
answers
132
views
How does the multivariate normal distribution work with multiple conditions?
The following multivariate distribution is given:
$\left(\begin{array}{c}X_1 \\ X_2 \\ X_3\end{array}\right) \sim N_3\left(\left(\begin{array}{c}1 \\ 1 \\ 1\end{array}\right),\left(\begin{array}{ccc}1 ...
3
votes
0
answers
213
views
Correlation matrix as maximum likelihood estimator under constraint
The Problem
Although it seems to be straight forward I am struggling to prove the following statement.
Assume, we have $p$-variate Gaussian observations $\left\{x_1, \ldots, x_N \right\} \subset \...
3
votes
0
answers
122
views
Proving Equivalence between Multivariate distributions and Gaussian Bayesian Networks
I am studying Probabilistic Graphical Models by Daphne Koller. In Chap 7, the authors say the following.
I can't convince myself of the highlighted part. Induction typically has a statement for n, ...
3
votes
0
answers
60
views
If normally distributed random vector $X$ has a PD covariance matrix, then any conditional distribution induced by $X$ has a PD covariance matrix?
Suppose I have a random vector $X$ whose distribution is joint normal. I know that the covariance matrix of $X$ is positive definite. I wonder if I partition $X$ in any way: e.g., $X=(X_1, X_2)$, then ...
3
votes
0
answers
43
views
Inequality of Probability
This problem is from Matrix Analysis for Statistics by James R. Schott.
Problem. Show that if $x$ and $y$ are two independently distributed random vectors with $x\sim{\rm Normal}(0,\Omega_1)$ and $y\...
3
votes
0
answers
401
views
Efficient computation of marginalized multivariate normal posterior distribution
In general,if we know that the marginal Gaussian distribution for some variable $\textbf{x}$ and a conditional Gaussian distribution for some $\textbf{y}|\textbf{x}$ of the forms:
$$p(\textbf{x}) = \...
3
votes
0
answers
676
views
What is the distribution of the CDF of a sample drawn from a multivariate normal?
Introduction: Lets say we have a random variable $X$ that follows a normal distribution, $X \sim N(\mu, \sigma^2)$ , with a CDF function $F_X(x) = P(X \leq x)$. Then we draw some random samples $S_1$, ...
3
votes
0
answers
228
views
Covariance matrix of integral of multivariate normal distribution
If $t = [t_0, t_1, \dots, t_{N-1}] \in \mathbb{R}^N$ with $t_i \sim N(\mu_i, \sigma_i^2)$ and its covariance matrix $C \in \mathbb{R}^{N \times N}$ where $C_{ij} = Cov(t_i, t_j)$ is given
If I define ...
3
votes
0
answers
137
views
Joint distribution of log hazard ratio estimates for two outcomes
I have a study where individuals are randomized to a treatment or control group, and there are two time-to-event outcomes $S_i$ and $T_i$ measured for each individual $i$. The two outcomes have some ...
3
votes
1
answer
60
views
Problem with two correlated random normals
Imagine you have a two-dimensional multivariate normal random variable with $\mu = [0, 0]$ and $\Sigma\ = \begin{bmatrix}1 & r\\r & 1\end{bmatrix}$. (Conceptually, you have two random normal ...
3
votes
0
answers
104
views
How to characterize the effect of $(\textrm{Diag}(\Sigma^{-1}))^{-1}$ badly approximating $\textrm{Diag}(\Sigma)$
I have an almost singular covariance matrix $\Sigma\in\mathbb{R}^{n\times n}$ that has a few large eigenvalues, followed by many many comparatively very small ev's.
If I were to try to approximate ...
3
votes
0
answers
79
views
Equality of two multivariate normal CDF's
Let $\pmb{X} \sim N_d(\pmb{\mu}, \pmb{\Sigma})$ and $\pmb{Y} \sim N_d(\pmb{\nu}, \pmb{\Omega})$; $\pmb{\mu} \neq \pmb{\nu}, \pmb{\mu} \neq \pmb{0}, \pmb{\nu} \neq \pmb{0}$, and $\pmb{\Sigma}\neq\pmb{\...
3
votes
0
answers
2k
views
Distribution of quadratic form of multivariate normal with linear term
Suppose that $A$ is a symmetric non-random matrix and $X\sim N(\mu,\Sigma)$ and $b \in R^n$ is a non-random vector. Then what is the distribution of
$$X^tAX+b^tX \quad ?$$
The distribution without ...
3
votes
0
answers
594
views
Truncated mulitvariate normal: first two moments
Let $X\in \mathbb{R}$ be a univariate random varible for which it holds that $$ X \sim N(\mu,\sigma^2).$$
where $\mu\in \mathbb{R}$ gives the expected value and $\sigma^2>0$ is the variance.
If ...
3
votes
0
answers
418
views
Argmax order statistics for multivariate normal
The Problem
Given a gaussian random variable $\mathbf{x} = (x_1, ..., x_N)^T \sim \mathcal{N}(0, \Sigma)$ what is the probability that $i = \underset{i\in\{1, \dots, N\}}{argmax}\{|x_1|, \dots, |x_N|\...
3
votes
0
answers
230
views
transformation with mvtnorm
I'm struggling to understand why the following R outputs don't yield the same value.
...
3
votes
1
answer
67
views
Creating multi-variate normal with constraint:
Is it possible to create multivariate normal from the same normal variable?
That is i have a normal variable $x_1 \sim N(0,1)$, and $p$ normal variable $y_1$ which their marginal distribution is also ...
