Questions tagged [multivariate-distribution]
Probability distribution over vectors (as opposed to univariate distributions that are over numbers).
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1answer
26 views
Generate a multivariate normal vector
I'm working in R and I was wondering, let's say I want to generate a random vector $X \in \mathbb{R^p}$, with $X \sim N(0,I)$ where $I$ is the identity matriz in $\mathbb{R}^{p \times p}$. The ...
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17 views
Combining marginal bivariate probability distributions into a single multivariate one
Consider a set of 3 correlated random variables $X$, $Y$ and $Z$. I have calculated bivariate marginal distributions over any pairs of these random variables. That is, I know probability density ...
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0answers
10 views
How to determine if a multi-dimensional vector comes from a multi-variate distribution or not?
I have a complex multivariate distribution given by a multivariate random number generator (black box). In other words, whenever I call the "generator" I get a random vector containing ...
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27 views
Simplifying assumption for regular vine distributions
I am trying to understand vine copulas and specifically, conditional bivariate copulas. Let $c_{X_1,X_3|X_2}(\cdot,\cdot | \cdot)=:c$ be the probability density function of a bivariate conditional ...
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2answers
74 views
Can a random variable be uncorrelated with its product with a correlated random variable?
I have a random variable $X.$ I want to find a random variable $Y$ such that $Y$ is correlated with $X,$ but $Y$ is not correlated with the product of $X$ and $Y.$ Is it always possible?
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25 views
Conditional Expectation of a normal distribution [duplicate]
say we have a multivariate normal distribution with ${\boldsymbol Y} \sim \mathcal{N}(\boldsymbol\mu, \Sigma)$
The conditional expection is $\overline{\boldsymbol\mu}=\boldsymbol\mu_1+\Sigma_{12}{\...
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21 views
Test goodness of fit of a multivariate distribution via testing GOF of linear combinations of components
I am modeling a continuous bivariate distribution of a random vector $(X_1,X_2)$ using a copula. I would like to assess how well I am doing. Given a data sample, I could probably do a bivariate ...
4
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1answer
168 views
KL divergence for joint probability distributions?
I have a pair of joint probability distributions. I want to measure their similarity/dissimilarity. If they were single-dimensional probability distributions, then I could measure the Kullback–Leibler ...
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1answer
73 views
What is a generalisation of t-test to the case of multivariate distribution?
When we have a sample of numbers, one of the most basic tests is the t-test, in which we check the null hypothesis that the population mean is equal to zero.
I am interested in a generalisation of ...
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23 views
Mutual information relationship to copula entropy is borked?
I have posted a related Question based on a paper, Ma, Jian, and Zengqi Sun. "Mutual information is copula entropy." Tsinghua Science & Technology 16.1 (2011): 51-54.
In the paper, they ...
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38 views
Copula entropy: calculation is borked?
I came across a pretty cool paper whose idea makes a lot of sense to me.
Ma, Jian, and Zengqi Sun. "Mutual information is copula entropy." Tsinghua Science & Technology 16.1 (2011): 51-...
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1answer
153 views
What is the best way to sample points from an arbitrary 2D distribution?
My problem is the following: I want to randomly sample points $(x,y)$ according to the Himmelblau function
$$f(x,y) = (x^2 + y - 11)^2 + (x + y^2 - 7)^2$$
which I treat as a kind of multivariate ...
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0answers
14 views
Covariance matrix in multivariate is Wishart random variable
I'm working on sample variance distribution, on page 111 of Methods of Multivariate Analysis, it says
"The joint distribution of these $p(p + 1)/2$ distinct variables in $W =(n−1)S = \sum_i(y_i − ...
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33 views
Is it reasonable to look at the output of simulating from a multivariate distribution as univariate distribution? If yes, what is this called?
Suppose I have $X_{n} \sim MVN(\underline{\mu},\Sigma)$ where $n$ is large (several thousands). However, the $\mu_i's$ and the elements of $\Sigma$ are such that almost every simulation from $X_n$ ...
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28 views
Measuring entropy/concentration in multivariate datasets
I am interested in measuring how concentrated or widely dispersed a certain binary property is among a population, which is defined by a number of categorical variables.
For instance, let's say I have ...
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23 views
Multivariate Bernoulli Logistic Regression on OpenBUGS
I am new to WinBUGS/OpenBUGS, and am trying to solve this Multivariate Bernoulli Logistic Regression
$$P(θ1,θ2|X,Z) ∝ P(X|θ1) P(θ1) P(Z|X,θ2) P(θ2)$$
Is this valid on OpenBUGS?
This is my OpenBUGS ...
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11 views
Does the t-copula capture serial nonlinear dependence?
Say $\mathbf{Y}=[y_1,\cdots,y_n]'$ with elements that are uncorrelated, yet serially dependent. As a result, in the literature, the means of allowing for nonlinear serial dependence for processes ...
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25 views
PDF for Bayesian Decision
I have the discrete-time sequence $X(n) = 1$ with probability 3/4 and $X(n) = 0$ with probability 1/4 for $n = 1,...,10$. The observed signal is $Y(n) = X(n) + N(n)$ where $N(n)$ is an AWGN with ...
4
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2answers
126 views
Interpretation of multivariate conditional gaussian function form?
I've been reading over this Multivariate Gaussian conditional proof, trying to make sense of how the mean and variance of a gaussian conditional was derived. I've come to accept that unless I allocate ...
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0answers
29 views
Kth Order statistic of a multivariate distribution
The Kth order statistic for a univariate is equal to its kth-smallest value. For instance, given $\{6,9,3,8\}$, the 2nd-smallest value would be the 2nd order statistic.
How does this concept ...
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0answers
36 views
Multivariate Gamma parameter estimation
Consider $X$ a d-dimensional random variable with positive values, mean $\mu\in\mathbb{R}_+^d$, and covariance matrix $\Sigma\in\mathbb{R}^{d\times d}$. If I have $n$ samples $\{ x_1, ..., x_n \}$ ...
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25 views
Derive expectation of the log determinant of a precision matrix from a Wishart distribution
I'm reading through section 21.6 of Murphy's Machine Learning: A probabilistic perspective where they derive the variational bayes algorithm for fitting a mixture of gaussians. One of the steps ...
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0answers
15 views
plotting bivariate normal distribution samples
I have generated two samples $\underline{X_i}$ and $\underline{Z_i}$
$\underline{X_1}$ $\underline{X_2}\dots \underline{X_{5000}}$ , while $\underline{X_i} \sim N_2[(1,2)^T,\begin{pmatrix}2&1.5\\ ...
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2answers
36 views
Why is there covariance instead of simply variance in multidimensional normal distributions?
Maybe it is just because I'm only experimenting with 2 dimensional normal distributions, but multi-dimensional normal distributions for me seem like just multiple one dimensional normal distributions.
...
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0answers
48 views
Probability distribution associated with nuclear norm?
The $\ell_1$ norm of model parameters is often added to loss functions because it induces sparsity in the solution of the overall cost function:
$$ c(\theta) = \log L(x|\theta) + \lambda ||\theta||_1$$...
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12 views
Help in calculating diagonal covariance matrix for generative model for binary classification
I am given this data. I want to fit a generative model $\cal{N}(\mu_0, \sigma_0^2 I_2)$, $\cal{N}(\mu_1, \sigma_1^2 I_2)$ for the $0$ and $1$ classes respectively using $\textbf{MLE}$ and plot ...
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0answers
12 views
Condition Random Variable on Range of Another Random Variable [duplicate]
Assume that $v_s \sim N(\mu_s,\sigma_s^2)$ and $v_b \sim N(\mu_b,\sigma_b^2)$, denote their correlation by $\rho$, and assume they are jointly normally distributed. How would I assess $E[v_b|v_s\leq c]...
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113 views
Multivariate Normal distribution Cramer-Rao bound
How to compute the Cramer-Rao lower bound for $\mu$ of a multivariate normal distribution $\mathcal{N}_{p}(2\mu,\Sigma_0)$.
Also I would like to know the minimum variance unbiased estimators for $\mu$....
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0answers
14 views
Bivariate random effect problems in selection models (Mixture Cure model)
I am currently working on a mixed effects selection model.
The selection model is a logistic model with a Gaussian random effect.
The principal model is a survival model with a Gaussian random effect (...
2
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1answer
122 views
Find expectation of conditional normal distribution
I am struggling with some finding expectation value question .
the question is to find $E[Y|X]$ from the result $P(Y|X)$
with given mean and covariance
$$\mu=[\mu_x, \mu_y]^T$$
$$\Sigma=\begin{bmatrix}...
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0answers
28 views
What is the entropy of multivariate data multiplied by a vector?
It is a general rule that for multivariate data $\boldsymbol{X}$ and a matrix $\boldsymbol{A}$, their entropy is
$$h(\boldsymbol{A} \boldsymbol{X}) = h(\boldsymbol{X}) + \ln |\det \boldsymbol{A}|$$
(...
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0answers
113 views
What is the distribution of the CDF of a sample drawn from a multivariate normal?
Introduction: Lets say we have a random variable $X$ that follows a normal distribution, $X \sim N(\mu, \sigma^2)$ , with a CDF function $F_X(x) = P(X \leq x)$. Then we draw some random samples $S_1$, ...
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1answer
96 views
What is the fisher information matrix of the multivariate t distribution?