3
votes
0
answers
167
views
Marginalizing multivariate-normal distribution canonical form
Regarding the problem of margenalization of canonical forms of multivariate gaussian distribution it was mentioned in probabilistic graphical models text book that
$$\int{C(X,Y;k,h,g)}dY$$
is ...
3
votes
0
answers
109
views
Does a "pruned" i.i.d. multivariate sample behave as the i.i.d. sample?
Let $z_1,\cdots,z_n$ be $n$ points drawn i.i.d. from $\mathbb{C}N_p(0,\Sigma_n)$. The distribution of the covariance $S_n=\frac1n\sum_{i=1}^n z_i z_i^*$ is well known in the limit as $n,p(n)\to\infty$ ...
3
votes
0
answers
96
views
A question concerning distribution of $\mathbf{Y}/\|\mathbf{Y}\|_2$ where $\mathbf{Y}\sim \mathcal{N}(\boldsymbol{\mu},\mathbf{I})$
I know that when $\mathbf{Y}\sim\mathcal{N}(\mathbf{0},\mathbf{I})$, $\mathbf{Y}/\|\mathbf{Y}\|_2$ is distributed uniformly on the unit sphere. But to my surprise, I failed to find a simple closed ...
3
votes
0
answers
62
views
Multivariate distribution for products of random variables
Suppose I have an $n$-dimensional complex, zero mean normal distribution with covariance matrix $\Sigma$, which is not diagonal. Denoting each of the random variables as $x_1, \dots ,x_n$ I would ...
3
votes
0
answers
152
views
Sampling distribution of variances of multivariate normal RVs
Is there an analytical expression for the distribution of variances of MVN RVs? I mean if $X=[x_1, \dots, x_D]\sim \mathcal{N}(0, \Sigma)$ where $\Sigma$ is a $D$-dimensional covariance matrix, is ...
3
votes
0
answers
555
views
Composition of Multivariate Gaussians
This question is teasing my intuition for a moment :
$X \sim N(0, S_1)$
$Y|X \sim N(X, S_2)$
Does $Y \sim N(0, S_3)$ with some $S_3 = f(S_1, S_2)$ like (for instance) $S_1 + S_2$ ?
What I found is ...
3
votes
0
answers
1k
views
Clarification on LDA and the multivariate Gaussian
From my understanding, to calculate the posterior probability of a sample $x$ belonging to a class $k$ using Linear Discriminant Analysis you would first calculate the eigenvector matrix $W$ required ...
2
votes
0
answers
45
views
Notation for distributions on subspaces
Question
What notation should I use for a normal distribution on a subspace $U \subset \mathbb{R}^n$?
Motivation
I have a geometric intuition for Bessel's correction which goes something like this:
A ...
2
votes
0
answers
18
views
Probability of N-dim Gaussian marginalized over a plane, and a likelihood of a point on a plane
Context: I'd like to fit a series of coordinates, each with their uncertainties, to a (hyper)plane. I am trying to derive the likelihood for a point on a plane observed at a point. I need a ...
2
votes
0
answers
87
views
Relationship between univariate and multivariate normal distribution
I have a portfolio consisting of N assets with known average historical returns ($r_1$, $r_2$, ...$r_N$) and a known set of weights ($a_1$, $a_2$, ...$a_N$) subject to $\sum_{i=1}^N a_i$ = 1
I am ...
2
votes
0
answers
211
views
Conjugate Prior for Multivariate Normal Variances and Correlations
Is there a way to separately specify conjugate priors for the variance and correlations of a multivariate normal? The inverse Wishart is conjugate if you want to specify the covariance, but covariance ...
2
votes
0
answers
131
views
linear transform of a random variable follows multivariate normal, what is the distribution before the transform?
$x$ is a $n\times 1$ random vector,$A$ is a $m\times n$ matrix. Given that $x$'s linear transform $z = Ax$ follows a multivariate normal distribution:
$$
z = Ax \sim N(\mu_z,\Sigma_z)
$$
The question ...
2
votes
0
answers
137
views
Joint distribution of sample correlations of variables taken from a multivariate normal distribution
Let us assume multivariate normal vector $(X_1, \cdots, X_n)$ with mean vector $\mu$ and variance-covariance matrix $\Sigma$.
A sample correlation will not exactly equal its population parameter, but ...
2
votes
0
answers
496
views
Henze-Zirkler Multivariate Normality on R and Python
I am trying to run some ANCOVA analysis but beforehand, I would like to show that data follows multivariate normal distribution, which is one of the assumptions. On R, there is ...
2
votes
0
answers
427
views
Using multivariate normal likelihood when determinant of covariance matrix is zero
Estimating a multivariate normal (MVN) model requires minimising the negative log-likelihood of MVN (constant term dropped):
$$
\ell(X|\mu,\Sigma)=\frac{n}{2}\log|\Sigma|+\frac{1}{2}(X-\mu)^T\Sigma^{-...
2
votes
1
answer
254
views
Conditional distribution of multivariate cauchy distribution
In the example of multivariate normal distribution,
$$
\begin{bmatrix}
\mathbf{x}_1 \\
\mathbf{x}_2
\end{bmatrix} \sim \mathcal{N}\left(\begin{bmatrix}
\mu_1 \\
\mu_2
\end{bmatrix}, \begin{...
2
votes
0
answers
191
views
Best measure of similarity between multivariate Gaussian distributions?
I am working to describe differences in covariances between a "baseline" multivariate Gaussian random process $\mathbf{x}_0$ and other processes $\mathbf{x}_i$. All are discrete and ...