$\newcommand{\bx}{\mathbf{x}}$
$\newcommand{\bSigma}{\boldsymbol{\Sigma}}$
$\newcommand{\bE}{\mathbf{E}}$
$\newcommand{\bD}{\mathbf{D}}$
Consider the multivariate central t distribution with p.d.f.
\...
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4answers
2k views
Is this causation?
Consider the following joint distribution for the random variables $A$ and $B$:
$$
\begin{array}
{|r|r|}\hline & B=1 & B=2 \\
\hline A=1 & 49\% & 1\% \\ \hline A=2 & 49\% & 1\...
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0answers
13 views
Is the mvt (multivariate t) adjustment for multiplicity appropriate for repeated measures and longitudinal trials?
As in the title. If I have a longitudinal experiment with t0...tn timepoints, and want to verify the Dunnett contrast (all versus baseline), can I use the mvt adjustment for multiplicity? I think it ...
4
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1answer
139 views
Correlation Coefficient for Hypergeometric-type Distribution
Problem Statement: A box contains $N_1$ white balls, $N_2$ black balls, and $N_3$ red balls
$(N_1+N_2+N_3=N).$ A random sample of $n$ balls is selected from the box (without replacement).
Let $Y_1,Y_2,...
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1answer
455 views
Probability distribution of the product of two dependent random variables
It is well known that being $X$ and $Y$ two independent random variables with distributions $f_X(x)$ and $f_Y(y)$, respectively, then the probability distribution of the multiplicative function $z = ...
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2answers
182 views
What does i.i.d. mean for multivariate case?
When we say a random variable is i.i.d., it's often used to describe the dependency between the observations of that random variable, which I call the row dimension, indexed by time if it's a time ...
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0answers
27 views
How does the maximum entropy principle affect joint entropy, mutual information, and other info measures?
The maximum entropy principle says that we should use the probability distribution of a univariate dataset that has the highest level of entropy because it offers the lowest information. How does the ...
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0answers
81 views
Rescaling multivariable normal pdf and normalizing constant
I am trying to understand change of variables for a random variable and how it changes the pdf and the normalizing constant.
Let $\mathbf{x}$ be $N$-dimensional normal variable and let $\mathbf{\...
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0answers
27 views
What are the best known techniques to verify that a GAN samples correctly from a given distribution?
I would like to know what are the best known techniques to check that a generative adversarial network (GAN) samples from the correct distribution.
Naively, I would say it all boils down to a ...
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0answers
18 views
Multivariate rare events
My data is something like this:
I have U urns, and I have taken a bag of $n$ objects from each urn. Each urn has $N$ objects, and I have sampled $n$ with replacement.
$n$ is comprised of coloured ...
2
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0answers
45 views
Algorithm to generate non normal (discrete and continuous) correlated variables using a correlation matrix
What I am trying to do here is, given a dataset (let's say n observations of N variables), and thus a correlation matrix M that result from said dataset, I would like to write a Python algorithm that ...
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0answers
41 views
Empirical conditional density of continuous variables
I have a dataframe, with data of several continuous variables. The variables are not independent.
My goal is to sample from the distribution that generated this data.
What's a relatively easy and ...
0
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1answer
57 views
Multivariate distributions
Let $x$ and $y$ be random variables with the following joint density function:
$f(x,y) = e^{-x}$ for $0<x< \infty$, and $0<y<1$
If $z= x+2y$, what is the joint density function of $x$ and ...
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1answer
16 views
How to choose what feature vector to plot in multivariate regression analysis?
I'm new to the field of machine learning and I have been having this doubt for a long time now. If we want to plot a scatter plot, we plot it as, x as a function of y where x is a 1-D array. But in ...
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1answer
1k views
Does the covariance of i.i.d. random vectors/multivariate random variables have any zero terms?
If we have i.i.d. random variables, $X$ and $Y$, then $\text{Cov}(X,Y)=0$.
But let's say we have i.i.d. random vectors $\boldsymbol{X}$ and $\boldsymbol{Y}$, where $\boldsymbol{X}=(X_{1},...,X_{p})$ ...
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0answers
46 views
How to draw huge high-dimensional normally distributed vectors in a memory-efficient way
I would like to simulate a few hundred multivariate normally distributed vectors $H$ of dimension $10^6$ with covariance matrix $\Sigma$, preferrably in R. The individual entries of $\Sigma$ are ...
0
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1answer
282 views
What's the best function to test multivariate normality when sample size more than 3000?
Before MANOVA, I need to test multivariate normality.
Then I tried MVN::mvn() function in R, output as below:
...
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0answers
14 views
bivariate conditional joint sensor model
I am struggling to find $P\left( V_t | z \right)$ from $P\left( V_t | z , V_p \right)$. Here $z$ and $V_p$ are independent variables.
...
